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0, there exists a positive integer n such that THEOREM
(33)
I ?(n+l\
-ai;<£
(1 < • < * ) .
X% + * —1) : There exist positive constants c = c(a,t) and X& = Xa(a,s) such that in any interval 1 < n < X the number of n satisfying (33) is greater than eX/logh+1X whenever X > Xo.
49
Functions
ip (•»•)• ( T ( » ) and >8(n)
211
Proof. To begin with, by similar arguments as in the proofs of lemma 3a and of theorem 1 [4], we can choose Ao, At, ..., Ah, depending on a'ts and e only and satisfying the same conditions as in theorem .1, such that .....
{3i)
9»(4i)Mi
^)jA~l-
ai{<
t:
,
,
Wld
2
cpjiA^jiAj
e
^(.--D^^-DA;:;-**'
For those J.^s we assume that (x0, ....,#&) is a solution of (1) satisfying (2) with Z = Xc>. If we take Aoxo = w, then £ 4 ^ = n+i (1 < i < h). Since (a?,-, A,-) — 1 (0 < i < 7i), we have rp{n)
fp(Aoxo)
(35)
[
n
tp(iAi)liAi _ _
On account of * ; < c9X (0 < ?'
From (34) and (35) we can choose X3(a, e) > X2 such that if n = Aoxu > X3, .then w(n4-i)
. - f i - T - ? - - « , i < *•
(i < i < h).
q>(n+i—l) i Thus we have proved that from every solution of (1) satisfying (2) with Z = X"1 and such that AQx0 > X3 we can define a positive integer n — A0x6 satisfying (33), and that evidently different solutions correspond to different n. It is clear that the number of solutions of (1) satisfying (2) with Z = Xci and such that AoxQ < X3 is less than X\+l. Hence, by theorem 1, there exist positive constants Xo and c, depending on a'i& and e only, such that, if X > Xo in any interval 1
The proof of theorem 3 is analogous to the preceding one but based on the proofs of lemma 3b and theorem 2 [4].
50 212
A. S o h i u z e l and Y. W a n g
§ 8. Theorems on the function 0. We now prove THEOBEM 4. For any given 'positive integer h, there exists a constant b== b(h) such that for-any given sequence a of h integers alt a2, ..., ah > 1 there exisU a positive integer n such that (36)
e(i)Ui < 6(n+i) < bd{i)ai
(1 < i ^ li).
There exist positive constants c' = c'(a) and X'o — X'0[a) such that in any interval 1 =$C n ^ X the number of n satisfying (36) is greater than c'Xjlogh+lX, whenever X > X'^. Proof. We can choose Ao, A1} ...,Ah, depending on a^s only and satisfying the same conditions as in theorem 1, such that 8(Ai) = «., (1 ^ i ^ 7t). For those A[s we assume that (x0, ..., cck) is a solution of (1) satisfying (2) with Z = XCK If we take Aoxo = n, then iAiXi — n+i (1 < i ^ h). Since (Xi, iAi) = 1 (1 < / < h), we have 9(n+i) = 0(*4,-)0(iei) = Q(i)a.fi{Xi)
(1 < * ^ h).
As in the proof of theorem 2, we can choose X'2{a) such that the number of prime divisors (identical or different) of each x,: (1 < i $: h) does not exceed c10 for X > X'2. Hence for X > X'2 0(£)av < d(n+i) < 6(i)ai2c™ = b()(i)a,-
(1 < * < h).
Further proof is analogous to the proof of theorem 2. From theorem 4 we can directly obtain T H E O R E M 5. For any given sequence of h numbers alf...,ah, where sequence of natural Oi = 0 or + c o (1 ^ i ^ h) there exists an infinite numbers ) i - j , n 2 ! . . . such that
9(nk+i)
fc-oc 0{nk + i— 1) In the course of publication theorems 3 and 5 were proved independently by Shao Pin-Tsung (to appear in Shou Hsueh Chin-Chan (Progress of Mathematics) and Pei-Ta-Hsueh Pao (Transactions of Peking University)) (cf. [5]). References [1] H. Rademacker, Beitrage zur Viggo Brunsohen Methods in der Zahlentheorie, Abh. Math. Sem. Hainburgischen Univ. 3 (1923), p. 12-30. [2] A. Schinzel, Quelques theorimes stir Us fonctions
51 Functions [4] — On functions p. 4 1 5 - 4 1 9 .
213
cp(n) and a(n), Bull. Acart. Polon. Sci.. Cl. I l l , 3 (1955),
[5] Shao P i n - T s u n g , On the distribution of lltu values of a class r>i arithmetical functions, Bull. Aoad. Polon. Sci., Cl. ITT, 4 (1956), p. 569-572. Keou par la Redaction le, 7. 3. 195G
Note added in t h e proof by A. Schinzel. Theorem 1 easily results from the following theorem of G. Rieei (see G. Ricc.i, Su la congetiura di Goldbach e la iionstante di Schnirehnnn. Annali ilella R. Souol.i Nnrinalo Superiore. rli Pisa •> (2) 1937, p. 83): Let, a-jK + frj,
diffe-
(«xa;H- 6a) (tt2w +ft g )... {a,fx + bf)
and put fljtt + frj = d1l\, «2w-(-i, = d2P2,... .a/.v-^-b/ = d/Pf, -wkererf1f/!>...rf;— D. The number of natural numbers x «C f such thai all integers P X ,P 2 Pf have no prime factors < J1/(1+2T(/» is of the same order of magnitude as f/log'f. In fact, in virtue of lemma 2 [4] thorn twists .1 natural number m such that. Ai\'»i + i.
(Ai. {m + i)iAi)
--• I
( ( ) < ; ; < I'-)
;uid in v i r t u e of t h u f o r m u l a s i l ) - ( 3 ) of [ 4 ] UlA\...A%,m)=Aa.
{AlA\...Al:l,+i)
= iAi
(1 ^ i ^ h).
Pin 6, -
ar-AaA\...A\, ,H - A\A\...
w /,t B ,
A*/(i ~i)A{_t
hi •---. (-w-fi-
( I -- ; < > . . + ! ) .
i)j(i-\)Ai_l
We therefore get (a%, bi) = 1 {l^.i^.h+1) aud (Ao, 616a ... 6A+I) == 1 whence also ((ft+1)!, 6j6» ... ?>A+I) = 1. From the last equality it follows that the polynomial (u^e + bj) (aax + b2)... (ak+ix + bh+i) lias no fixed divisor > 1, since such divisor I> divides (7t+l)!. Putting in the above mentionod theorem of Ricci dt=l, a^x+bi -• Xi_ 1 (1 ==J i ^ h) we find that the number of natural numbers x =g; $ such that all the numbers xi ( O ^ i ^ / t ) have no prime factors ^ f1/(1+J*(*+1)) i* of the same order of magnitude as f/loeA+1f. Put I = -
A
0X-m
A A
l \--Ah
, ^ |1/(1+2»{A+1))
As the number xt satisfy the system of equations (1) and for x ^ $ the conditions (2), we get the inequality
wJiere e2 > 0 and X1 are constants depending only on Ai and «( i* an arbitrary constant <
, depending therefore only on h. 1 + 2 T ( » + 1)
52 ANNALES POLONICI MATHEMATICI XIX (1967)
Corrigendum to "A note on some properties of the functions
[j
(l-i/p.) < TCr(ft+i) < c3 < i
(i <j < i - D ,
log[i+(ft+i)n,]2 < c4iog^,,, < c5 [](l-i/ps)-1 7i
r
(i-i/f.)/](1-1W"1/](i-i/}')"1
c,[] /J
7 = 1 rj<s^rj~i
< cs f]
f](l-i/p)'1
s= \
p|^4
i
a-iip)- ig.
vs,z
Then by lemma 7
logiZ < log j[l + (A+l)r0] fj[l^(h
+ l)rnf(h+2)j
n=l
< c, f](l-llp)-1 v
CO
£d < CilogZ...
«-o
Itcfu par la licdaclion le 26. 4. 1966
53
A NOTE ON SOME PROPERTIES OF THE ARITHMETICAL FUNCTIONS ip(n),a(n) AND d(n)*
WANG YUAN Institute of Mathematics, Academia Sinica
1. Introduction Let f(n) be an arithmetic function and consider the problem of distribution of the ratios f(n + l)//(n) (n = 1,2,.. .)• Somayajulu [1], Sierpiriski [2] and Schinzel [3] treated the functions by their elementary methods. Hua Loo-keng pointed out the possibility of applying Brun's method to treat this problem, and then the author and Schinzel [4], and Shou Pin-tsung [5] obtained more precise results in this way, for example, they proved the following result: Let
i ^ ^ i. 1 < v < k.
The aim of the present paper is to treat this problem by Linnik-Renyi's method, and to improve the above result: under the same assumption, we shall prove that there exists a sequence of prime numbers {pj} such that ^ ^ + "
+ 1)
l
The proof of the above result depends on the following: Fundamental lemma. Let k be a positive integer and m
0
= (fc + l ) ! 2 < 7 o i - - - ( ? o t o ,
" i t = fti • • •
fcti.
l < i < k
(1)
be integers which are coprime from one another, where q^ (0 k + 1. When x > Z > (motni • • • nik)2, let Nz{x) denote the *Acta Mathematica Sinica, 8:1, 1958, 1-11
54
number of integral solutions (p, XQ,XI,...,
Xk) of the system of equations
(p+l=moxo, | p + v + 1 = vmvxVi
(2)
1< v
satisfying the conditions 1 < p < x,
and iip'\xv, then p' > Z 0 < v < k,
(3)
where p and p' denote prime numbers.* Then there exist two positive constants c\ and Xi, and a positive constant a depending only on m^s such that Nx°(x) >
k 2
log
Cl3;
x log log x
,
x>X1.
2. The Proof of Fundamental Lemma Let M = (mo,mi,... ,mfe)2 and A be an integer such that 1 < A < M and (A, M) = 1. Let p\ < P2 < • • • < pr < Z be all primes not exceeding Z and not dividing M and a^ (1 < i < r, 1 < j < k + 1) be positive integers satisfying the conditions: 1 < a^- < pi and aij1 ^ a^ for j \ ¥" h- When x > Z > M, we use Mz(x) to denote the number of prime numbers satisfying 1 < p < x,
L e m m a 1.
p = X (mod M),
p^
l
l<j
+ l.
(4)
There exist A and aij 's such that Nz(x)
Proof.
aij (mod pi),
>
Mz(x).
It follows by Sun-Zi theorem that the system of congruences y + v + 1 = mv (modml),
0< v
(5)
has a unique solution A in the interval 1 < y < M. Since mv\(\ + v + 1), we have (mv, A) = 1 by the definition of mv. Hence (A,M) = 1,
(mv,X
+
" + 1) = 1 ,
0 < ^ < A:.
(6)
+1.
(7)
Let aij=Pi-j, a
Hereafter we use p,p',p\,P2,
l
l<j
• • • ,Pi,P2< • • • t o denote prime numbers.
55 Take p satisfying (4) for the A and a^ 's. Then
jp+1 = moxo \p + v + 1 = m o x o + v ~ vrrivXv, 1 < v < k by (5), and
. fp+1 \ /A + l \ ,m0 = m ,m0 = 1, (xo,mo) =
V m0
/
V o
J
(xo,mi) — (moxo,mi) = (p+ l,m») = (p + i + 1 - i,mj) = (-z,mj) = 1, 1 < i < fc, (xi, mj) = (irriiXi, m3) = (p + i + 1, m^) = (p + j + 1 + i - j , rrij) = ( i - j,mj) = 1, i / 0 , i ^ j , j ^ O , / /. v finiiXi \ fp + i + l \ N (ixj, m 0 ) = [irriiXi, m 0 ) = (p + i + 1, m 0 ) = (i, m 0 ) = i,
i ^ 0;
by (6). Since i 2 |mo, we have (xj,mo) = 1 (i ^ 0). Therefore (x o a;i • • • Xfe, m o m i • • • mjt) = 1
(8)
((P + 1)(P + 2) • • " (P + k + 1), P! • • -Pr) = 1
(9)
and
by (7). It follows by (8) and (9) that for such p, we have a solution (p, xo, x\,..., Xk) satisfying the requirement in Fundamental lemma. The distinct primes correspond evidently to distinct solutions. The lemma is proved. Lemma 2. There exist a positive constant f3 depending on k and two positive constants C2 and X? depending on k and M only such that MXP(X) >
C2
log
T
*
x log log x
,
x>X2.
Fundamental lemma is clearly a consequence of Lemmas 1 and 2. For the proof of Lemma 2 the reader may refer to Renyi [6] for k = 0. There is no essential difficulty to extend his proof to the case k > 0, and we still give a sketch of the proof for completeness.
56 3. Brun's Sieve Method We suppose that M,\ and Oy's are integers satisfying the conditions stated in Sec. 2. Set
P(x,Q)=
g
a, = - _ £ _ + * , ( * ) ,
(1,0) = i,
p = I(mod Q)
Mz(x) =
5Z
V
p = A(mod M) p£ay(modp 4 ) (l
L e m m a 3.
Let r = r0 > r\ > • • • > rn > 1 be any given set of integers. Then
Mz(x)>
xE ——
R,
where
2n+l
_/l. , i \ 2 n + l V ^ V V V ••• V ^ yis.fi.) Z^ Z^ Z^ Z^ Zs
R=\RM(X)\
+ (k + l) Y, \RMPa(x)\
V\fv )
+ (fc + 1 ) 2 ^ Y l \RMPaPB(x)\
+ •••
a>0 2n+l
+(*+irEEEE-Ei«^*(4 or
a>/3>--->!/
''^'•n
The estimation of E: Take /i = (1.25) ^F1 andfto= -^e. Then there exists a constant 5Q> M such that for 5 > So, we have
)
^^
—7-7 < log /l0 = T, V(P)
I
1
V
T-T-
/
< /l 0 .
_^______
57
Let pTj (0 < j < t) be the largest primes
Take n = t + rt and rs = rt(t < s
ptM
where C3 is a positive constant depending on k and M only (see Wang Yuan [7]).
4. Several Lemmas We use XD to denote the character modulo D. Let D = p"1 • • -pf be the standard decomposition of D. Then XD =Xpa^ • • • Xpa' • If XPQ* is primitive, then we say that XD is primitive with respect to p^. XD is primitive in the usual sense if it is primitive with respect to all p"* (1 < i < I). Lemma A. Let q,A be two integers such that A > C4, where C4 is an absolute constant, and p be a prime number satisfying A < p < 2A and (p, q) = 1. / / 2 and fci
logf
log .4
then except O(p 3 / 4 ) primes in the interval A
the
holds for any x(n) modpg which is primitive with respect to p, where c$ is an absolute constant and 62 = (4 x 104 ks)~1 in which k% is defined in Linnik [8]. We refer Linnik [8]. Renyi [6] and Littlewood [9] for the proof. Lemma B.
Suppose that (I, Q) = 1 and Q < e^logx,
then
where e is any pre-assigned positive number, CQ and the constant in " 0 " are absolute constants, and Qi is the modulus of "exceptional" character x-
58 See Titchmarsh [10], Page [11] and Siegel [12]. Since ap < logo;, we have
Lemma C.
The estimation P(r O) <
2xl gX
°
holds uniformly for 1 < Q < ^fx. We denote by E the set of positive integers Q = p[ • • -p'uM satisfying Pi > • • • > p'mPi + M,p^ < zM
If Q £ qi,...,qu belong to Set K
l < i < 2t + 1,
V'i
i>2t
+ l.
E, we write Q =p' 1 ?i, q\ =p'2Q2, • • •, Qu-i =p'uQu, qu = M and call integers the diagonal factors of Q. It is evident that all diagonal factors of Q E if Q £ E. — n ^ p ^ t P- We omit the proofs of the following lemmas, since they can pfM
be easily derived from the definition of E. And we use cr, c%,... to denote positive constants depending only on k and M if there is no special explanation. Lemma 4.
Let v(Q) denote the number of prime divisors of Q. Then v{Q) <
10(fc + l)loglogx Lemma 5.
ifx>c7.
The number of elements of E is < KZ?, where f — 1 + T~Y'
Lemma 6. Suppose that Q £ E, Q = p[qi, p[ < q± and p[ > MK. Then the number of such Q is not exceeding Zgvlh" , where g is a positive constant depending on k only. Lemma 7. Let {p*} be a set of integers with the property. For any given number A, there are at most A3/4 elements of the set which are contained in the interval A
V —
E ~T^ = IT (1 + -T^—^) log* + 0(1). Lemma 9.
Ifn> 2, then
59 5. T h e P r o o f of L e m m a 2 Set K
XQ(X)
= Xl x< 3^ a Pp<x
Then for (I, Q) = 1, we have
( 10 )
P X
( >® = JQ)T,XQWK™W^
W )
(XQ)
It follows from Lemma 4 and the definition of E that
R<(k
+ 1) 2 " + 1 ^
< e ( 2 n + 1 ) l o g ( f c + 3 ) J2
\RQ(X)\
QEE
\RQ(X)\
QGB
< e10(fc+l)log(fc+3)loglogx J - |i? Q (a;)|.
(11)
Q€i5
If Q = pi 9 i € £7 and Q > e ( lo « x ' 2/5 , then p[ >Q^ by Lemma 4. Since 71 < pi
>2e ( l o g l ) 1 / 3 ,
x>c9
, we have
Jbi = ^ + l = | ^ ( r i + o f r i T ) ) < l l ( f c + l)lQgIagx, log^ log Pi V \l°gPi/7
x > c 1 0 . (12)
Take B = 2e (logx)1/3 and A = 2 n B(n = 1,2,...). For a given qlt we use Lemma A to the interval A < p < 2A and introduce the following two conditions: (i) If Q > e('°sx) 5, then we say that condition I is satisfied. (ii) If Q € E, Q = p[qi and p[ is not the exception with respect to q\ in the sense of Lemma A, then condition II is said to be satisfied. If the above two conditions are all satisfied, then it follows from Lemma A and (10) that P(x,Q) = — ^ 7 - P ( X , 9 I ) + O ( X 1 - ^ T i o g : E ). If <7i still satisfies these two conditions, then the process may be continued and so on until one of the conditions I or II is not satisfied. Suppose that such processes are performed 5 times. Then
p
' =*i^) p
where q0 = Q and ki = logq\jlog §• + 1.
60 If condition I is not satisfied, i.e. qs < e^°sx^2
5
, then we have by Lemma B that
If condition II is not satisfied, then
P(x, qs) =
+O tp(q.)]ogx
. , \V(««)/
by Lemma C. In conclusion we have the following four types of error terms:
^ W '
(")
/QN
'
(m)
^ ^
.
(IV) ^
^
logs.
We denote by Ri,Rn,Rni and i?iv for those terms in YIQZE \RQ{X)\ that belong to types I, II, III and IV respectively. (i) By Lemmas 4, 7 and 8, we have Ri < cnxlog^x/e1/4^^1'3,
x > c12.
(ii) By Lemma 8, we obtain Ru < cis xlogx (iii) Take e =
18/fc+1^og(-fc+3->
• e 0°g^) 2/5 -'=6Vl5^ ;
%>
Cu
- Then it follows from Lemmas 8 and 9 that
Rill < CIS Z l o g 3 * • e-18(fc+l)log(fc+3)1oglogx]
x >
Cie
(iv) We use R^ to denote the part in R\y with ki < 2. Take Z = x1/JV
and A ^ > ^ ,
where iV will be determined later. Then by Lemma 5, we have -Riv < cirzf
' xl~^ ^°ZX = cnxl~™ log a;, a; > cig.
61 (v) Let R$ be the part in Rw with ki > 2. Then kt satisfies 1v < ki < 2v + 2, u(fc+1) loslogx ] by (12). Since p\ < q\'v and p\ > 2e(loe*)1/3 > 2 v = l, 2 , . . . , [ MK (x > cig), we have [J£(fc+1) log logs] R
C
IV < 20
log2X,
Xl~^iZ^
J2
X > C21
by Lemma 6. Take
(13) We have R\v
< C22 a; log- 5 1 • e "(fc+iSiogiogz-s ^
a; > c 2 3 .
By the combination of (i), (ii), (iii), (iv), (v) and (11), we have R < c 24 3;e-8(fc+1)lQs(fc+3) '°s'°g' log3 a; < c 24
^ log *•
, ' a;
x>c25.
(14)
Hence it follows from Lemma 3: Lemma 10.
There exist c26 and X3 depending on k and M such that ^
{ X ) >
^ ~
X
'
X>X
"
where N depends only on k. The Proof of Lemma 2.
Set (3 = jj. Then
MxB(x) < log^ • e-2^MxJ(k
+
*)*lo*]o8*) +logx.e-^W>s*n(x) \ log a; J s^ ( x \
and /(fc + 3)xloglogo;\
c27x
M — ^ — ) - ^ ^ ' x>C2s b Since h = (1.25) "k+r > 1, we have limy_oo V^V depends on k only.
— 0- Hence the existence of d is proved and it
62 by Lemma 10. Let
y>X2.
The lemma follows.
6. Applications of Fundamental Lemma Lemma 1 1 . Let GQ = 1 and av = v (1 < v < k). For any given k non-negative real numbers ai,..., a^ and e > 0, there exist positive integers mo,..., m/t depending only on ai's and e such that
^::?Z^-^
(15)
Proof. Let pu = ^ (l 0 (1 < v < k) such that h
^-a/-^ dv
pu
< | . ^ i , 3 pv
\
i.e. 6i---^_i^^+i'--rffc _a bi • • • bv^idvdv+i
pv-\
• • • dk
pu
e /?„_! 3
p^
<
~
<
~
Let &i • • -bv-idvdu+i
• • -du
—
= ^_i,
. , ^ ; , i
l
bx • • • Ofcdi • • • d f c
Then
For any given e' > 0, we may choose positive integers m'0,mi,... , m t which are coprime from one another such that their prime divisors are all >k + 1 and
63 since 0 < r\v < 1 (0 < v < k) and f ] p (l — | ) = 0. Hence we may take e' = e'(di's,e,p) sufficiently small such that ~^~
„ Pv-i
<
£
P^± 2
a,P0
~^~
<-.?°.
Set mo = (A; + l)!2mQ. Then we have the lemma. Lemma 12. Lemma 11.
This lemma is obtained by changing the function
Theorem 1. For any given k non-negative real numbers a\,...,ak there exists a prime number p such that
and e > 0,
y(p + v + 1) _ ^
(16)
Moreover, there are two positive constants C29 and X4 depending on a{ 's and e only such that the number of primes satisfying (16) in any interval 1 < p < x is not less than C291—k+2 x,—; whenever x > X4. Proof. First we choose using Lemma 11 the positive integers mo, m i , . . . ,m,k depending on a^s and e only and satisfying the requirement of Fundamental lemma such that (15) hold. Then let (p, xo,... ,Xk) be a solution of (2) satisfying (3) and Z = xa. Since (xi, m o , . . . , mfc) = 1, we have ifip + v + l)
=
~ ip(crv-imv-ixv-i)
_
- avmv
~~ y(^-im,-!) O"y— \TTLu — 1
_ ~^r ip(a^-i)
_ p + v+ l p + i/
'
^
'
Xv—\
Since the prime divisors of Xj, are all >a; a and £„ < 2;, the number of prime divisors of xv is at most [^]. Hence
(18) and consequently, it derives from (15), (17) and (18) that there exists C30 such that (16) holds whenever x > p > C3o(a,e). This proves that we may obtain a prime p satisfying (16) from a solution (p,xo,...,Xk) of (2) satisfying (3), Z = xa and p > C30. Since the number of solutions of (2) with p < c 30 is at most c 30 , Theorem 1 follows immediately from Fundamental lemma. Theorem 2 is obtained by replacing the function tp(n) by
64 Theorem 3. Let k be a positive integer. Then there exists a constant 7 depending on k only such that for any given k + 1 positive integers ao,a\,... ,Ofc, there is a prime number p satisfying av
+ i/ + l)<jav,
(19)
0
Moreover, there are two constants C32 and XQ depending on av 's only such that the number of primes satisfying (19) in any interval 1 < p < x is not less than log*+^toglogs
Proof.
WfleneVer
X>XG-
Assume that a o , . . . , a^ satisfy 2 a "
Q
We take tv — av + 1 in (1). For a set of m^'s, let (p,XQ, ... ,x^) be a solution of (2) satisfying (3) and Z = xa. Since the prime divisors of xv are all >xa, xv is a product of at most [^] primes. Set 7 = 2- +1 (fc + l)! 2 .
(20)
Then d(p + v+l)
= d{avmvxv)
d(p + v + 1) = d(aI/ml/xu)
> d{mv)d(xv) <
<(k + l)\22tl'2li}
> 2%" > av
d(ao)d(mll)d(xv) <-yav,
0
The theorem follows. From Theorem 3, we derive the following Theorem 4. For any given a\,...,a,k, where ai equals to 0 or +00 (1 < i < k), there exists a sequence of prime numbers {pj} such that d{pj+U + l)
lim -^77 r-^- = av, j-HX> d{Pj + V)
l
Finally the author would like to ask a question: Is Theorem 1 still true if cp(n) is replaced by the divisor function d(n)7
References [1] B. S. K. R. Somayajulu, The Euler's totient function
65 [4] A. Schinzel and Y. Wang, On some properties of the functions, Polon. Akad. Nauk 3 (1956) 201; A note on some properties of the functions
66
SCIENTIA SINICA Vol. Vin, No. 4, 1959
MATHEMATICS ON SIEVE METHODS AND SOME OF THEIR APPLICATIONS* WANG YUAN (3L
5C)
(Institute of Mathematics, Academia Sinica)
§ 1. INTRODUCTION
For the sake of brevity, we denote the following proposition by («, b). Every sufficiently large even integer can be represented as a sum of two integers > 1, of which one contains at most a and the other at most b prime factors. In 1919, v. Brun ll] first gave an essential improvement of Eratosthenes' sieve method and proved (9, 9). Brun's method and his results were improved and extended by several mathematicians, for examples: (7,7) (6,6) (5,7), (5,5) (4,4) (a,b),
(H. Rademacher12'), (T. Estermann131), (4,9), (3,15), (2,366) (G. Ricci1'1), (A. A. EyxuiTa6t5]), (A. A. ByxuiTafiW)", where a + b<6 (P. Kuhn)19"1".
In 1947, A. Selberg112"141 published his new improvements of Eratosthenes' sieve method. By the combination of the methods of Brun, EyxurraS and Selberg, we have (3, 4)[15J. By the use of Selberg's method and some results in the theory of the Riemann C~ function, A. H. BnHorpa,noBt16'171 obtained (3, 3). The aim of the present paper is to prove (3, 3) and* (a, b) (a + b*^5). Moreover, using EyxurraS's method with more complicated numerical calculations, we have (2, 3)118"201, that is: *First published in Chinese in Ada Uatheniatica Sinica, Vol. 8, No. 3, pp. 413—429, 19581) C£. also B. A. TapTaKOBCKHft'7-81.
a 358
Theorem 1. Every sufficiently large even integer can be written as a sum of two positive numbers > 1, of which one contains at most 2 and the other at most 3 prime factors. We have also the corresponding results for the problem of twinprimes and the problem of representation of large odd number as a sum of two almost-primes15. Namely, we have the following Theorem 2. For any given even number ^, there are infinitely many integers n, such that each of n and n + k. has at most 3 prime factors and n{n + ^) is a product of not more than 5 primes. Theorem 3. Every sufficiently large odd integer can be represented as 2N + 1 = IP + Q (P > 1, Q > 1), where the number of prime factors of P and also of Q is not more than 3 and PQ is a product of at most 5 primes. In this paper, p, p', p", • • •, pi} p2, • • • denote primes. § 2. COMPUTATIONS
Lemma 1. Let J2(«) be the number of different prime factors of n, that is Q(n) = 2 1. If x > 1 and z > 1, then
? j £fe ii^, i n_^n(.-|)
1
(1 + f ) - ^
+ O(log22-loglog3w) + O((loglog3*)2) , where /*(«) denotes the Mobius function'21'.
Lemma 2. If 21 x and z> 1, then *<*
»
'«V
P-2/
8 P>2 P(P~2)
(».i)=l
"x
P
P>1
+ O(log ^ log log 3 xz). Proof. Let (P(r) = Tl (/> — 2). sp
1K»)]2 OW n / , ,
r<*
x
'
2
=
V
Then by Lemma 1, we have |K«)l2° ( r t
V
_2^
s<x/r
1) An integer is called almost-prime, if the number of its prime factors does not exceed a fixed constant.
68 359 __
|Kr)l2^
f_l_
(p - iy (p + 2)
p
z
+ O(log #.v log log 3 2#)> =
_ l n Q, - I ) 2 (p + 2) n _ ? _ . , „ . y l/x(r)l4^) 3 2 V ? ',, P + 2 " r ^ , | J (p2 _ 4 ) + O ( log #:*: log log 3 #x)
= ~ TT Cf " ^
8 p>i p(p — 2)
TT —~ PU
p
log2^ + O(log ** log log 3 «e) .
Thus we have the lemma. Lemma 3. For any given 7 > 0, there exists a positive constant *o — ^0(7) s u c n t n a t ^ e inequality log x
holds for x > x0, where d(x) denotes the number of divisors of x[Z2] Iog3x
Lemma 4. Let A ( * ) = e ^'««»*> where x > 1. Then there exists Cj > 0 such that
1
4
log 3J
i z^OCe""'1108'0891) .
Proof. Let /?,- be the ;th prime number. theorem we have
Then by Mertens
ir(i-|)--o(n(>-^r)-o( n (i-if) = O( log log 3*) .
Hence, it follows by Lemma 3 that
a(«-ir-{-^(.-ir+s>(»-in.^r p^fp*
69 360
A(*)
=
p
V
PI
V
f\x
J
pp'\x p*pf
TT (i - i.)
AGO
P\x\
pj
log3x
-c 1
= 0(e
U
'""«••) .
Thus we have Lemma 5. Let q—O(l) be a given integer and y be an integer. Let «(D = 1, « ( ? ) = — ( * I y), * G 0 = —(? + y), «G0 = TT g(p)and/(«) = p
P
Pi"
- f v - TT (1 - g(?)). If z > 1, then »
/C«)
4 p|,
V
?
P > 2 />(/> — 2 ) p|, y P — 1
pl?y P — 1
log log 3 yz/
p>i
Proof, (i) If z ~> A(y) > log 2^, then log log 3^ = O(log 2y) and log log 3y = O(loglog3x). By Lemmas 2 and 4, we have 2
»
WJOJ _ s J K ^ U ^ fk")
n
«•£*
(i - ±y
n
Pl» V
P'
+ 2M'--)-2>^=-n(i-ir P'+9
(»,«y)=l
+ 2 ^( 1 _4r( l _^.)- s >•
p'pf'ly P P
P'
^
P
/
i£«&Ln(i-|r+-
n
n
p\n
\
P/
>{i +y iz4:(i-4r+ s ^ ( , - Pi 'r ( i - -P v^ r + -3} y P P^ p' "ly P P 1
V
fSAW p'+«
IK«)|2^
2
K
f IP'P''.9>=1 P'=iy"
n
(1
+
(».«y)=l
+ O (log 2 2y log log 3 ^y) J
_2_)
V
70 361
4
p\q P P>2 pip - 2 ) p\*,p—l
x
Ioglog3y
\p\qyp~l
P>2
P>1
/
(loglog 3 y ) 2 /
pl«y ? — 1 P>2
+ O(log2 0y • (loglog 3 zyY) = JL 7 | £Z^ yj C P - 1 ) ; U £^2 l o g ^ + Q / ^ £ - 2 4
Pl9 ^ p>2 Pip ~ 2) f-«y ?— 1
Iog22gy \ ^
^pl«y?~l
P>2
Ioglog30y/"
P>^
On the other hand, we have n
I
ply \
?/
PJ V8 p>7p(.p-2)
P+q
=
p'\y p V
p]qy P>2
P
I
T ^ g 7 i T T ^ p ^ n ^ f i o g ^ + o(TT£nfSky?"1
4 pi, P p>2 /?(/> — 2 ; (,|,y/>— 1
lo 2 y
f : ).
Ioglog3^y/
p>i
P>J
This proves the Lemma. (ii) If A( y )
2^>{i + 24(i-4)"' + 2 ^(1-^(1-^)"
<•
p'ly P
-1 »
V
P'
»'«»|y P P ^
Pin X
"
P'
\
/» '
r '
(»,«y)=l
= J ^ n tzl\\ (* ~ iy TT 2=iw# + ofTT£=^ 4 i,, ? "»;(p-2)iU-i P>J
and •<- / ( » )
J »
l».9y)=l
^
O'lyP V
»
P'
p | » ^ P/
log22gy
"j
Vpi'y/'-l log log 3 «y/ P>2
Jff\, P P \
P> V
P ^
71 362
4 t u P P>IP(P — 2) piqyp—l
^piqyp — i
Ioglog3sry/
Thus the Lemma follows. (iii) If A(y) > z, then by Lemma 2, we have S J^2i = o ( 2 l£Wl^ n ( 1 -i)-)
V(log log 3y) 2 /
V ;,' y p - 1
log log 3 zy )
and 4 pi,
P
P> 2 ? ( ? — 2)
p]n
P— 1
= O(TT -^IA . _wii2_y ^pi«y P — 1 P>2
Hence, »<, / ( « )
4 •;,
p
+ 0(n^zA.
^ Play P — 1
log log 3 zy )
;;2
P{P
- 2) ;,, y p - 1
W2zy \ log log 3 z y /
Thus the Lemma follows by (i) (ii) and (iii). Lemma 6. Let jB > a > 1 be two real numbers. Then for * > 2, we have p § 3.
THEOREM A
Let 2 < y < * be two integers. Let a,q;
a;,bj (1 < i < r)
(w)
be a sequence of integers satisfying the following conditions: 2\q, <7 = O(l); if Pi\y, then«/ = *i (mod/>,), otherwise a,-^ £,• (mod/>,) ( l ^ « < r ) ,
(1)
where 2 < px < — <pr < f are all the primes not dividing q and not exceeding $ and where $ is a real number and $ > q.
72
363 Let Pul(x, q, £) be the number of integers « satisfying the following conditions: 1 <^ » < ar, » == a (mod q) , n ijfe a; (mod pi) , n Ej£ b; (mod £>,•) (K.•<>-). (2) It follows by the Chinese Remainder Theorem that each of the systems of congruences y = a; (mod pi)
(1 < i < r)
y = 6; (mod pi)
(1 < « < r)
and has a unique solution in the interval 1 < y < pyp2 • • • pr. Denote these solutions by a* and b* respectively. It is evident that the conditions (2) are equivalent to the following conditions: 1 <«<
«,
n = a (mod q) ,
(ji — a*) (» — &*) ^
0 (mod p;)
(\
Theorem A.
(3)
r
Let c > 0 and P = 11 /?,- = TI p.
Then the esti-
?+«
mation p;(*, ? , f) <
^——- + O^2-" log6 $) y |/*(w)
holds uniformly in (w), where ^(D = 1, «(p) = — (phi, P gin) = ]T g(?)
s(p) = — (P + y), P and /(») = ^ — TT (1 ~ g(?)).
Proof. If ^ IP, then (^, ^) = 1. It follows by the Chinese Remainder Theorem that the number of solutions of the following system of congruences (» - «*) (» ~ **) = 0 (mod 10 1 n = a (mod
73
364
£
l = 2««-«(*.,))
[JL]+
*R»-«'X»-i')
O ( 2 *«)
^- k.1 '
= *(*) — + O(2O(«) . Let
A *
IK")I / v M£i
v
tip
*(*> /CO x ^ / *
'<«>
X
« ^
^
'
where ^ | P . Since hx = 0 and Arf = 0 for «/ > $c, therefore
n^a (mod qi
< X <S0 < r
fl=d (.mod ij)
=•2
2 M<,
I*5
^)/2
2
(
VrfKCfi-a"1) (*-»•), p)
2*€
2
i
n=« Imod «>
= f - 2 2 ^sCG/i,*,})
where {^i,
-TT = 2 -fr Z ^)= 2 ^ £W
r|«
gU;
rf|r|»
ix\n gKX)
=
= 2 —r n a-gw)-z/(*). Hence, y
~
^
^
^ d'7r7J~7yT
S 2 ^
K\n d\k
g(jL)
74 365
= 2
2 KKgiddgUZ 2 /GO
= 2 /G0( 2 **«(*))'. Let
5=
V IK"»)1
Then, we have for \\P
m|/
m|/
and
2 !•«(*) = 4- 2 2 ^ - ^ 5
'1*1*
' 1 * 1 ' «m<«'/*
*<•"**•>
= LK£L
s fid) •
Hence,
By the Mertens theorem, we have
-((2WJW(S5^"J) *l?
vv
«*
*l/
TT (1 - A V
= O(£fclog«£). Thus we have the theorem.
y
vv
t<^
*}
75 366 §4.
APPLICATIONS OF THEOREM A
Let / be a positive integer and / < c ^ / + 1. Then by Eratosthenes' sieve method we have V
IK")! _
i
f(»)
\Kn)\ _
y
/(«)
1
y
y
\Kn)\
1
(
/(*)
fin
*
2
2
/C»)
K.<««
+
l^i-+-•••+(-!)'
f
pjp^? 17
+
A
^
(4)
1M(")1 _ +.. /(-)
(».pip»«)=i
(-1)'
!
2
If
1 ^ c < 2, then
\ftM\
^
«
1°.
s
^B>
^ « f W ,
S ! y ,
2
^r.p; (».p1-p/«)=i
1 < B <
for sufficiently
/(»)
large
*, we take
i
x2*
f = -—^— ( > q) and write 2c = d. Since
o +o o £^-,£}= (,£i) (|i)= (T) s P+y
V X V" ) ' thus by (4) and Lemma 5, we have y
_IK^)i = y
^
K») i
\Km)\ _
/(»)
y
_2_ v
(<%<
1A*(«)1 •i.^flogM
P £„ Km)
+o
\-^r)
xnf=|iog^ + o(TTf4--^-).
76 367
Therefore, it follows by Theorem A that P-(*. q, ^V") < P»(*, q, j ^ 5 - ) \ log x J
< ^ ) ^ Tfog4* - + °p\qy ( nP ^ - I^-) — I T log log * / 3 2
(5)
X
P>2
A{d) = 2e» \ £—1 (2 < d < 4) , [G*-l)>-2(|)log|J
1
p>2 V
(6)
(/»— 1) ^ P I , P — 1 pl,y? —2
(7)
P>2
where the constant implied by the symbol "O" is an absolute constant, and y denotes Euler's constant. 1
2°.
x2c If 2 < c < 3, then we take also f = -—r— and write 2c — t/. log 5 a:
Since V
1
y
1^(")| _ y
1
y
1^(»)1
and V ^ pp'<Se
1 _ y J_ = V (_?_ 2 _ 2 2 \ 'W /(?') , ^ , » ' f <^, V-2 ' t-2 P-2 p J pp'
pp'
therefore by (4) and Lemma 5, we have
2 !/*(»)! = y I/*C»)I _ y 2. v !/*(")! «<«« '(»>
«ci. / ( » )
«
p
»
77 358
l
+ 2 — 2
4 wK?-2)
pi,
f
pi,y?—1 p>/
V
pi«>?-1 P>J
-^+o(^)
log log «/
Hence it follows by Theorem A that for any given e > 0, there exists a positive constant *0 = xo(e) such that (5) holds for x > x0 and AW=2t»(
f
-—
\ ( 4 < i < 6 ) , (8)
where 5(«)
=
Hm
^ _
2
- ^ r Iog2-il- (« > 2) .
(9)
PP'<€"
Eq. (5) also holds for x > x1(e)> 6 < d < 8 and
(6<J<8),
(10)
where K(a)=jto
8. 2
-^>Iog'-|l; (a>3).
(11)
«
In the same way we have the expressions of A(d) for d > 8. Let us put A(d) = A(2) for 0 < J < 2, then it is evident that (5) holds for 0 < d < 2. We may estimate 5(a) and X(a) by Mertens' theorem. For example,
Hm 2 i->Hm( 2 ^ I
-i+
78 369
+ Z | +
2
-V+--- + 2 \
2 A-
2 - ^ 2 -V)
_—, log 1.01 ! n1 ,log 1^19 4-log, . 1.02. , ,log 1.09. 1.11 log1.18, 4---- + log 1.1 1.01 1.09 1.08 1.09 +, .log 1.095.log1.105.> nnnCiC1 0.00561 1.09 1.095 We have also the estimations of 2 «f
pp
T and
2
T (2 < i
t
PP'
&
(2-'
< 9). It follows that *(3)>4 1im(o.81 2 - V -f- 0.64 2 - V + • - •
+ 0-01
2
—^-r) > 0.087202 .
f
§5.
THEOREM B
Theorem B,. Let 1 < a. < /3 < 15 be two given numbers. If there exists a non-negative and non-decreasing function A(d) (0 < d < 14) with the property that A(d) has at most a finite number of discontinuities, such that the following inequality holds uniformly in (w) and d: P-(*W,s 1 ") < / i ( * ) - ^ - +o(TT - ^ ^ •, 2
log AT
V f,;, p — 2
2 2
,y ,
),
log x log log */
then xi/^,1/-
^
/
VJ—«
V« + i /
P+y
+ o V( ^ _ _2 £ ? y ^ _ _ _ ) log * Jog log x ' holds uniformly in (vf).
«»
/ log2*
79 370
Proof.
L e t n = [ V l o g x ] a n d u, = « + A l l l / - 1 ( 0 < / < « ) .
Then, by L e m m a 6
P+ y
p+ y
I <
I I
/ p log2 *//»
log A:
A ( - ^ ± L \ SSZJL (i O g ifi±L \ » ; + 1 + 1/ log2* \ W;
^2
+ 0(
V log2 */p log log */)
+ "<-« - »') + o (^S3^L) Ul+lUl
'
\ log* X /
(iog^±L+ »m-«A\
^log x log log a; V
«/
«;+i «; / /
Since -—•— is an increasing function of x for x > 1, hence* 1 +x ° '
"f ^(_fti^i_){los^±L + »/ + i-^}-r i y l (_PA__)^±i i 2
< l (" C-^TI) - H ^ T ) ) .JIS. Sr ^ <• It follows that
P+ y
+ o(
c 2
s3±
).
Mog *loglog*/
Thus we have the theorem. Theorem B2. Let 2 < « < ) 3 < 15 be two given numbers. If there exist two non-negative and non-decreasing functions A(z) and
80 371
A(z) (0 < z < 15) with the property that each has at most a finite number of discontinuities, such that Pu(x, q, XW) >*(«) - ^
e
+o(
log x
) (0< *
*>*
(12)
Mog x log log */
and P-(*, q, xVz)< AGO £«£- + O ( log x
/«»*
) (0< * <15), (13)
''log a: log log x/
where the constants implied by the symbol " O " are absolute constants, then [0, = \
AI(«)
if 2 < « < r ; f8 -i
z
J a—l
. , 2
(14)
Z
and ^li(«) = A{&) -2 f8"' A(») iL±J-rf«, J«-i
2<«<j3<15
^r*
(15)
have the properties of the functions A(z) and A(z) respectively. Proof. Let n = [v/log^r ] and u, = « + ^ ~ " / (0
The
w
difference of P»(x, q, ps) and P^x, q, pr+J) is the number of integers satisfying the following conditions l^ra^tf,
Let
TJE=0 (mod q),
(1 < « < r) ,
nE£= a; (mod pi),
n ^ ^/ (mod pi)
» = cr+i (mod f r+1 ) (c r+1 = ar+1 or ^>r+1) .
(16)
and 3«, ^; W, %Y> (KKr),
(Sr)
where a(r), 2*r), a,(r), ^,-M, S/M and £/ r) denote the solutions of the congruences mpr+i + ar+1 = a (mod q) , wp r + l + * f+1 = o (mod ,•) , »j/>f+1 + a r+1 = */ (mod />,•), mpr+i + i>r+: = <*/ (mod />,),
and wp r + 1 + *r+i = ^« (mod pi)
81_ 372 respectively.
Hence {P«-r(~ , q, Pr) + P= r (— , q,Pr), if Pr+l+yl
P«(x,q>Pr) — P«(x,q,Pr+D<\
. \P<*A-JL—, q,pr\ , V J I Pr+l _I_
1
If the primes in the interval x " Pl
< xv-,+% <
/+1
if pr+1\y .
< p < x"'
are
< . . . < ^ < ,4/-, <
fl+1
?/+i ?
then PM(*, 2
+ o (*
1-J_
P u , (-iL , ^ , ;,,._,)
»)
>2A(«,-1)*
£i
^
2 l
+ pf
2
j _ p log x/p
c x
2
"
Vlog log .v _ i _
-
)
j ^ _ p log2 *//>/
, 14
> 2A(«, - 1) ^ ^ (log t^±^-1 lOg^X \
«,_!
+ of^a) + o ( — £ ^ Mog3x/
+ _i
-J:—)
M; — 1
«, + 1 — 1 /
(log*'*' ~
Vlog 2 xloglog* V
x
+ _J
«/— 1
«;—1
JL_^ « J+1 — 1 / / '
Hence
p-c*,«?, * v o - p-(*, \J«-i
sr2
/ log2x
Vlog"x/ +
O(
So* ) Mog 2 xloglog*/
82 373 This proves (15).
In the same way we have (14). § 6.
THEOREM C
Let 2 =SJ y < x be two given integers. a,
q;
(» =
a;, b;
Let
1,2,---)
(a>)
be a sequence of integers satisfying the following conditions: 2I<7J
<7=,O(l);
otherwise
a,- ^
^
Pi\y>
then
£,- (mod />,)
a; = (; =
b{ (mod />,-),
1, 2 , • • • ) >
(17)
where 2 < px < p2 < • • • are all the primes not dividing q. Let 15 > ? > « > 1 be two given numbers. set of integers n satisfying the following: 1 ^
X,
» ^
n == a (mod ^ ) ,
n ^ a; (mod £>,-) ,
n^ a1+j (mod pl+j) ,
n^. bi (mod p,-)
(1 < « < f ) ,
(1 < ; < / — s) ,
n ^ b,-fj (mod pl+j)
where p , < xu" < p , + 1 and />» < xu" < p , + 1 . of SO? is denoted by M(x, xl/% xVu).
Let 93? denote the
(18)
The number of • elements
The purpose of this section is to prove the following Theorem C. The number of elements n of 9H satisfying most m of the following congruences n =. c,+j (mod />,+,-)
(1 < ; < / — s , c J + i = a,+i or b,+i)
at (19)
is not less than '^'
lm+1 J«-i
\g + lJ
z2
/log 2 *
Vlog'xloglog*/
The proof of the Theorem depends on the following two Lemmas.
Lemma 7. M(x, x1", x1'") = Pa(x, q, xv') + O (x1") + O (xl~v') .
Proof. P u (*,? ) * 1 ")-M(*,* 1 ",* 1 / ") y « - / (. «<* "="/+/ ( m o d » J + / )
»<* J »^» / + ,('»oJ * J + f - )
83 374
Thus, the lemma follows. Lemma 8. There exist two sequences of integers (&>;) and (5,) (1 < / ' < / — s) such that the number of elements of SDI satisfying at least / congruences of (19) is at most
-f { 2 P - ( — >
let Tj be the subset of 3Dt whose elements
n = c,+i (mod £,+/)
(^+; = «,+/ or i>s+j) .
(i) If />,+/1 y, then aJ+i^bJ+i (mod p,+i). the solutions of the congruences at+j + mp,+j = a (mod + ?»?,+/ = a; (mod p,) , aJ+j •+• mps+j = 3; (mod />,-),
(20)
For 1 < / < s, we denote
(1 < i < (7) , (1 < m <
?/)
,
(1 < m < /?,) ,
by a(i), aiU), biQ) respectively. It is evident that a^ = b\n (mod pi) for /7,-|y, otherwise a^^b? (mod />,•). Let
, q, x1/v) . Then the number of elements of Tj is at most Z3.,, ( ' \ P
bJ+j + mpj+f = a (mod
( 1 < » » < ^ ) J
^ / + / + nipj+j = a,- (mod pi)
( 1 < w < /»,•) ,
. ^ + / + mp,+l = *,- (mod />,-)
(1 < m < />,•)
by 2 0 ) , 5, (/) , ?,-(n respectively. Let 2 ( ' \ q; ~aP; IP. Then the number of elements of F,- is not more than
(«,)
84 375
P"i ( — , <7, *"') + P*, (—
, q, **") .
Since pJ+i > xv", therefore
If the element' of SDt satisfies at least / congruences of (20), then it belongs to at least / different sets T,-. Hence the number of elements of 3K satisfying at least / congruences of (20) does not exceed
| { 2 (P., (— > i, *») + P3| (—,«, *»)] + | 2 r~,(—.i,*") x
t i
/>/+/
'
= 4 " 2 (P-, ( - ^ - > * v ') + P*, ( — > ^1/K)) + o &-1"). Thus, we have the Lemma. The proof of Theorem C. It follows by Lemmas 7 and 8 and Theorem Bj that the number of elements of 9H satisfying at most m congruences of (19) is not less than
Ps+i+y
' V Ps+i
//
= p. (—, -?, ^") - -4rr 2 (P., (-^-, ^ ^ ) + p S/ f_£_ , ^, «i/")) + o (*1~1/0 + O (a"")
+ 0(
So*
).
MOg 2 * log log AT/
_ ^ _ _ _
85
376
Thus, we have the Theorem. § 7.
THE PROOF OF THE MAIN RESULTS
Let A(«) and A(a) (0 < « < 10) be two non-negative and nondecreasing functions with the property that each has at most finite discontinuities such that A(«) ^f
+O(
log x
-) < PB(*, q, *"')
/*>*
Mog x log log xJ
e < A(a) ^f+O ( »* ) (20) log' x Mog * log log x/ holds uniformly in («) and a. Denote these functions by Ao(«), Aa(a); Aj(«), A^a); — . Let x, = 3.5 + 0.01 i (0 < / < 210) and *210+I- = 5.6 + 0.1 / (1 < i < 34). For sufficiently small e, we have A(Xi) (0 < / < 232) by (6), (8) and (10). By the use of ByxuiTaS's methods and his estimations of A(10) = 100.02073 and A(10) = 99.98181[5>6), we have A(x,) (233 < ;' < 244) and A (A:,) (0 < / < 244). Define
f A0(x)
— A(xi+1),
if
x; < x <
I ^(*)
= A(*,),
if
Xi < AT < «:,•+!.
xi+l;
Now we list the values of Ao(*) and A0(x) for integers: a Ao(a)
10
9
99.98181 | 79.78469
ylo(a) 100.02073
82.7207
8
7
I
6
5
60.88817 j 43.51554 j 26.70925 68.52511 I 54.39352 43.0082
4
9.18109 34.89666
0
(21)
29.39023
Divide the interval («— 1) < x < (jB—1) into « sub-intervals «,•< x <«,-+]. Since A(a) and A(a) are increasing functions, hence 1
A(z)
\
A(z)
-—
— dz,
and
J'-i
Take aI+, — «,• = 0.01.
T~
— dz<. 2L Mtts+O sr
,=t
J"t
dz
-
s
It follows by Theorem B2 that
c r | l O 9 8 7 J 6 ••• j 0 < a < 5 . 5 3 Au(a) 99.98181 80.892035 63.59931 47.471252 31.004145 ••• j — Aa(d) 100.02073 j 81.11841 j 64.4031491 50.529826 41.01897. ••• j / l u ( J ! ) = / I « ( a )
(22)
86
377
I.
Let x = y be an even integer. Let <*=1>
9 = 2;
a; = 0, bi = x (« = 1, 2 , • • • ) .
(w.)
Let /»,- be the zth odd prime, (i)
By (21), we have
P^*, 2, «") - (i- P A, M M ^ - <**) - ^ V2Ji
\« + l/
/ log2*:
^
+ o(—~^—) Mog x log log */
> (60.88817 - A- U ( 7 ) P iL±_lrf«+ yio(6.4) P - ^ ± 1 - ^ z
+ ^co r ^ * + M5.4> p *±i A +4,(4.5) r *±i j , r ^ ^ J2 2T2
+
0(
Jl.21 2T2
Jl
z1
i
log 2 ^
^ ) ^log AT log log * '
> 0.56125 - ^ - + o(—-^^ ) >3 log a: Mog x log log x/
for x > x0. Hence, It follows by Theorem C that for x > x0, there exists an integer n such that 1 < n < (x — 1) and n(x — n) have no prime divisors < xus and have at most 3 prime divisors in the interval xm < p < xu\ Hence, «(*—•») is a product of not more than 5 primes. Since x = n + (# — «), thus we obtain (a, £), where a + b < 5. (ii) By (21), we have
> {26.70925 - -?- U(5) P f<
+ 4,(4.8)
^
*4-1
J 4.6/1.4
£ + 1 ^ + 4,(4.6) JJ2
^ + A.(4.9) p " ' 1 £ ± i ^ f4.6/1.4 ' 4 . 1
J 4.4/1.6
f4.4A.6- 1 1
i±i*r + ilo(4.4) Jf2
•
J 4.2/1.4
iiirf. ^
1) Here, we use the following simple fact: If g(x) is a non-negative function and / ( * ) is a .non-negative and non-decreasing function in the interval a ^ * *S i , then
PtfCO/OV* < /CO P ,«(*) <** + /Cof^C*)'* holds for any a < (c - <J) and e < *.
87
378
for x > xQ. Hence, we have (3.3). (iii)
By (22), we have
1
\ 3 hn
) log2 x
z2
\z+V
Mog2 x log log.tr/
4.5+o.ttty+o.m
> (1 (8) - 128 y >iu(4.5 + 0.02y) r^-o-^-o^ ^ W s» V" 3 y = 0 (4.5 + 0.02y)2 J"4,5+o^y ^+l/log2^ 3.5-0.!Cy
+ of
£^—)
Mog * log log AT/
> o.43 - ^ ^ - + O ( log *
^^
) >3
Mog x log log */
for x > x0. Hence, we have Theorem 1. II.
Let x be an integer and y = ^ be a fixed even number. Let a=l,
q = 2;
ai = 0,
Let /?,• be the /th odd prime.
b,- = - £
(* = 1, 2 , • • • ) .
(«»2)
By (22) we have
\3 J9/7 \z + l/
J log2*
z2
+ o( * ) > 0.43 -&tfL + O ( _ _ ? ) Mog2 a; log log x/ log * Mog A; log log xJ > 0 . 4 - ^ ^2 log * for x > xa.
Hence, it follows by Theorem C that for x > #0> there 2
\ - integers n in the interval 1 < n < x such log2* that «(« + ^) has jio prime divisors < x1/a and has at most 2 primes in the interval xm < p < / / w . Thus we have Theorem 2. exist not less than 0.4
III.
Take x = y be an odd integer. a = x — 2,
q = 4;
a; = 0,
Take
£,• = *
0"=l,2,---).
(.^3)
1) It follows by ByxiHTa6's"J result that ——~— is a decreasing function in 5.53 ^ z ^ 10.
Since ^ O ) = A W for 0 < 2 < 5.53, henc. Jbj&L < ^ - | ^ ° 4.5 + 0.02j; < 2 < 4.5 + 0.02y + 0.02.
o
f f
for 0
88
379 Let pi be the ;th odd prime.
By (22), we have .
\3J9/7
+ o(
/«*
\2 + l /
22
/log2*
)>3
Mog x log log #/
for x > x0. Hence, it follows by Theorem C that for x > x0, there exists an integer n such that 1 < n < x — 1, 2\n(x — n), and has no prime factors < x1/s and has at most 2 prime factors in the interval xm < p < x1/u. Thus, we have Theorem 3. § 8.
OTHER APPLICATIONS
I. First of all, let us state the grand Riemann hypothesis as follows: The real parts of all zeros of all Dirichlet's L-functions L ( J , x) are «SS —. (Ji) 2 1241 From (R) we may derive the following : Let (\, I) = 1. Then
0=/ Imod *)
where li x = I
J* log/
and the constant implied by the symbol " O "
is an absolute constant. Assuming the truth of (/?*), we have118'251 the following: 1°. Every large even integer can be represented as a sum of a prime and a product of at most 3 primes. 2°. There are infinitely many primes p such that p + 2 is a product of not more than 3 primes. 3°. Every sufficiently large odd integer can be represented as 22V + 1 = p + 2Q, where p is a prime number and the number of prime factors of Q is at most 3. 4°. Let Z2(N) be the number of twin primes (p, p + 2) not exceeding N. Then
z 2 (N)<8n 2 ( 1 -(T3i)i) 1 -^ + o ( l o g J i V 1 ^ l o g i v)-
89 380
II. If F(x) denotes an irreducible-integral valued polynomial of degree ^ without any fixed prime divisor, then we have126'271 the following: 5°. Let n(N; F(x)) be the number of primes represented by F(x) as * = 1, 2, — , N. Then *(2V; F W ) < 2S to - ^ - +o(-2-) log TV
VlogZV/
,
w h e r e the constant fiP and the constant i m p l i e d by the symbol " O " depend on F(x) only and y denotes E u l e r ' s constant. 6°.
Let B
_I*
*
+ 1
if
»
l * + «/,
K*<5;
if
k>5,
where w is the least integer satisfying ,
1
-^ 5.64527 ,
U> + 1 ^
3.65 , 1-
5\ — w log
i
• .
4.8396 4.8396 w+5 Then there are infinitely many integers A: such that F(x) is a product of at most n primes. III. For the problem of the distribution of consecutive almost, primes, we have1271 7°. For sufficiently large x, there exists always a number (i) in the interval x — x10/u < n < x with at most 2 prime factors; (ii) in the interval x — *20/49 < n < x with at most 3 prime factors and (iii) in the interval x — xv$ < n ^ x with at most 6 prime factors. 8°. Let v be a positive integer. Let w be the least integer satisfying the inequality 4.8396
4.8396
w+ 5
Then for sufficiently large x, there is always an integer between x — xl/' and x which has at most \ = w + v prime factors. REFERENCES
[ 1 ] Bran, V. Vid.-Selsk. S\r. 1. M. N. KL. 3, 1—36. [ 2 J Rademacher, H. 1924 Abh. Math. Hamb. Univ. 3, 12—30.
90 381 [ 3] [ 4] [5] [ 6] [7] [ 8] [ 9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
Estermann, T. 1932 / . reine Angew. Math. 168, 106—116. Ricci, G. 1937 Annali della R. Scuola Normale Superiore di Pisa (2) 6, 70—115. ByxniTaS, A. A. 1938 MameM. c6 , 4(46), 375—387. . 1940 fiAS GGGP, 29, 544—548. TapTaKoBCKHH, B. A. 1939 RAH GGGP, 23, 122—126. . 1939 RAH GGGP, 23, 127—130. Kuhn, P. 1942 Norsfc Vid. Sels^. Fork. Trondhjem 39, 145—148. . 1953 Tolfte Skandinaviska Matematikerkongressen, Li:.nd. 160— 168. . 1954 (Sept.) Abstract from the Proc. Inter. Math. Congress, Amsterdam. Selberg, A. 1947 Norsfc Vid. Sels^. ForAdi. 19(18), 64—67. . 1950 Proc. Inter. Congress Math. 1, 286—292. . 1952 Den 11-te Skandinaviska Matematikerkongress, 13—22. Wang, Y. 1956 Ada Math. Sinica, 3, 500—513. BHHorpa,uoB, A. H. 1957 MameM. co. 41(1), 49—80. . 1957 MameM. c6. 41(3), 415—416. Wang, Y. 1957 Science Record (New Ser.), 1, ( 1 ) , 9—12. . 1957 Science Record (New Ser.), 1, (5), 15—20. . 1958 Acta Math. Sinica, (3), 413—429. Shapiro, H. N . & Warga, J. 1950 Comm. Pure Appl. Math; (3) 153— 176. Hua, L. K. 1955 An introduction to the theory of numbers (in Chinese). ByxuiTaS, A. A. 1937 MameM. c6. (2), 1239—1245. HyaaKOB, H. T. 1948 RAH GGGP, cepun MameM. (12), 31—46. Wang, Y. 1956 Acta Math. Sinica, (4), 565—582. . 1957 Prog. Math. Academia, (3) 416—426. . 1957 Science Record (New Ser.), 1 ( 3 ) , 1—6.
91
SC1ENTIA SINICA Vol. XI, No. 12, 1962
MATHEMATICS
ON SIEVE METHODS AND SOME OF THEIR APPLICATIONS* WANG YUAN (3£
56)
(Institute of Mathematics, Academia Sinica) 1.
STATEMENT OF RESULTS
In this paper, we shall give the detailed proofs of the following three theorems (cf. [ 1 ] ) . Theorem 1. Let F(x) be an irreducible integer valued nomial of degree ^ without any fixed prime divisor. Let j k + l , if »= i •• \ + w, if where w is the least integer
poly-
1<*<5; k>5,
(1)
satisfying
, . ^ 5.64527 , 3.65 , 5\ — u> /-,\ w + 1^ + log —i (2) 4.8396 4.8396 w+ 5 Then there are infinitely many integers x such that F(x) will be a product of at most n primes. Theorem 2 . Let \ be a positive integer. Let n be an integer satisfying (2) and (2). Then for sufficiently large x, there is always an integer between x and x + xVK which has at most n prime factors. ber
Theorem 3 .
For sufficiently
large x, there exists always a num10
( i )
in interval x < m < x + x" with at most 2 prime
factors;
20
(ii)
in interval x < m < x + xA9 with at most 3 prime factors; and i
(iii)
m+
in interval x < m < x + x *
with at most 103 prime
factors.
* First published in Chinese in Ada Mathematics. Sinica, Vol. IX, No. 2, pp. 87—100, 1959.
92 1608 No complicated numerical computations are needed in the present paper. As to the history of the above problems, we refer to Brun [2 \ Rademacher t3 \ Ricci[4] {5\ Kuhnf6:i t7] and /IHHHHKCS). In this paper, p, p', p", • • •; pi, p2, • • • denote primes. 2.
SEVERAL LEMMAS
Let F(x) be an irreducible integer valued polynomial of degree £ without any fixed prime divisor. Let ">(/>) denote the number of solutions of the congruence FW=0(mod?), where p>\-
(1 < * < p ) ,
Let a>(«) = JJ ^(p), where n is a square free number tin
and its prime divisor is always greater than \. Lemma 1. According to Nagellw RGO = 2
*
SM-
2
K
^^^-logf = log^ + Oil), P
^-~logJog* + p, + o(.J^), Mogx/
P
*
P /
loga;
Mog2*/
where pF, f*-F and the constants implied by the symbol "O" depend only on F(x). Lemma 2 . / / n is a square jree number and its prime divisor is always greater than ^ and if &>(«)#0 and / ( « ) = J J /(/>)> where
2 P\»^t>k
^
= «flog* + o(log*),
93
1609
where «* = H
IJ 7
1
(
«#
V TV
symbol "O" depends only on
F(x\
Proof. Since «*(/>) ^ ^ and n is a square free number, therefore OJ(«) <^CCn)==O(«e)j where £ ( « ) denotes the number of prime divisors of ft, e is any given positive number and the constant implied by the symbol " O " depends only on e and F(x). Since the series V1,
—, A / is absolutely convergent for s>0, hence we have
PI" ^ p>k
fl--^ v
p ;
On the other hand, let a(s) =
^J p\n
for j>0.
"^TT"'
^ p>*
Hence « ( » = a F log ar(l + 0(1)).
(Cf. [10]).
(I-A) 1) It follows by Mertens' Theorem that =
p
Ttlcn
TI
-.
n (»-})
7-^rr = lim -~
pr • Hence af = (_er ^ F ) ~ l , where T denotes Euler's constant.
jr-^-
=
94 1610
Lemma 3. Let l < a < | 3 be two given numbers.
Then
where the constant implied by the symbol UO" depends only on F(x). Proof. It follows from Lemma 1 that R(») = log* + ?•„,
rn =
O(l).
Hence ^ l
*
^
=
2!=
>p
u>{p) _
l
^
logn-log(«-l)+
E
r
^
- ~ r - ; = Z i + Z»;
—-+o(*^) =
= f^
^
+ O (x'hllog * = J2S*-) = V
J ^logdog^ =
R(n) — R(n - 1) _
I logArJ a -i
Vlogx
h O (,*• P J = 2
inlog 2 /!/
*
+ i ;
log-1log* «— 1
Vlog2^/
hO(i:
p
;,
Vlog2*/
Thus we have the Lemma. 3.
Let Si= 2
\-i"\ {l^i^m), m
exceeding \.
THEOREM A
w h e r e q1} ••-,<]*
Let X = I I <#+1 and
are a l l primes not
95
1611 K a , - , •••Jaf«{f.)
a, K;
(« = 1, 2, • • • ) ,
(a))
be a sequence of integers satisfying aih is= aiJ2 (mod p,-) (ji^j2), where pi
(1 < i < r)
has an unique solution in the interval K n < p ! ••• pr. Denote this solution by «,-. Hence P»(x,g) is equal to the number of integers satisfying the following conditions 1 = ^ « = ^ * , « = a(mod K) , (n — ai) • • • (n — o^) ^ 0 (mod pi)
(4)
(K^
Theorem A.
Let c>0, P~±i
p-(*, e) <
holds uniformly
y
pi. 2/
N
g\rt)
Pi"
square free number
«(») = 11 <*(p)*0p\n
+ o(^iog6*f)
in (<«>), where g ( l ) = l ; g(p) = —~—for «>(p)*eO and
p>\; and £(«) = II i(p)> fW^-rr
Proof.
Then the estimation
and its prime
II (l~g(p)) f°r
n
being a
P\«
divisor all greater than \,
Let
(or, » ) = 1
with
96 1612 w h e r e n\P
and S =*J "^. ~77~T~ • ~, 1\.n)
Then
n\P
p-(*,o = =
where {dud2}
E
K
"<*
JLJ
2_i
<
<<2|P
^h2
I
E
*
rfi(n-ai)—(ii-a.)
I
1 =
2_j Wi, <s)|(»—«i)—(»— o^)
— * V
V
— T7
ZJ
ZJ
E f
n<*
1 3 A
'IA',
U)({^i, ^2}) , r ,
7T
H
denotes the least common multiple of dx and d2.
=
E d|P
Since
for »IP, hence
*
r\T
/wi l
E <(|n|P
^00 r. '
97 1613
Then
ll«r
; y vl«r n (i - })J j
= o (log2* e • ( S ^ T ) = o (log2** • ( £ < * « ^ )2) =
where ^(n) denotes the number of divisors of n. Thus we have the Theorem. 4.
THE ESTIMATION OF THE UPPER BOUND OF P»(X, f )
Let £>K. Let / < c < / + l, where / is a positive integer.
+ c- D'
i
y
f<,.rU/oo---/(p<'>)
v
M2(^)
^., ( / ) /(»)•
J_
1°.
Take € = T~1T log* x
v «(P)#0
ior
-L- - y »(P)#0
M
K^<2-
Denote 2c = d.
Since
^> = v
">2(^)
(±\
»(P)#0
therefore from Lemma 2, we have
"•I*1
t».K)=l
"(P)#0
(m,K)=I
=< (2^ — 1 — clog<:)«Flogf + o(logf).
= o
Then
98 1614 Hence it follows from Theorem A that P-Gr, **) < P u (x, -^—) < 6
+ aFK [d — 1 — — log — logS V 2 2/
+ c (-?-) = A(d) -&*- + o UL.Y \ log xJ
K log x
(5)
V log x'
where the constants implied by the symbol uo" depend only on F(x)> and A{d) =
d-i-AiogA
2
er (2 < d < 4).
(6)
2
I
2°.
x*
Take f = -—rrfor 2<£-<3. Denote 2c — d. It follows from 5 log **
Theorem A that for any given e > 0, there exists a positive constant .r0 depending only on s and F(x) such that (5) holds for x>xe and ^
A(d)= J
1
* 1
d
d — 1 — — log 2 2
,T ( 4 < K 6 ) ,
-— .
s
( d
(7>
\
h8I— — e \ 2 /
where *(0 = Km
E
^ ^ 4 ^ log ^1/log?
( c >2).
(8>
"Cpp')#O
By the same way, we have the expressions of A(d) for d>6. 5.
THE ESTIMATION OF THE LOWER BOUND OF Pa(x,
?)
For simplicity, we denote PM;pi,
• ••,Ps)=
S
l
l
> ?"(^) = ff=
2
!»
1<*<«aimod d)
where aijx^aiu (modp;) (/i^/z), k
99 1615
Theorem B. The estimation ?FX + o (
—
6 5
x
\
Mog2*/
P«{x, x - ) > 6.4524 Kiozx
holds uniformly in («), where the constant implied by the symbol "O" depends only on F(x). Before the proof, let us state the well-known lemma. Lemma 4 . Let r > rx > • • • > rn > 1 be a given set of integers. Then PM(*, f) >^E - R, where ^
1
y
;
"
-+-
P"
y> ,
/
" • " ~r
i
<-
• •
P°Pp
2 »+ 1
2-j 2-j 2-i 2-i'''
«< r ^
2-i
"<,•„
papB-'-pu.
12 = (l + ^r)(l + ^ r 1 ) 2 - - - ( l + ^ r j 2 . The proof of Theorem B. Let e > 0 be a given sufficiently small number.
Let h = —~ + e. Then there exists ^) depending only
on e and F(#) such that 2
»&- < log (A + e) < 0.452 = r ,
TT ( l - JUizlY1
<
1.572 = jl,
for 5>5 0 . Let pr—pr^ be the greatest prime not exceeding x65. If 2 < / M ^ i
. * + l, then -p,m denotes the greatest prime not exceeding **-5 *m~l, where pr,+l
has the property that P*l+1<8o
Let n be an integer such
100 1616
that 2n>2t + rt+1. {11])
Let pTm = pTf^t
+ 1 ^ m <«).
Then we have (cf.
PM, g) > (i - i : A "T* + H T] -) - n (i - ^) + V
^
(2» + 4) !
35
+ O (*
+
o
p.
/
M>
> (1-0.0073193)— > 6-4524^ ^riogx
/ X ,'J, V
IT
fl - J ^ " N ) + O f-^-^ >
/_*_\ \io g 2 x/'
Thus we have the theorem. 6.
Two THEOREMS
Let A(a) and ^1(«) (0 < « < 6.5) be two non-negative and nondecreasing functions such that
A(«) -BE*- + o (_L_) < p . (,, J ) < A(«) - f £ - + o (_iL_) (9) K log *
\ log x/
K log ar
\ log xf
holds uniformly in (w) and a, where the constants implied by the symbol cto" depend only on F(x). For the existence of A (a) and A (a), see §§4—5. Theorem Cx. Let A(a) and A(a) (0
f A(/3) - f"1 AGO — ,
r
and Aj(a) = A(JB) - ( ^ A(«) — , ^flftf /^tf ^»«
a O < 6.5
as those of the functions
A(a) and A(a)
101 1617 I
65
Proof. Take x such that x ' >\.
Now, we estimate the lower
ia
i
i
bound of P»(x, x ). Arrange the primes between x" and x" as follows: I i P, < X5 < p,+i < " " • < pr < x" < p,+l.
Then Pa(x, xJ) =Pu,(x, p,) and Pu(x, x~°)=P»(x, pr). i
i
If ;*r , + i<* a , then the difference between Pa(x, pf) and P u (*, /7J+i) is equal to the number of integers n satisfying one of the following conditions: 1 < n ^ x, n = a(mod K), n ^ a,-;(mod p,) (1 ^ i ^ s, 1 ^ ; ^ w(p,)), » — «/+i; (mod p i+1 )
( 1 < / ^ w(p J+ i)) .
For l ^ i ^ j , \^K
K7<w(p / + i), each of the following
™ps+i + a,+\j = a (mod K) rap^+i + as+li = a,7(mod p,)
has an unique solution. spectively. Let
(10)
(1 ^ m <J K) ( 1 ^ m ^ f>,)
Denote these solutions by asi and aljit re-
asj, K; asjil{i. < i < j 3 1 < ; < o>Gvn), 1 < / < £<>(?,•)), a,+i;(l < / < <»(p,+l)).
(a>f/)
Then it is evident that the conditions (10) are equivalent to the following conditions: 1 < m < * ~ a'+li ,
m
= asi(mod K) , m H S,,;/(mod pi)
Ps+i
( K * < r , l < ) < « ( ? r t i ) , 1
2
"^y I" x
/=i
=•/»-(*, p,) - 2 P-(x, Xs) = PB(», x?) - 2
P
2
P
^ S P*) = P,+i
-,/(-^-»P0
'
+ o(i),
P - . , ( — , p.) + O (xs).
(11)
102 1618 Let n = \-\/log x ], u, = a H Lemma 3 that
TI=
— / (0 < / < « ) .
Then
it follows by
n
S
2>-.,(—^ '°g n+t
<
S'-,(T--(T L -)
s I
<
+
<*'"'^
1
£ -i-
+O
-L
A(«m - i) *^£^- + o (x K^log—
I
S -L-
^^-\< -L ?log —i
< 4(«1+1 - 1) - ^ l o g ' i - ^ i + o (_L_log!Ji±Llli) + O ( - ^ - ) . Klog.v Mogx «/ — 1 / Mog^x/ J(/ — 1 Since
«/ — 1
TTa
Ja
~1
w
= g f"'+1"1 (A(»m - 1) - AGO) ^ < < 2{A(«,+1 - 1) - A(«, - l)}log^±i^i < /=o
<
max
«/ ~ 1
log "'+1 ~ 1 g U K . - 1) - A(«, - 1)) =
-°(-r=)« Vlog* y therefore
P.(*, O > P-(*a xh - S 7/ + OO") ^ > A O ) r ^ _ + O(_L_) _ g A(«/+1- l)log?^±^ •
103
1619
. _i!£fi_ + o (_fl_ V i^imziS] Wog* , = 0
Klogx
\
J «-l
+ o (-?!-) ^ Mog 2 */
«/ — 1 /
i< / log X
V log XV
Define AQS) - f " A(«) -^-, if AO) - T 1 A(«) - ^ > 0; J"—1
«
o, if AO) - r
i
Jo—I
tl
A ( « ) — < o.
J o—1
U
Hence Aj(a) satisfies the left hand of the inequality (9). The other hand of the inequality (9) may be proved similarly. Theorem C2. Let l < a < | 3 ^ 6 . 5 be two given numbers. the inequality
i i i ^
p
'VpJ
/
VJ—i
V* + 1/ * /Klogx
Then
VlogJ
/^o/«/y uniformly in (api), where A(z) satisfies (9) and the constant implied by the symbol "o" depends only on F(x). Proof. Let « = [ v V T * ] , «, = a + ^ i : : £ L / - l ( 0 < / ^ » ) .
tog*
,"'+I+1
\« / + 1 + 1/K log a:
u,
Mog2*-/
\log*
M/ /
Then
104 1620 Hence
i
i /- t
;_0
KlogX
<([
H
\J.-i
A
'
/
^P
\z< ;+1 + 1 /
1=0
MOg2*/
«;
(Mii)j-_
\« + 1/ z/Klogx
+ o
MogX , = 0
K,/
fM
\logxJ
Thus we have the Theorem. 7.
THE SET 3Jt AND SOME OF ITS PROPERTIES
Let 1 < a < ^ ^ 6.5 be two given numbers. of integers satisfying the following conditions 1 ^ n ^ x, n = a(mod K),
Let 3DT denote the set
n ^ a, ; (mod /»,-) (1 ^ «! ^ s, 1 ^ / ^ o>(p,)),
« ^ a J+lV (mod ?J+I-,-) (1 < i < r — j , 1 < / < w(pt+i)) i
where pt^x"
i A
p,<=x"
(12)
The number of elements of 3Jt is
denoted by M(x, x", x"). The aim of the present section is to prove the following Theorem D. Let A(«) and A(a) ( 0 < a < 6 . 5) be two functions satisfying (9). Then the number of elements of 9JI satisfying at most m of the following congruences n =flj+.vCmodpJ+i)
( K ; < w(^ + ,))
(13),-
w not less than V ^
OT+1
J«-l
\«+l/
2T / X log*
\l0gxZ
where the constant implied by the symbol uo" depends only on F(x). The proof of the Theorem depends on the following two Lemmas.
Lemma 5. P u ( * , * ' ) = M ( * , * ' , * " ) + O(xl~") + OCx'X
105
1621
Proof. P^x,x')
-M{x,x",x")<
2
Yi
TJ l
= o ( 2 -2) + o (s
I) = o (x -h
1==
+ o (A
u
«>x"
n^x
Lemma 6. There exist sequences of integers (w,-;) such that the number of elements of 3JI satisfying at least m congruences of {13) is at most , _ , »u>, +1 o
-! V m
<• = 1
**^
V p f— i = 1
V
Pi+r'
'
Proof. For l < i < « - i , let T; be the subset of 3K whose elements satisfy the congruences (13),-. Denote the solutions of the congruences ps+;m + as+n = a (mod K)
( l ^ m ^ K)
and pi+,m + as+il = aw(mod pa)
(1 < m < pa)
by 5,;(1 < i < / - s, 1 < / < <»(ps+!)) and 5,^(1 < / < ^ - s, l^y<w(/7 B )) respectively. Suppose we have 2,7, K; a,-^,
(l<«
w(pB)).
l^u<sr (w,/)
Evidently aimji H S,-Wl,(mod f>a)
for aap^aUVt
(1 < M < r)
(mod/7B). Hence the number of elements of V{ is at most
If the element of SOt satisfies at least m congruences of (13),-, then it belongs to at least m different sets T}. Hence the number of elements of 2R satisfying at least m congruences of (13),- does not exceed
The proof of Theorem D. It follows by Lemmas 5 and 6 and Theorem C2 that the number of elements of 3ER satisfying at most m congruences of (13),- is not less than
106 1622 1 1
1
U< " ^
x" *"") —
Mix
V
; >
V
/
P
x
r
I—
\
x" 1 =
= (AW - - J — f""1 A {-*!-) **-) -1&L- + o (-iL.). V
OT+
1 JK-1
\ « + 1/ z / j ^ l o g X
Vlogx/
Thus we have the Theorem. 8.
THE PROOF OF THE MAIN THEOREMS
Let A(a) and A (a.) (0
^£££_ + o (^LJ) ^ P^Xy J) ^ Klog.r
\\o%x)
_EI^ +
A(a)
K log A:
o
(_f_) Vlogar/
holds uniformly in (u») and «, where the constants implied by the symbol "o" depend only on F(ar). For the existence of A(a) and vl(«) see §§3—5. By the method given in § 3, we have the following table: d
5.5
5.2
/100
5.65
5.372
d
4
A(jT)
4.42
3.8
5
3.6
5.197 3.4
3.2
4.8
4.4
4.2
5.036
4.7085
4.56
3
2.8
2.6
2.5
4.288 4.161 4.049 3.941 3.848 3.76 3.6834 3.65
0<<^<2
. (H)
3.564
Then it follows from Theorem B and Theorem Q that A(5) > 6.4524 - f"il(*) — > > 6.4524 - A(5.5) ["-^L J5.2
2
A(4.2) T' 2 — > 4.8396. J4
0
(i) For ^ > 5 , let flj,', •••,«„(,.),- denote the solutions of the congruences F (y) s= 0(mod pf) (0 < y < pj)° 1) For the sake of convenience, suppose the congruences F(x) = 0 (mod p) and F^x") = 0 •(mod p) have no common solution.
107 1623
respectively, where k. < Pi < P2 < *' ' are all the primes greater than \. Let bP be the integer satisfying (0 < y <
F (y) ^ 0 (mod
anc
—T
^ ^ ^ ' * ' <«7»>^^ a r e a ^ t n e primes not exceed-
ing ^ . Denote the solution of the system of congruences y = bi (mod s#+1) ,
( 1 < *<»»)
by #. Let «, X; «„• (« = 1, 2, • • •; 1 < j < o)( ? ,)).
(5)
Then 9Jt is the set of integers satisfying the following conditions K n < x, Fin) ^ 0(mod qi) ( K i < m) , Fin) ^ 0(mod pi) (1 < * < s) , F(n) ^ 0(mod p\) (s < i < *). (15) If «/ is the least integer satisfying (2), then from the table (14), we have
S.4.S3P6
\-V A(-*1_\*Lw+ 1 J | \« + 1/ «
- yi(2) P
^ > 4.8396 - — L - (A(4) f - ^ +
5K-W
+ A(3.8) f3'16-^ + A(3.6) ( " *L + A(3.4) f " ^ + J2.57 i(
J2.12 «
Jl.77 Z
5
+ A(3.2) P *L + A(3) P- i l + A(2.8) f* *L 4Jl.5
«
Jl.27 if
Jl.08 J2
+ A(2.6) fU08 * . + A(2.5) f ^ - 1 - ^ 1 _ f^ Ji > 4.8396 -
«
J2 « J
5 64527
' 3.65 ^ 5 ^ - ^ ">+ 1 w+ 1 w+5
For x sufficiently large, | ^ - + o ( - ^ ) > ^ K log *
\ log x'
*. >
w+ 1 j5«±U_i z = 2)5
L
.
>
0
Then it follows
log *
from Theorem D that for AT sufficiently large, there are not less than integers y in the interval K J I < J ; such that F(y) has no prime
lo
1)
The existence of b; is duetothe fact that -F(j>) has no fixed prime divisor.
108
1624 X
X
divisor < xr and has at most w prime divisors in the interval x*< y < #5(*+l). Hence F(y) is a product of at most n — \-Yw primes. Thus we have Theorem 1 for the case \ > 5. In particular, take F(y)=xk + y (Ky<x). We obtain Theorem 2. (ii) From the table (14), we have
«5)-iK-fi)^>°' ««-T4<-TI)^>° and 5 JJS
\z + V z
Hence it follows from Theorem D that we have Theorem 3 and also Theorems 1 and 2 for the case ^ < 5 . Remar\. Making use of Theorem 2, but with more complicated numerical calculations (cf. [12]), we may refine table (14). REFERENCES [ 1]
Wang Yuan, 1957 On sieve methods and some of their applications, Science Record, 1:3, 1—5. [ 2 ] Brun, V. 1920 Le crible d'Eratosthene et le theoreme de Goldbach, Vid.Selsk,. Sly. 1. M.N.KL., 3, 1—36. [ 3 ] Rademacher, H. 1924 Beitrage zur Viggo Brunschen Methode in der Zahlen-theorie, Abh. Math. Hamb. Univ., 3, 12—30. [ 4 ] Ricci, G. 1933 Ricerche aritmetiche sui polinomi, Rend. Circ. Mat. Palermo, 57, 433—475. [ 5 ] Ricci, G. 1937 Su la congettura di Goldbach e la constante di Schnirelmann, Annali della R. Scuola Normale Superiore di Pisa, (2) 6, 70—115. [ 6 ] Kuhn, P. 1942 Zur Viggo Brunschen Siebmethode I, Nors\e Vid. Sels^. Fork. Trondheim, 39, 145—148. [ 7 ] Kuhn, P. 1953 Neue Abschatzungen auf Grund der Viggo Brunschen Siebmethode, Tolfte s\andinavisl(a M.atemati\er-}{pngressen, 160—168. [ 8 ] JIHHHHK, KD. B. 1950 3aMeqaHHe o npoH3Be,neHHH Tpex npocTbix «mceji, RAH CCGP, 72, 9—10. [ 9 ] Nagell, T. 1921 Generalisation d'un theoreme de Tchbycheff, Journ. de Math. ( 8 ) 4, 343—356. [10] Widder, D. V. The Laplace Transform, Princeton Press. [11] Wang Yuan, 1956 On the representation of large even integer as a sum of a prime and a product of at most 4 primes, Acta Math. Sinica, 6:4, 565—582. [12] Wang Yuan, 1959 On sieve method and some of their applications ( I ) , Acta Math. Sinica, 8:3, 413—429.
109
SCIENTIA SINICA Vol. X, No. 1, 1961
MATHEMATICS
ON THE LEAST PRIMITIVE ROOT OF A PRIME* WANG YUAN (3E
%)
(Institute of Mathematics, Academia Sinica") I.
INTRODUCTION
Let p be a prime number15 and g(p) denote the least positive primitive root of p. Let «(«) denote the number of different prime factors of n. BHHorpa^OB111 first proved that g(p) <2'V y 2 log ? ,
(1)
where m — <»(p — 1). The historical records of the estimations of g(p) £re as follows: g{p) < 2mp1/2 log log p of BuHorpaAOB121; g(p) < 2a+W~po£ Hua[31; g{p) = O(pU2 log17p) of Erdos1^ and g(p) =0(m'pm) 151 of Erdos and Shapiro , where c is an absolute constant. On the assumption of the grand Riemann hypothesis, Ankeny[yI proved that gCp) = O (2- log2 p log2 (2'" log2?)). (2) As to the lower estimation of g(p), Turan161 has proved that *O0 =Q(losp).
(3)
In this paper, we give in detail the proofs of the following two results (cf. [8]). Theorem 1. For any given positive number e, we have * t o = O (*."""),
(4)
where the constant implied by the symbol " 0 " depends on e only. Here the main order 1/2 is now replaced by 1/4+ s. Theorem 2. On the assumption of the grand TZiemann hypothesis, we have g(p) = oCm6 log2 p). (5) * First published in Chinese in Ada Mathematica Sinica, Vol. IX, No. 4, 1959. 1) In this paper, p, q, pi, pi, ••• denote primes.
no 2
Hence, the upper bound on (5) cannot be improved beyond O(m6 log p). I am indebted to Professor Hua Loo-keng for his valuable suggestions and assistance. II.
CHARACTER SUM
(I)
The aim of this section is to prove the following Theorem A. Let 8 < — be a positive number. exists a positive constant P(8), such that
Then there
2 zoo <-, holds for all p> P(S),
(6)
H> p1M+* and any integer
and X is a nonprincipal character modulo
N, where rj = — 6
p.
The proof of Theorem A depends on the following three Lemmas. Lemma l. [iU01 Let [p] denote the finite field of p elements. Let fi(x), ••-, fn{x) be different normalized^ polynomials, each irreducible over [p]. Let f^1} — , \n denote the degrees of these polynomials, and let K = \i + • • • + \n. Let X1} • • • ,Xn be nonprincipal characters of [pi with the convention #(0) = 0. Then
£ ZxC/iGO) • • • Z.O.GO) < ik - D A / 7 , x
where the summation is over a complete set of residues (mod p). Lemma 2. Let r be a positive integer, and h be a positive integer satisfying \
S&*) = 2 *o + «). m=l
Then 2 |SAO)|2' < O-ry-ph' + 2r
where X is a nonprincipal character modulo
p.
1) A normalized polynomial is one in which the leading coefficient is 1.
m 3
Proof. If X is a character of order d{> 1), then
si5Awi2'= 2 ••• 2 2 - i 2 « ( » + ».)--(* + %)) *
(7^=1
mr=l
» =1
»r=l
*
*((* + « > • • ( * + «r)). Divide the set of integers {m : , • • •, mr, «1} • • • , « , ) ( 1 ^ ? M , - < A , 1=S tij^h, K ; ^ r ) into two classes <7X and
each of {«iJ+1, — ? «;,)
^ ^w/^+i> — ? ^/V^ consists of at most ——•—
distinct integers, each occurring dl times ( / > 1 ) , where (i1} ••-,;',.) and (/i> —> fr) are permutations of ( 1 , 2 , • • • , ? • ) . The other sets are in <s"2- Thus 2
\SA(.X-)\2r= 2
2
+ 2 2
Z ( ( x + « 1 ) - - - ( * + « r ) ) ^ ( ( * + »l)---(*+»r)) +
x
-1
^((* + «!)•••(* + ™r))*((* + »i) •••(* + »r)) =
Since the number of elements of ^ does not exceed 2
r-inC.A'
'-«
l
"
J
2
l)
/it-H-r^]
<
;
\
»= i
/
= 4r6' 2 A'A l [ l 7 i ] <(2r) 7 'A', therefore
2j<(2r)V.
i = i
\
h —
^-) < 1
/
m 4
Also, we have
where Kl,
< h2'(2r — l ) \ / y < 2rA2rVT. Thus we have the Lemma. For any of the integers H > 0, q > 0, t and N, we define the interval l(q, t) as consisting of all integers z satisfying N+tp ?
<x
+ H+,p
(7)
1
Lemma 3. t n l Let q run through a set of distinct positive integers, Q in number, all satisfying qx
and all being relatively prime in pairs. Suppose that 2Hqt < p. Then (for given p, N, H) it is possible to associate with each q a set T(q) of integers t, with 0 ^ t < q, their number being q - Q, in such a way that the intervals l(q, t), for all q and all t in T(q), are disjoint. Proof of Theorem A. It follows by Polya's Theorem1121 that we may assume pyW
Suppose that
<
H
< pm+a^
(8)
113 5 for some N and H satisfying (8); we deduce a contradiction if p > For q < p, we have N+H
4-1
2 *(») = 2
»=AT+ I
r=0
N+H
1-1
2 zoo = 2 2 zoo.
n=N+ 1
t ~ 0 r^f(
Hence
2
2 z« >^.
C9)
Now we apply Lemma 3, taking the set of q to consist of all the primes in the interval (10)
pv<-£l
It follows by Ingham's Theorem"31 that the number of q is Q
= ^ 4 ) - JP™ - -4*) = i £ ^ l (i + oil)). \
Hence
2
2
p7 /
(ii)
log p
2 ZGr) >^-Ji2HQq-l>
where the summation is over all primes in the interval of (10). We rewrite the result as
2
2*0) >mz.
Since the number of I(q, t) is at most pwQ and
2 2 Z(* + ») = A 2 Z(») + 2 c^OT,
» 6 l m=l
r£f
BI=1
where |^ m ] <2m, therefore
2 2lS*(»)|>^-2P"W. Let
L 8p' J
C12)
114 6
Then
Since the number of integers in l{q,t) it follows by Holder's inequality that
is at most 3 p~u*H, therefore
Since the intervals / are disjoint, then
Combining this with the result of Lemma 2, we obtain (——YHQA 21 - < p(2ryrhr + 2rKj~Phlr.
Take r = [y] + l. Then
and
(i^r H Q k i r < 3r^h2r
(i3)
for p>P2{S) > Pt. v^n the other hand, it follows by (8), (11) and (14) that (13) is impossible for p>P3(S) > P2, in virtue of (l 4 ) S - 2rV - 2q = 8 - 2 (T—1 + 2 ^ > < 5 - i - < S - < S 2 > A VL5J/6 6 6 Thus we get a contradiction and this proves the theorem. III.
CHARACTER SUM (II)
The purpose of this section is to prove the following Theorem B. Let X be a nonprincipal character modulo p. Then, on the assumption of the grand Riemann hypothesis, we have 2 AGOZGO/^ = O(*" log p),
(15)
U5
7
where A-iri) denotes the von Mangoldt function. Proof. We need to mention the following well-known facts: (i) There exists Tm such that m < Tm < m + 1
and
y-(tr + iTm,X) = O ( l o g 2 ? ( | r J +1)) (l/3<
If l/3<<7<2 and \t\>\,
then
r « = o(*"»'" |r|- w ). (iii) have
On the assumption of the grand Riemann hypothesis, we y- (1/3 •+ it, X) = O(log p(|*| + 1)).
Hence, we have
r + '"rto*'— (*,*)* = o(x»f" ^ % 2 ^ ) = o(l) (asm->oo),
J
2+.T m
V
I.
J
rra
/
r
2 A(«)z(«)tf - = - -I__
p
1
= 2 * ro) + -^
f 1/3+ i oo
r w * ^ (,, z) ^ = 7
j
*rw ^ a, z) & =
= 2 * T < » +O(rv3log?), p
where p runs over all the nontrivial zeros of L(s, X). Since the
H6 8
number of zeros of L(s, X) satisfying n ^ y = J?(p) ^ n -t- 1 is O ( l o g p ( | « | + 1 ) ) , hence
2 xT(p) = O ( ^ 2
2
ITC1/2 + ,y) l) =
e^" Iog ? (n + 1)) = O(* w logp).
= o(xV2'Z Thus we have the theorem. IV.
SIEVE METHOD OF BRUN
If Q is a positive integer and (Q, px • • • />,) = 1 , then let us denote by P(Q> Pi> ' ' ' > Pt) t n e s u m °£ nonnegative numbers an, where n satisfies the following conditions: n > 1, ind n ^ 0 (mod p.)
(1 < * < s) , ind n = 0 (mod £))°.
(16)
Especially P ( T ) denotes the sum of ao, where n satisfies « > 1 , ind» = 0(mod T ) .
(17)
It is evident that P(Q'y pi, • • •, p,+i) is equal to the difference between P(Q> Pi> " ' > P') a n < i t n e s u m °f an, where n satisfies n ^ 1, ind n qfc 0 (mod fO (1 < * < s) , ind n = 0 (mod ^>) , ind » = 0 (modf>,+i).
(18)
Since the conditions ind n EEE 0 (mod ^ ) and ind n = 0 (mod pt+i) are equivalent to the condition ind n = 0 (mod Qps+i) j therefore we have P ( 0 ; ? i , • • • , px+i) = P(.Q;
pi,
••,
pi) — P(.Qp,+i;pi,
• • •,
pi).
1) If jr is a primitive root of p, then for any integer n in the interval 1 ^ n < p there exists a positive number a such that g« = B(mod p ) . We write a = indf n or briefly a = ind n.
117 9
Using this formula successively for r times, we get r
•••» pr) = P ( e ) - 2
P(Q;pi,
P(QP«;PU
•••,-?.-!).
(19)
Let r = r0 ^ rx ^ • • • > r, be any given sequence of t positive integers. that
Then we have by (19)
r(0; ft,•••,*) > P ( 0 ) - 2 P(ep-) + + 2
2 PiQp'p',;
pi, •••, p-x-i).
(20)
Using (20) successively for t times and observing that pCQp.p^,
• • •, p . , p p , ; p i , • • • , pt>,-D
< PCQp.,
•••,pfs,'),
we get PC.Q; pi, • • • > pr) > P(Q) - 2 P(ep-) + 2 2
- +
2 2
P(.QP«P^
2 + 2 - - - 2 PtQp.p.x, ••-, Pl>x
«<<• «l
-
(21)
If there exist M and N such that p(r)=^+oGv),
(22)
where the constant implied by the symbol "O" is an absolute constant, then we have by (21) and (22) that PCQ;Pi, ••-,Pr') >~--KR\N\
(K>0),
where
* =i-2|+22?V- + -"o>«!
- 2o
«>a!>...>p.
r r<
a
H8 10
and R
«<' °l
2
2-2K
"*ir a 1
0f
< r IT (1 + rd\ i=l
By the suitable choice of r, ru • • •, rt[W, we have
Lemma ^4. / / (22) holds, then there exists a positive constant & such that (SM
P(Q,;PI,
r
.
••-,Pr')>~Q"[\ U-—)+o(rf-*|2v|). ,=l \
V.
Let x=[p1M+t}.
p;/
PROOF OF THEOREM 1
Then we have by Theorem A that
!, z w -°(7) (f = 7>
(23
>
where X is a nonprincipal character (mod p). If q\p — \, then let us denote by Xq the character of order q for modulo p. We have by (23) that
2
l=72 2W=7+o(4
(24)
q
i»d n=0(mod 9)
Let q\P-\
q\P-l
Next, let N(x) denote the number of primitive roots not exceeding x. Then
*(«)=
2 1> 2 «<*
U<JT
(,i«d n,rt=l
(ind ». P j ) = l
I" 2 4lP—1
2 »<*
4 > l o g , ^ i»d ISO (mod 4)
1 = 2,-2,.
119 11
Hence, by (24) we have V
«IP-I
«>Iog2p
4'
X
«|p-i P ' «>log2p
Mogp/
Let fl, i £ « < x ; lO, if w > * . Then, it follows by Lemma 4 and (24) that
£,>** |T (i-i-)+o(iog^-4)> > , °* ( « > 0 , p>P 1 (e)). log log p Hence iV0O>£-^ log log p
O>0,p>P2(e)>PJ).
This proves Theorem 1. VI.
PROOF OF THEOREM 2
If d 1 p — 1, then let us denote by nd the residue modulo £, which satisfies «2 = 1 (mod />). Then
where 2 ' denotes the omission of the principal character.
Hence,
we have by Theorem B that
G-GO = 4 2 ^ ) ^ + ° OWl°g P).
(25)
120
12
By partial summations and MefibiiueB's Theorem", we have n
2
AMe~
< c*-»(e3 + l)x
Cc3 > 1) ,
2
A GO*"* < ffjcar, (c3 > 1) ,
(26)
n
(27)
and *Z A(fl)e~ > e-lcxx ( c 3 > l ) .
(28)
Let P=
IT
J] q,Px=
«IP-I
?(^4>2);
q\P-l q^Cttn log 4nj
and n n
where the summation is over all the positive primitive roots modulo p. Hence, it follows by Theorem B that N(*)=
2
= M GO -
2
A(n)e~>
(ind «» P)=l
il(*)^-
(ind»»Pi)=l
2
9IP-1 q>c.m lop 4m
2
G,(*) =
q>ctm log 4m
— HGO + O U
w
^
log p),
(29)
where MGO =
E
n=l
A(n)e~t HQO = 2 A(»)ff~. »=1
(30)
Let =\A{n)e~,
a
"
I 0,
if
+ B/; iip\n.
?
1) There exist two positive constants a and e» such that e\.x ^ S A(n) = ^(x) ^
121 13
Then, we have by (25) and Lemma 4 that Mix) > a |T ( l ~ —) Hix) + O{.{c<my-mxw log p) > f1^
log ajn
+ O (Urn)" 91 * 1 " log p)
i* > 0)
(« > 0),
(31)
where « (and j3 and y used in the sequel) is an absolute constant. On taking c3 sufficiently large, we have by (26), (27), and (28) that n
n
H W < 2 Ain)e~ + 2 A{n)e~ <2c2c3x and n
Hix)> 2 Air$e~ > e-lax. »
Take c4 = ct(c3). N
(» >
Then we have by (29) and (31) that
«*~ lg i* _ _?£_£3fL. + 0(( C 4 ^) J - 9 9 1 x ) / 2 log ? ) > log
^ + Oiianz^x^hg > ~ log c4 • log 4m
p)
OS > 0).
(32)
Write
2V(*)=
2 n »
A(«)e~^+
2 n
^ 0 0 ^ = 2 , + Z3.
" > < r s : r lo K 4<»
Then, by (26) we have 2
, < ffa*-"1"'*- (ffs log 4m + l ) * .
Take c5 = c5(c4). Then it follows by (32) that 2
>.
T£
+
oCCc^m)""**' log p) (y > 0).
log c4 • log 4/»
Take ct = c6ics) and x =c 6 m'-"log 3 ? .
(33)
122
14
Then we have
5\>o. That is, g(.P) < ^5 * log 4m.
Hence, we have Theorem 2. REFERENCES [ 1 ] [ 2 ]
Landau, E. 1927 Vorlesungen ilber Zahlentkeorie 2, 178 Leipzig. BuHorpa^OB, H. M. 1930 ZLA.H CCCP, 7—11.
[ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [10] [11] [12] [13] [14]
Hua, L. K. 1942 Bull. Amer. Math. Soc, 48, 726—730. Erdos, P. 1945 Bull. Amer. Math. Soc, 51, 131—132. Erdos, P. and Shapiro, H. N. 1957 Pacific Jour. Math., 7, 861—865. Tiiran, P. 1950 Math. Lapo\, 243—266. Ankeny, N. C. 1952 Ann. Math., 55, 65—72. Wang Yuan 1959 Sci. Record, New Ser. Ill, 5, 174—179. Weil, A. 1948 Pub. Inst. Math. Strasbourg (N. S; Nr. 2), 1—85. Davenport, H. 1939 Acta Math., 71, 99—122. Burgess, D. A. 1957 Mathematica, London, 4, 106—112. Polya, G. 1918 Nachr. Wiss. Gott., 21—29. Ingham, A. E. 1937 Quart. Jour. Math. Oxford, 8, 255—266. Ricci, G. 1937 Ann. del. R. Scu. Nor. Sup. di Pisa, 6, 70—115.
_ _ _
123
SCIENTIA SINICA Vol. XI, No. 8, 1962
MATHEMATICS
ON THE REPRESENTATION OF LARGE INTEGER AS A SUM OF A PRIME AND AN ALMOST PRIME* WANG YUAN {Z
%)
(.Institute of Mathematics, Academia Sinica)
§1 In this paper, we shall give the detailed proofs of certain results obtained upon assuming the truth of the grand Riemann hypothesis. (Cf. [1], [2]). First of all, let us state the grand Riemann hypothesis as follows: (R) The real parts of all zeros of all Dirichlet's L — functions L(s,X) are < 1/2. From (R) we derive the following131 (R*) Let (/,O = 1- Then «(*;*, 0 = 1
V
{'
where h x = \
X
i = - ^ + o(*l/2log*),
dt
J 2 log*
.
Now we state the results as follows: Theorem 1. Under the truth of (R*), every sufficiently large even integer is a sum of a prime and a product of at most 3 primes. Theorem 2. Under the truth of (R*), there exist infinitely many primes p such that p +2^ is a product of at most 3 primes, where \ is a given positive integer. Theorem 3. Under the truth of (R*), every sufficiently large odd integer N can be represented as N — p + 2P, where p is a prime number and P is an almost prime of not more than 3 prime divisors. * This paper has been published previously in Chinese in 'Acta Math. Sinica, Vol. X, No. 2, pp. 168—181, 1960, but the Appendix is added during translation.
124 1034
Theorem 4. Let Zk(x) be the number of prime pairs of the fbrm (p, p + 2/() not exceeding x. Then
z*G0 < s TT ^ 4 TT (i - T-^-TV 2 )T4- + ° ( T ^ — h* 1O* A p]2kp
— 2P>2\
P>2
(p — iyJ\og2x
\ log3 x
J
Theorems 1,2,3 improve the results which were obtained independently and simultaneously by the author1-"11 and A. H. BnHorpa,a.OBt5]. Our original results were obtained by replacing 3 by 4 in these Theorems. It is well known that if K(X; \, I) is represented by
PEE/(mod (•)
then Theorems 1, 2, 3 may be derived from the following weaker hypothesis (#**) (R**) Let X be the character mod D. Then L(s,X) has no zeros in the domain 2 In this paper, p, p', p", • • •; pu pz, • • • denote primes. §2 Lemma 1. If x>l and z > l , then
2
i ^ = ^£)log2 +
o(loglog3,).
(»,x)= 1
(Cf. [4]).
Lemma 2. Let f(k) =
P~ 1
Ky<x, then
i S r = TTT & J = 1 Tn(i + T - 1 - ^ ) log * + odogiog 3,).
2 (*,2y)=l
p>2
?roo/. Let ^ ) = IT L=1^. Then
v
i£<£± = y j^i) n (i + -±-\ *± V -&& T - i - =
,^<. /U)
W.2y)=l
. i s . 9(0
«,2y)=l
PI! ^
p - 2/
, ^ . POO ^
«,2y)=l
125 1035
"V ^C?) y » /"2CO = 7l>(q) t
=
(«. 2y)=l
(', 2«y)=l
««* =
JE (2 2 O
^
J
q
^Ilog*+O(loglog3*) =
(«.2y)=i
= J^y) 2y
yy A P +2y^
= V TT ^ 2
1_^Vog2
+
+ 0 (l0gl0g3^)
=
pCp — 2 ) /
^ TT(l+
fly P — 1 p>2\
,
!
J l o g g + O(loglog3x).
P{p- 2)/
f>2
Thus we have the Lemma. §3 Let 2 < y < x be two given integers. Let ((/>)
a, q; «,•
(1 < * < r)
be a sequence of integers satisfying the following conditions: (1)
1 ^S x, ( a , q) = 1; if p , | y , then a,- ^ 0(mod pi), otherwise «/^0(mod pi)
(,Ki
where 2 < px < • • • < p, ^ f are all primes not exceeding £ and not dividing #. Let Pm(x, q, f ) be the number of primes p satisfying the following conditions: (2)
K * J ?
S
aCmod q), p =£ a,(mod p,) ( K « < r ) .
It follows from the Chinese Remainder Theorem that the system of congruences y = a,-(mod pi) ( l ^ i ^ r ) has a unique solution in the interval 1 < y < pi • • • pr. Denote this solution by a*. Hence Pa(x, q, £) is equal to the number of primes satisfying the following conditions: C3)
P < * , P = a(mod q), p ^ «*(mod pi)
(lKKr~).
126 1036 r
Theorem A. Let c > 0; P—\]p;. (R*), the estimation H
p_(x, q, e) <
Then under the truth of
+ o(xmiogx
*
•
$*we)
Ao/^i uniformly in (w), where f(\) =
.
PI* P — 1
Proo/. Denote g(4) lows from (R*) that 2
I£
=
(^y) = l
and
^ 1 ^ then it fol-
1 = , ! ! % „ + O(^logx).
p<x
p=a(mod ^) p=a*C°»od ^)
Let
/i*1
c*.
*l? (*>y)=i
».y)=i
where d\P. Then P
P<*
(>=o(mod q) (<>-<•-, f ) = l
"V
p=a(mod 9)
V* 3 3
Wl.»)=l (i,.»)=l
VICp-o*. f) (i(, y ) = l
"V
'
1 —
(>=a»(mod / * )
"jjp 4,IP (<*i.y)=i (<*i.y)=i
+
»(^.(Siw))-^» + ». 4|f
Let
«. y)=i
127
1037
Then 3 efh^ —
1
V
A»2("») -
f^Al
1
V
f(.mlQfi(.m)
(OT, y ) = l
For (^/, y) = l, we have
•lit If
^
•'I*IJI
m<^/i
T\f»KJ
= .1 V ff^) y 1 u ^ = JL . f^H r|P C»-,y)=l
Hence ^
rf^f
i2lp
"Hi"*!.*,)
«!.y)=l W2,y)=l
(
(t,y)=i
By Merten's Theorem, we have
for
Hence
§4
;? = o(* i/2 io g * -e^iog 2 ?). Thus we have the Theorem. §4 Let £ > 2 . (»• y)=l
Let / < < : < / + 1, where / is a positive integer. p|»
(»• ««yl=I
Then
128 1038
+
...+r-iv
V P'-V"«'
(n, 2«y)=i
(p.«y)=l
V Mill.p'V'u
(n, 2pyg)=i
(4) (»,2p'—p V^)* 1
(p'—p" . «y)=i
If 3 ^ « < 6 and *1/"<«7
1°. <: =
< 1 . It follows from Lemma 2 and (4) that
^ Too" ~ .t1. Too" ^ Ji p~^ i'A BIP (B,y)=i
P > 2
(n, 29y1=I
K^=^)^
g
+ O(loglog3x) =
=^
8«
n^
n (i + - r ^ — r W * + ooogiogsx).
#|,y?— lp>2^
P>2
pip — 2)/
Hence by Theorem A we have \
(5)
< A(«)
log" x)
^^^ + O ( c°yX - l o g l o g y V
9(^)log2A;
\
log3 * /
where , ^ ~. A(«) =
8«
r
er,
u — 2 (6)
_ x / s x c 4y = (? - r | T | T ( 1 — —. r r ) , f denotes Euler's constant. tUyp — 2 p > 2 \ (/.— I ) 2 / f>2
2°. c=
If 6 < « < 1 3 and ar1/B
< 3.
Since
\g^^o(£>o(gjH(j),
129
1039 l
"v
(.n, 2<7y)=l
p-tqy
1
v if!W _ v C + «y
_of y
(»i 2p«y) = l
/(») ^
V ^ /W ^ p-f-«y
^2(») =
v
(». 29y)=i
fin
i
-y ?(»))_ O(iogx\
P+«y
(». 2«y)=l
therefore
.-S- ? i i ' w ^ ^(?) »SP Kra) ( B , 2
vs J
'
(»> 2p9y)=i
p-tqy
From Lemma 2 and (4) we have
(n, 2«y)=l
n\F (». y)=i
(», 2«y)=l
(
/'""1).2iogg + o(iogiog^).
=±(2^-i-ciogo n £ ^ n 2
Pky P
P>2
l f > ! ?VP — 2)
Hence it follows from Theorem A that (5) holds for (7)
er
A(«) = 4
2
(6<«<13).
4
Let £/ denote the root of the equation 4
2
4
Then <M(») _ yru){ >0> if 1 3 > « > U ; l < 0 , if U>u>3. du Hence A(u) is decreasing in the interval (3, U) and increasing in the interval ([/, 13). By numerical calculations, we have 7.35 < L7<7.4.
130 1040
§5 Let z/>4. 1°. I £ ? = a: 1/ 'and2<«<
*
t
2 l =
^
f
SinCC
l ^ -
"If C», y ) = l
, then we take * = f ( j - ^ ) ^ 1
2
P > 2
(», 2<jy)=l
+ O (log log*),
therefore from Theorem A we have
< (8)
n
8M
« —2
pXq3
P>2
P~
log 3 *
/
)
^
( ? — I ) 2 / 9 ( ? ) log2 x
p — 2 P>2 V
+ O f*^loglog*_\ ^
1
* yy (i -
= A (K)
£ii£
yC^iog2*
+ Q
+
/^ ggy loglog^\ V ^c^iog3^
where A1(«)=-^^^.
(9)
« — 2
2°.
If # = *
1/B
and
• J__A 2
<«<
2
J_-± oo
" * 1/ "
f / 1
then we take $ = -—T— and c = — I log3* 2 W
»!'
(". 2«y)=l
2£/L
\2
X TT ^
1 \
• — I. uJ
Since
(n, 2«y)=l
«/
2 \2
«/
2 \2
TT ^ " ^ l o g x + OOoglog*),
pl«y P "~ 1 P>2 P.KP ~~
P>2
(*<8),
l
)
»/J
r
131
1041
therefore it follows from Theorem A that (8) also holds with
(10) \ 2
«/
2\ 2
«/
2\2
«/
Since
4(«) = —>ii(«) = <*«
- —(1 - log*) (2c - 1 — clogc)2
< 0 (if
1 < c < 2)
for given 4 < v < 8, hence Aj(«) is a decreasing function for 1
u^
-1 — i_ ' 2
f
§6 Theorem B.
Under the truth of (R*), the estimation
P B (x, g, x1/13) > 25.8096e-r fj ^ ^ 1
n ri -
^i
p>i P — 2 )p\iy
y
'
+ o (XCqy \
^o/^ uniformly in (w), where q is a given integer. Lemma 3. Let r > rj > • • • > rB > 1 ^f a given set of integers. Then under the truth of (R*), the estimation FXx,q,Pr)>^--\R\
(P a .y)=l .
(Pa*p, y)=l 2B+]
s
_yyyy... v___i «
ji
••
R = O((l + r)(l + r:)2- • •(! + r.)V«logjr).
132 1042
Proof. Let P^iq; pu • • •, pr) = Pu(x, q, pr). Especially, we have P»(q) = n(x, q, a). The difference between P»(q; pu • • •, pr-\) and P»(q;pi, ~'',pr) is equal to the number of primes p satisfying the following conditions: p ^ x, p = a(mod q), p ^ a,(modp,)(l < i < r — l ) , p = a,(mod ^ r )_ It follows from the Chinese Remainder Theorem that the system of congruences (y = ar(mod pr), ly ^ a\mod q) has a unique solution a* in the interval 1 ^ a* ^ qpr- If pr\y, then (a*, qpr) —1; otherwise (a*,qpr) > 1. For the sake of brevity, we write all the (,) as (w). Hence P»(q;
p\, •••, pr-0 — P»(X; f = P^qPr',
\<1,
Pi,
py, • • •, pr) =
•• , Pr-l),
Upr^fy',
if pr\y, r
P»(.q',
Pi, • • , Pr) = P»(q)
P
— 2
-(.4P°>
PIK * * *
5
P°-i) — ^ J
O<0< 1. Using this formula r times and with the restriction j3 < rx, we have r
P«(q;
2
pi, •••, pr) > P«(q) -
+ X
E
«= 1
P«(qp°)
+
PP-I) - Kr + rn), o < 6 < l .
P»(qp°pp; pi,---,
(taPp< y ) = 1
Let r > rx > • • • > rn > 1 be a given sequence of integers. P»(qp°-
• -fv; ?i5 • • • 5 PM-I) ^
Since
Pu>(qp*---p,>),
therefore we have PXr, pi, •'•, Pr) > P«(q) -
X (i- a . y)=l
2 S^-S »>P>->)»
P
P
^IP«)
+ 2 2
F
»(.
«>P (.PaPfi' y ) = 1
"<^- • •P,)-(1 + '-)(1 + '-I)2- • -(1 + r,,)2.
133 1043
Since we assume the truth of (R*), hence r-(*;?i, • • • , P r ) > ^ -
\R\.
Thus we have the Lemma. The proof of Theorem B. Let e be a given sufficiently small positive number. Let A = — + s. Then there exists So such that 2
—7"T< log(* + s ) < 0.452 = r,
TT (1 — -4"T) ' <
h
+ s < 1.572 = A
p-f«y
for 8>80. Let pr = pri be the greatest prime not exceeding xvn.
If 2 < ^ ^ 1M X
^ +1, then we denote the greatest prime not exceeding x *~ by pr , where p,i+i is the least prime with the property that p1/^ < 80 < />f|+1 • Let « be an integer such that 2n>2t +r, +1 . Let r4 = r, + 1 (/+ K ^ < « ) . Then we have (cf. [4]) g H
P Ca; a xuus) >
* — I 7?
P(?)
where £ > (1 — 0.0073193)
> 25.8096*"' TT ^ ^
]7
(1 — — — ) >
TT (1 - •
)—
+ O (-**-),
P>2
R = O(*; 7 + l l + n * + "* 7+ "'log a :) = O(* T + l l + 1 3 ^- 1 > log AT) =
o(-^-\ Mog3*/*
Thus we have the Theorem. §7 Theorem C1# Let a} j3 be two positive numbers satisfying 8>JB>4 and J3>a>2. Then
134 1044
£
PX., P,, *>") < (-ff-i'M^l *,) * + o teiog log,),
p-tqy
where q is a given positive integer. Proof. Let
«=[log*], «,==« + ~ T ? /(0
p (v f,CT v1^) = "v
"v
p (x x^pq
y
( =o j _
zup
log I
"-1
xuii~) = V 1 T, ;=o
j_
I
1
1 p+«y
<
2
Y
A/-1^)
+ O l^f log log, log^).
*"'+1
p+«y
Since ^ ( « + o ( ) ) = /li(«) + O () and Ax{u) is a decreasing V Mogx// Mog,/ function, therefore T, < /ii(«,)
/'f, l o g ^ + O ( - ^ 5f log log, log^±i). (pC^)log2, «/ Mog , «; /
Since y . A1{UI) log ^
- ( ' ^ ^ ^« < s
\»/
(>ii(«/+i) - M « / ) ) ^ ^ i o g ^ =
Vlog,/
hence
i=i
^?>(^)- )a
This proves the Theorem.
"
/log 2 *
Mog3*
/
135 1045
Theorem C2. Let 3=^«<|3<13 be two given numbers. Then
P-(*, q, r1") > PX*, 9, ^ ) ~ - T ^ f r - T — < * «
+
"
O (£s£loglogxY
+
Mog *
/
where q is a given positive integer. Proof. It is evident that we may assume a < U < j3. We estimate the differences />„(*, q, xm) - />„(*, ^, .r1/y) and Pu(x, q, x1/u) /»„(*, q, xl"). The difference between Pa(x, q, pm) and P^(x, q, pm+1) is equal to the number of primes satisfying the following conditions: p < x, p = a(mod q), p ^= a,(mod p,)(i ^ m), p s a m+1 (mod p r o + i).
If pm+1-|'y) then by the definition we know that it is "equal to Pu,{x,qpm+Upm); otherwise it is equal to 0 or 1. Hence P » U , ^ , f m ) — P - ( « , ? , Pm+l)] . , .,
,
Arrange the primes between x1/t; and A:1/a as follows: pt < x1/u < pl+1 < • • • < p, < xya < ps+l.
Then PXx, q, xvu) = PXx, q, xl") +
P
^
»(*. P/+i» ?» ?/) + ° ( D .
7-7
Let»=[log*].
«m = a +
And put
miO^m^n). o-l m=0
«<
Since p;
and .!(«) is decreasing in the interval (a, £/), log p ;
Tra=
2
P-(*. «P/+i,
i
< ^(«m)
ft)=
l
y 9w)l°8T*
2 l
log%ii «m
+o ^
_i_
P.(^r, ??,+i, ^r 1 "^) <
I o g Iogrlog^)i Mog3x «m /
136 1046 Since
2 A(«m) log-?** - T ^ f l du = O (-J-Y therefore r
\ q>(q)Ja
'-du\—-+ol-^-log "
log x).
\\og3x
Ao^x
1
Hence P^x, q, *^) < P^x, q, x"°) + (-^["MHld,,)* \
ti
/log 2 ^
it
/log 2 x
+
+ O (-^LloglogA Mog x
I
Similarly, we have V cp(q))u
+ o(^f
\log
A;
log log x). /
Thus we have the Theorem. §8
Let 4 O < 8 , Ku^v be two given numbers. Let 9H denote the set of primes satisfying the following conditions: (11)
p < x, p = a(mod q),p^
a,(mod pi)(i < y ) ,
where pt<^xv"
p = «, + ,(modp, + ,)(l
is at most
i- 2
*S<>, *fc+/» *V').
137 1047
Proof. Let F ; be the subset of SDT whose elements satisfy the congruence P = a!+i(mod
p!+i).
Now, we estimate the number of elements of F,-. If ps+i\y, number elements of F,- is equal to 0 or 1. Assume ps+!^y. the solution of the system of congruences
then the Denote
(n = af+,-(mod ps+j) , <•» = a(mod q)
by as+j.
Let
Pa.(x,qps+hxv°).
Then the number of elements of F, is not more than
If the element of 301 satisfies at least / congruences of (12), then it belongs to at least / different sets F,-. Hence the number of elements of 3H satisfying at least / congruences of (12) does not exceed
«
}
Thus we have the Lemma. Theorem D . The number of elements m congruences of (12) is not less than
p,(x, q, *-) - -LJVAM**^
+ o Us*.bg log,).
/"* Jf /(J3(,^)log i x
m + 1 \J«
Proof.
of 9H satisfying at most
Mog**
/
Since
i^u^», 9}
X
)
Mw^X,
X
, X
) —
/
i
/
i
1
==a
»<«-* p=o,+y(modpj+/.)
< E (4- +1) = o(*1/a) + o U~h,
therefore it follows from Lemma 4 and Theorem Cx that the number of elements of 9K satisfying at most m congruences of (12) is not less than M-(*, xl", * 1/8 )
i—• X ) P»y(*5 *PH./, x1") + 0 ( 1 ) > "> + 1 ; « _,
138 1048
> p.ix, q,*-) - -!-(['AM**-)
y
m + 1 \J«
# /
+ o (£*£ log iog,). Mog3x
/
Thus we have the Theorem. §9 It follows from Theorem B and Theorem C2 that P B (*, g, x1'6) > (25.8096 - f" 4L«L i«) V
J6
+ O ( ^ 3f log log,) > 8.4 Mog *
(i)
g
«>*
9 ( 4 ) log 2 *
/
^
/
M
+
+ O ( ^ f3l o g log*). Mog *
/
Let x = y be an even integer. Let
(a)!)
a = 1 , j = 2 ; a,- = x ( » = 1 , 2 , - • • ) .
From (9) we know that there exists a positive constant x% such that
P^x, 2, x") - l ( f AW ^ W2 + o (^£3 l o g log,) > 3 Vb
«
/log *
Vlog x
/
> (8.4 - 6.588) ^--2 + O f - ^ - l o g log*) > ^ 2^ > 1 log * Mog * / log * for x>xx. Hence it follows from Theorem D that for x>xx there exists a prime p such that \
Let x = y be an odd integer. Let
(wz)
a = x — 2 , q = 4 ; ai = * ( « = 1 , 2 , • • • ) .
From (9) we know that there exists a positive constant x2 such that
P^x, 4, *"') - iff'ilW A\_£**_ + 0 (^f_ kg,) > ^1£5_ >2 2 5 5 3 log 2 3VJ3 u
for ar>^2-
/Hotfx
Vlog *
21O8 *
Hence it follows from Theorem D that for x>x2
exists a prime number p such that p<x — 3 and 1/<s
divisors <# p'<,xVi.
/
there
has no prime
and has at most 2 prime divisors in the intervals x1/s<
Hence —-— is a product of at most 3 primes.
have Theorem 3.
Thus we
139 1049
(iii)
Let \ be a given integer. Let
(
a = l , g = 2 ; a, = - 2 *
(* = 1 , 2 , • • • ) .
It follows from (9) that there exists a positive constant x3 such that
P^s, 2, *-) - \(l±M.*»)^+ 3 VJ3
«
/ log x
O (^-log log,) > f&Mog x
/
log x
for x>x3. Hence from (9) we know that for x>x3 there exist not less than ——?— prime numbers p in the interval 1< p < x such that log2*
r
r
p + 2\ is a product of at most 3 primes. Thus we have Theorem 2. From Lemma 2 and Theorem A with c = \, we have V
log**/
pi2* p — 2 f > 2 \
P>2
(p — l ) v l o g 2 *
Mog3*
/
Since
Iog8T p+2k=p'
<s n ^
log 3 ! p+2l{=p'
Io n (i - -7-h^rr2 + o (rr ^ ^)' 3
*.!» ? - 2 f > A
(p — l)Vlog *
P>2
Vlog *
/
we have Theorem 4. REFERENCES [ 1]
Wang Yuan 1957 On Sieve Methods and Some of the Related Problems, Science Record, Vol. I, No. 1, 9—11.
[ 2 ]
Wang Yuan
1959 On Sieve Methods and Some of Their Applications,
Scientia Sinica, 8, 375—381. [ 3 ]
MyflaKOB, H . T. 1948 O KOHeraofi pa3HOCra RJIH 4>VHKUHH
[ 4 ]
Wang Yuan 1956 On the representation of large integer as a sum of a
HAH OOCP, cepufi MameM., 12, 31—46. prime and a product of at most 4 primes, Acta Mathematica Sinica, 6 ( 4 ) , 565—582. [ 5]
BifflorpaflOB, A. W. 1957 npHMeHeirae Q ( j ) Mam.
C6; 4 1 , 49—80.
K pemeTy
SpaToafceHa,
140 1050
APPENDIX 1°.
We state the generalized weak Riemann hypothesis as follows:
(R>) The real parts of all zeros of all Dirichlet's L-functions L(s, X) are ^d" 1 , where 1<
(1,3) may be derived from (RsJ, where S{^-
( 1 , 4 ) is the consequence
and
of (Rs2),
3.237 2 2 3 7 "
where
All the following results are obtained under the truth of 2°. tant.
Let 7 =
.
(1) holds uniformly in («>), where A(u) = -&-er
(if
U— 7]
7
<«<37)
and (3)
A(«) =
3°.
(4)
tlog
'2rj
2V
Let v>2rj be a given number and q = x1/u.
P.(,, q, J") < Mu) --^f—
holds uniformly in («>), where AM--&-S u — rj
Then
+ O (X'"]"k*X)
(5)
3 7 <«<7^).
(if
^ '- — 1 V
(Rs).
Let xVa < q < c ^ ' " , where c0 is a given cons-
d —1 Then the estimation
(2)
2.475
V
cpiqnotfx I
141
1051 for 7 < u < -
—, and
V
v
(6) \ij
2\7
uJ
«/
2\V
tt J
for
1 V
4°.
i
(if
*>4?);
oo
(if
j.^4^)^
1 . — _1 v
Let q be a given integer.
Then
PXx, q, *1/6-5') > X(6.5rj) - r ^ - r - + O (^f), (p(^)log2 x WogxJ
(7) where (8)
\(6.5r)) >2rjX 6.453306.
The proof is similar to that of Theorem B with the essential difference that here we put r = 0.452 and I = 1.5715 so as to obtain the more exact estimation V *<-"[(*+ 1 ) T ] * " < >hA*-H[U + l)r]tt +4 £i (2^ + 4)! ^0 (2/^ + 4)! +
i!(8ry! | j ( ^ f ? X « « ) * < 0.007183682. 18! f?i V 4 6080/
5°. Let 9<«
p.(*,ff,*"-) ^ pM(*, q, x»n (9)
+
fy
Let q denote a
, T ^ - *« +
9(^)10^* J« «
+0f^.l0gl08rA Mogf*
/
6°.
Let u, v ht two given numbers satisfying 2rj < v < IO7 and Let 9H denote the set of primes p satisfying the following conditions:
T}
p < x, p = <*(mod ? ) , p ^ a,(mod ?,)(» < f ) , (10)
p^as+i(modp*,+iXi
— s),
142 1052
where ps^xu"
fS^modc+iXKf^t-j)
is not less than (12)
PB(x, q, *"') - — M T M s ) -^-) y + O (-£*£. log log*). m + 1 VJ" 2 /(p(#)losr# Vlog3* /
7°.
Computation of integrals.
(l)
v
At = 1
• ^« = 2eT \
Jii
u
J5i
J r
5.5
J . . .
™L w
4
» — n
' —1
= w
M/_1__log_
, 7 1
« — n,
("5.5
2ve
r
h
i log
« — 77
i
j(w)du,
2
.
< 2^ (Xl /(4 + 0.02O + XI /(5.46 + 0.02;)) < 27
(ii) A
2
Let v = 5rj in 3°. Then
_ f57 yii(«) d
k *
=
u
f
Z^f^
du_
%,(±-i-)-i-4±-j-W4i---L) -
\i} uI 2\y u I 2 \V = 207?^ (2 ^~ = 2077
uI
<20?,? r (XU(l + 0.02O + 2 ^ 1 . 1 8 + 0.02;)) < 2077
8°. (to)
The proof of Theorem I. Let x = y be an even integer. Let a — 1 , q = 2 ; «,- = x
(« = 1 , 2 , • • • ) .
(i) Let 7 = 2.475. Then from 2°—7°, we know that there exists a constant xx such that
P.(«, 2, *^) - Iff11 ^ i « ) ^ - + O ( ^ f log logx) >
143
1053
> P_ (x, 2, ,-,,) _ ( p AGO du + JL(* AGO , \ ___£.2 + 2 VJs,
«
^5^r "
'log *
+ o ( - ^ -2l o g log*) > 2q[6A53306 — l o g AT
MOg X
/
> 0.05 - ^2 + O (^flog log*) > 1, 3 log *
Mog *
/
for x > *!• Hence it follows from 6° that for x > xx there exists a prime p such that p < x — 1 and x — p has n o prime divisors < x 1 / 5 ' and has at most one prime divisor in the interval x 1/5 ' < p ' < x 15' . Hence i ^ - p is a product of at most 3 primes. Thus we have ( 1 , 3 ) . (ii) Let ^ = 3.237. Then from 2 ° — 7 ° , we k n o w that there exists a constant x 2 such that
p.(*, 2, *1/5') - -Iff5' A^> <*«)-£*! + o ( - ^ l o g log*) > 2\J^_
> o.Ol ^f+O 2 log *
«
/log2*:
Vlog3*
/
f-^-log log*) > 1 3 Mog *
/
for x>x2. Hence it follows from 6° that for x>x2 there exists a prime number p such that p < x — 1 and x — p has no prime divisor <x1/5» and has at most one prime divisor in the interval x1/5' < p' <= ii-i
x ^ . Hence x — p is a product of at most 4 primes. Thus we have (1, 4). 9°. It is well known that in the proof of Theorem I, the hypothesis (Rs) may be replaced by
where ,4 is any given positive constant and the constant implied by the symbol "O" depends only on 8 and A. Similarly, (/?«) may also be replaced by (8.)
2 D^T/,
A W Max /(rnodD)
P(*, D, /) - _ _ £ «p(D) l o g *
= O (—^-), \log^*/
144
^ ^ ^
1054
where P(x,D,l)=
2
logp-e * (cf. [1], [2]).
P = I (mod D)
3 41
^ BapoW ' first proved (JRi.2). Later, Pan Chin Tongt21 obtained (R\s) independently, from which he deduced (1,5). From Theorem I, it can be easily seen that (#1.5) implies (1,4). In other words, we have proved Theorem II. "Every sufficiently large even integer is a sum of a prime and a product of at most 4 primes. Remar\. (1, 4) has also been proved by Pan and BapfiaH independently, but their proofs are more complicated than that given here, in fact, their proofs are based on (2?i.6) and (JR 16 ) respectively (cf. [5]). I am grateful to Messrs. Pan and BapdaH for their kindly informing me of their results. REFERENCES [ 1] [ 2 ]
[ 3 ] [ 4 ]
[ 5 ]
PeHbH, A. 1948 O npe,ucTaBjieHHH ^eimix qnceji B BHfle cyMMH npoeroro H IIOITH npocToro racjia, EAR CGGP, 2, 57—78. Pan Chin Tong, 1962 On the representation of large even integer as a sum of a prime and an almost prime, Acta Math. Sinica, Vol. 12, No. 1, 95—106. BapfiaH, M . B. 1961 ApHtpMeTHHecKne $YHKUHH Ha pe^Knx MHOJKecTBax, flOKjiadu AnadeMuu HayK Y3CGP, 8, 9—11. BapCaH, M. B. 1961 HoBbie npHMeHeHHH Sojibiuoro peuieTa IO. B. JInHHHKa, Tpydu Hucmumyma MameMamum, UM. B. H. PoMaH08CK010, Bbin. 22JIHHHHK, KD. B. I960 AcHMnTOTHqeCKaa cpopMyjia B aflflHTHBHOH npo6jieMe FapflH JlnTTJibByAa, HAH CGGP, TOM 24, N? 5. 629—706.
145
ON THE ESTIMATION OF CHARACTER SUM AND ITS APPLICATIONS*
WANG YUAN Institute of Mathematics, Academia Sinica Received 25 September 1962
1. Introduction By the use of Weil's [1] famous result on analogue of Riemann hypothesis for algebraic function fields over a finite field, Burgess [1] first proposed a method for the estimation of character sum which gave an improvement of the well-known Polya's inequality [3] for the case of real primitive character modulo p (prime number). Later, the author [4] and Burgess [5, 6] himself gave some further generalisations, modifications and applications on Burgess's estimation. His method may be stated as follows: Theorem A. [6] Let x be a primitive character modulo k. Let rj be a given positive number and r be an integer > 1. If k is a square free number or r = 2, then the estimation N+H
J2 X(n)
n=N+l
holds for any pair of integers N and H (H > 0). We have the following two consequences: Corollary 1. [6] Suppose that \{n) = (£) is the real primitive character modulo f, where (£) denotes the Kronecker symbol. Then rj
Y,(i)
*Shuxue Jinzhan 1 (1964) 78-83.
'
146
Corollary 2. [4] number). Then
Suppose that x is a non-principal character modulo p (prime "
H
5>(n)
71=1
*
holds for H > p*+r>.
The aim of present paper is to apply these estimations to the problems for estimations of the least solutions of Pell's equations and the nth non-residue modulo p. The latter problem will also be treated under the assumption of generalised Riemann hypothesis (GRH). Let d be a positive integer not a perfect square, d = 0 or 1 (mod 4). If the integer point (a;o,2/o) (xo > 0,yo > 0) is a solution of the Pell's equation x2 - dy2 = 4, such that XQ + \/dyo attains its minimum, then it is called to be a least solution. Let 6
Theorem 1.
_ x0 + Vdyp ~ 2 '
For any 5 > 0, there exists C4(5) such that \ne<
(j+5\Vdlnd,
holds for d> C4 (S). This gives an improvement of a result due to Hua Loo-keng [7]. Let n > 2 and n\(p — 1). If the congruence xn = c (mod p),
1< x
has no solution, then c is called an nth non-residue modulo p. Otherwise it is called an nth residue modulo p. Denote by N(p, n) the least positive nth non-residue modulo p. Theorem 2. Let S be a given positive number. Then for a sufficiently large prime p, we have (i) N{p,n)
,
(ii) N(p,n)
n>2, n > 21
and
(iii) N(p,n)
4hm ,
n >e .
147 (i), (ii) and (iii) are the respective improvements of results due to Vinogradov [8] and Buchstab [9]. 2. Proof of Theorem 1 Write a{a)
Lemma 1.
= y2(-)
and K(d) =
T(-)-.
Let r be an integer > 4 and r = £. Then W(a)\ < c5{T)ad-T2/6
holds for a > di+T. Proof.
Let d = fm2, where / is the fundamental discriminate. Then
-s»G)s:(9 V
k\m
/
X
n
'
and
K«)I<E
E(Q-
k\m n
'
Therefore by Corollary 1 we obtain k(a)| < y " c 6 ( r ) ( ^ ) k\m
N
r+
/*+47I7+TT
<
05^)0}-^d^+^+^
7
< c5(T)ad~*t^rV~7FTT>+£:
+ 2r
<-r+v = c5(r)ad~4r^+1>
< c 5 (r)arf~ T2/6 .
The lemma is proved. Lemma 2.
For any given 6 with 0 < 5 < | , there exists CT{5) such that
K(d)< Q+jVnd, hofa!s for d > CT(6).
148 Proof.
Let r = £ < | < 2T, where r is an integer. Then
1 1!
-
^
l
n(n + l) -
^
n+1
l
By Lemma 1 we derive
Finally it follows from Polya's theorem that |cr(a)| < ^
v
/
7
l n
/ < V^lnd
k\m
and therefore
The lemma follows. Theorem 1 is the consequence of Lemma 2 (see [10]).
3. Proof of Theorem 2 Lemma 3. [9] Let ty(x,y) be the number of integers in the interval 1 < n < x such that the prime divisors of n are all < y. Then l
/
X
\
V(x,x«) = p{a)x + O[ -== ) , Vvlnx/
149 where the constant in " 0 " depends only on a, and 1,
ifO
i_ r ^ i + r r i-1 ^ ^ +(_1)M
r/•-.../*-""'5I^M,
lta>1.
Therefore p(a) is a continuous and non-increasing function of a such that p(a)>e-a(lna+lnlna+aJ^). L e m m a 4. Let R be the number of nth residue modulo p in the interval 1 < c < H. Ifn\(pl),n>2 and H > p*+n(r) > 0), then R=-
+ S, n
where \S\ < Proof.
c9(V)Hp-^2/Q.
Let x{x) = e 2 7 r i I n d x / n . Then by Corollary 2, it yields that H
1
"
IT
1 "-
1 H
V
where \S\ <
c9(V)Hp-o2/e.
The lemma is proved. Lemma 5. Suppose that n > 2, n\(p — 1) and p(a) > ^ + S(5 > 0). Then there exists cio(a,S) such that N(p,n) < p*s, whenever p > Cio(a, 5). Proof. Take H = [pi+r>] + 1 (77 > 0). If N{p,n) > p&, then the integer in 1 < n < H such that it contains only the prime divisors < p ^ is an nth residue modulo p. If rj = 77(5) is sufficiently small, we have R > *(H,p£) > p((l + 5V)a)H +
>(1-4)H+O(4=)
o(-^=)
150 and v
n
'
\n
2)
\yflnpj
by Lemma 4, where the constant in "O" depends only on a and 5. This is impossible if p is large and therefore the lemma follows. Now we proceed to prove Theorem 2 as follows: (i) Take a = e^~T{\
> r > 0). Then p{a) > 1 - I n a = - + T . n
Hence there exists cn(n, r) such that N{p,n)
,
whenever p > cii(n,r) and n > 2. Here 5 > 0 and lim T ^ 0 5 = 0. (ii) Take a = 3. Then
p(3) = l - l n 3 + / 3 r " 1 ^ ^ > 0.4804 > 1 . Hence there exists C12 such that iV(p, n) < pA holds for p > c12 and n > 21. (iii) Take a = lnln°T[t+2- Then we can choose 5 = 8{n) > 0 for n > e33 such that p{a) > - + S.
n
Hence there is a Ci3(n) such that In In n + 2
N(p,n)
41n
"
holds for p > Ci3(n) and n > e33. 4. Conditional Result Lemma 6. [4,11] v"*' ^
-it W A W
Under the assumption of GRH, we have f x + O(x2 lnp), if x = Xo ?s principal character modulo p, I O(xi ln P), otherwise.
Hereafter the constant in "O" is an absolute constant.
151 Theorem 3.
Under the assumption of GRH, we have N(p,n) = O(ln2 p),
Proof.
n>2.
Since there is a cu such that *(a;) = E
A n
( ) ^ ci*x>
x>2.
We derive from summation by parts that E
A(n)e-*
71>C15X
holds for a constant c 15 . Consider now E
R(x)=
Xo(rn)A(rn)e-^,
r n
where X^rn
-
rn>l
^
A(m)e^
m>cl5x
> TXo(m)A(m)e-^(l--Te27riaIndm/n) m=l ,
V
-. >.
= (1 - V
n
/
a = 1
oo
n - 1 oo
E Xo(m)A(m)e-^ --TJ2
" / m=l
-ci4e-Cl5(ci5 + l)z
n
~ cue~^ {cl5 + l)x
A(m)xo(m)e 2 - oIndm /" e -v
a=lm=l
> ( 1 - - - Ci 4 e" Cl5 (ci5 + 1) )x + O(x? lnp). Take x = c\& log2 p. Then R{x) > 0, whenever ci 5 and ci6 are sufficiently large, i.e. N(p,n)
152 [3] P. Polya, Uber die Verteilung der Quadratischen Reste und Nichtreste, Nachr. Ges. Wiss. Gottingen (1918) 21-29. [4] Wang Yuan, On the least primitive root of a prime, Acta. Math. Sin. 4 (1959) 432-441. [5] D. A. Burgess, On character sums and primitive roots, Proc. Lond. Math. Soc. XII (1962) 179-192. [6] D. A. Burgess, On character sums and L-series, Proc. Lond. Math. Soc. XII (1962) 193-206. [7] Hua Loo-keng, On the least solution of Pell's equation, Bull. Amer. Math. Soc. 48 (1942) 731-735. [8] I. M. Vinogradov, On the bound of least n-th non-residue, Dokl. Akad. Nauk 20 (1926) 47-58. [9] A. A. Buchstab, On estimations of the numbers in an arithmetic progression whose prime divisors being less than a given order, Dokl. Akad. Nauk SSSR 67 (1947) 5-8. [10] Hua Loo-keng, Introduction to Number Theory (Science Press, 1957). [11] N. C. Ankeny, The least quadratic non-residue, Ann. Math. 55 (1952) 65-72.
153
ON THE MAXIMAL NUMBER OF PAIRWISE ORTHOGONAL LATIN SQUARE OF ORDER s (APPLICATION OF SIEVE METHOD)* WANG YUAN Institute of Mathematics, Academia Sinica Received 25 September 1964
1. Introduction In this paper, we shall give in detail the proofs of results announced in [1] with a slight modifications. Let a set of s distinct symbols, for example, 1,2,..., s be arranged in an s x s square in such a way that every symbol occurs exactly once in every row and once in every column. Such a square is called a Latin square of order s. Two Latin squares are called orthogonal if, when one of the squares is superposed on the other, every symbol of the first square occurs with every symbol of the second square once and only once. We denote by N(s) the maximal number of mutually orthogonal Latin squares of order s. Euler proposed a famous conjecture on N(s): N(s) = 0, whenever s > 10 and s = 2 (mod 4). Bose, Shrikhande and Parker [2] made a great contribution on Euler's conjecture. They proved that N(s) > 2 (s > 6). By the combination of their method and Brun's sieve method, Chowla, Erdo's and Straus [3] established that there exists a constant s0 such that N(s) > -ssr
*Acta Mathematica Sinica, 16:3 (1966) 400-410.
( 5 > S o ).
154 The author improved |s5T to s^s in [1]. In this paper we give the following
Theorem 1.
There exists a constant si such that N(s)>s^,
s>si.
In Sec. 2, we shall generalise the method of Chowla, Erdos and Straus, and give the relation between N(s) and sieve method. Theorem 2 shows that the estimation oiN(s) depends on the estimations of Pu(x, q; £,??) (£ < rf) a.ndPu(x,q;£) (see Sec. 2 for the definitions). The upper and lower estimations of Pu,(x, q; £) may be obtained directly by the use of sieve method, and then the upper and lower estimations of Pu{x,q;£,,r)) are derived by the simple inequalities
PM,Q\v)
(1)
In this paper, we obtain a more precise estimation for Pu(x,q;£) by the combination of the sieve methods of Brun, Buchstab and Selberg instead of the simple application of Brun's method (see Theorem 4). The combination of these methods first appeared in author's works [4-7] on Goldbach conjecture. In this way, we can establish that if s is sufficiently large, then N(s) > s&. To obtain more precise result, we need to give a more accurate estimation on Pul(x,q;^,T]) (£
a,q;a,i,bi,
i = l,2,...
be a set of integers satisfying 2\q,
q = O(xCl) ai^h
(modPi)
(0
qi---qt=O(l),
(i = 1 , 2 , . . . ) ,
where 2 = q\ < q? < • • • < qt are all prime divisors of q and pi < P2 < • • • the prime numbers not dividing q. Let Pu{x, q; f, 77) (f < 77) be the number of integers n
155 satisfying the following conditions n^bi (modft)
(pt < 7?).
(2)
In particular (3)
PM,Q;Z>Z) = PM,Q;O, i
(4)
PM,*,t> n) = PM;Z,v)Theorem 2.
Suppose that „ / \^
f2x\°\ \q J J
c(q)x gliT I
f
x \ \q\n'xj
(5)
and ^
i
i
i\
c(2)x
(
x
\
. .
6
PUJ(x-Xo,xT)>c3-^-+o[-^-) In x \ln xj hold uniformly in (u>), where q = O(x°+ T ), a,b,c are constants b>2,(a + l)b>c>b, and
satisfying
a > 2,
*> = «•-* n^nO-srhpHt). <7> p>2
in which 7 denotes Euler constant. Then for any positive number e there exists a constant SQ = So(e) such that N(s) > s<-a+»nb+v~£. whenever s > soTo prove Theorem 2, we need the following two theorems:
Theorem A. [2] If 1 < k < N(m) + 1 and 1 < u < m, then N{km + u)> min{A^(fc), N(k + 1), 1 + N(m), 1 + N(u)} - 1. Theorem B. [8] (i) N{uv) > (ii)N(pr)=Pr-l.
min{N(u),N(v)},
(8)
156 Proof of Theorem 2.
(1) Suppose 2/s. Take k such that
(
s \ *+^
-J
/
r
i
i
IN
k = - 1 fm0d2Ua + D(6+2) lo82 2] \
,
fc^O, -l(modp)(3
f
j .
(9)
Then it follows from (5) that the number of k satisfying (9) is not less than C4
a —2
> 1,
In s if s is sufficiently large, and therefore from Theorem B, we have N(k + 1) > 2[<»+1>V2)log2 *1 - 1 > s(-.+i)1(i.+2)-£ + i ; ^ ) "+ 1 J
+
- 1 >S(a+lKi,+2)-£+l)
(10) (11)
if s is large. Set s = si + s^k, 0 < s\ < k and u = s\ + u\k, where u\ satisfies the following conditions:
l<«i<(^J
,
(12)
ui = s i + 1 (mod 2),
(13)
3
+2
ul^s2(modp)U
j ,
(14)
(15)
in whichfcpdenotes the solution of ky = 1 (modp). The existence and uniqueness of kp (modp) for 3 < p < (|)^+rfe+5T a r efollowedby (9). Suppose 3 < p < (|)
/ S \ ( a + l)%+2)
/S\3Tl\
It is clearly f/s\W2
/s\-^m
fs\^\
157 whenever b < c < (a+l)b. Hence we derive by (6) that the number of ui satisfying (12)—(15) is not less than c*:-2- > 1, In s if s is sufficiently large. We know 2 / u by (13) and vt« yields by Theorem B that N(u) >
by (14) if 3 < p < (|)c+i>V'>. Therefore it
- 1 > s(a+D(i.+2)-£
- I
(16)
holds if s is large. Let m = ^ ^ = s 2 — u. Then m>sE
+ 5 _ ^ _ j + i > ^_j
^_j + i >u>i,
(17)
if s is large. Since 2/w, 2/A; and 2|«, we have 2 / m . It follows from (15) that the prime divisors of m are all > k, and therefore by Theorem B, we obtain N(m) >k>
(18)
N(k).
Prom (10), (11), (16)-(18) and Theorem A, we derive that there exists a constant so = so(e) such that N(S)
>
S (a
+ lK6 + 2 ) - e )
s
>
So
.
2) Suppose 2jfs. Take k satisfying ( 9 H 1 1 ) - Let s = s1+s2(k and u = si + ui(fc + 1), where ui satisfies the following: 1 < «i < [_)
+ l),0 < Si < (k + l)s
,
(19)
ui = s 2 + I(mod2),
(20)
Ul
£ -Sl (k + l) p (modp) (3 < p < ^£y o + 1 K b + 2 ) j ^
ui # s2 (modp) f 3 < p < (l V + j .
(21)
(22)
Since 2|(fc + 1) and 2 / s, we have 2 / s and so 2 / u. We know also that the prime divisors of u are all > ( | ) (»+DW2) b y (21). Let m = f^f = s 2 - u x . Then by (20)
158 and (22), we know that the prime divisors of m are all > k. On the other hand, m >u> 1 if s is sufficiently large. The theorem follows from Theorem A.
3. A Recurrent Formula The aim of this section is to prove the following Theorem 3. Let a,P be two numbers satisfying 2 < a < /3 < 15. If there exist two non-negative functions X(u, v) and A(u, v) (0 < v < u < 15) with the properties that if one of the two variables u or v is fixed, these functions are increasing with only finite number of discontinuities in another variable such that 1
l
Cl"
l
T
\
P u (s;s* ) *=)>A(/?,a)- 5 - + o ( r 2 5 In z \ln xj
(23)
and
Pu(x;xt,xi) < A(/?,a)-^- +o(-^-) In i
\\n
(24) xj
hold, where
(25) in which 7 is Euler constant, then the functions
Xttf, a) = max U, X((3,6) - j ' ^ A ( ^ j ^ ) ^
d z
)
(26)
and
A^,a)
= A(P,6) - J' * X ( ^ Y ^ ) ^dz
have the same properties as A(/3, a ) and A(/3,a), a<6 <(3.
(27)
where 5 is a number with
To prove Theorem 3, we need the following two lemmas.
159 Lemma 1. such that
Suppose £, < ( < rj. There exists a sequence of sets of integers (ujni)'s
P«frt,V)=Pufrt,OProof.
E PunJ^^lZ'Pni-l)v Pni J c< Pni <^
(28)
By Eratosthenes sieve we obtain
C
+ E
E
«Pn1
«Pn2
- E
^p»1p»a(^^.o
E
E
C
where PQPn
Vn
p^^^u)
+ ---, (29)
(
(x; ^, C) denotes the number of integers satisfying
0 < n < x,n = a (mod 2),
n ^ O j (modpj),
n = bni (mod pni),...,
n =
bnt(modpnt),
n^bj(modpi) (p» < 0-
(30)
We may assume that 0 < ai, bi < pt. Let a n i ,Oj n i and fojni be the respective solutions of congruences mPm +bni =a (mod 2), mpni + bni = ^ (mod p ^ and mpni + bni = bi (modpi) whenever pi =/= pni. Let K,) and Pu>niPn
Pn
(x',£,0
ani; aini, bini
i =/= ni
be the number of integers satisfying
0 < n < x,n = ani (mod2),
n = bn2m ( m o d p n 2 ) , . . . , n = bntni (modp n t ),
n ^ aini (modp^ (p<e),
n ^ bini (modpi) (p» < ().
(31)
160 Then
= p^,vn2...Fnt ( ^ ^ . c )
p^..,nM^0
(32)
and consequently
+ E
C
E ^ i r . j P B ,(^«-c)--
C
_
(x-bni.<.
4-
V^
V-
\
c
V
P
p.
(x-bni V
C
(x-bni.r
f
c
\
\ 7
= PWB1f^^;^Pn1-i). \
(33)
/
Pni
Substituting (33) into (29), we have the lemma. Lemma 2.
[9] Let a, f3 be two numbers such that f3 > a > 1. Then
^ ^ p l n
2
!
\7a_i
^2
/In2*
Vln3a;y
(34)
Proof of Theorem 3. Set £ = x? ,C, = x$ and r\ = xi in Lemma 1. Let n = [%/Ex] and uj = a + ^ J - 1 (0 < i < n). Then
x"i+i+i
E
P^fyxKp^+Oix^ + Oix1-*)
= T', + O(x-) + O(x 1 -J),
(35)
161
<
V
"
, ^
-
U+i + 1
//?Inf lnfX
,
CT
J
, ^
< UJ^,.,+1) r L V«i+i + l
/A;
z
25
p'z +1 \
, pln2f
Vln 3 i/
i±ij 2
x
Vln xA;
llnx 'lnPJpln^
+
/
+O
*2^J
+0(').
2
\\n x
pel
\\r?x)
Since ^
Vui + l + 1
) Jul
L VUJ + 1 + 1
^iui
Z2
Ja-l
/
\Z + 1
\Z + l
)
Z2
J\ Z2
(37)
= oh)=O('), we have
V! ll
p ± <XQ
fx-bm.xl
\
\
/
Pnx
n-1
= E^ n-1
= ^2 fi + O(xi Vlnx) + Oix1-^ Vlnx) l=o
(38) and therefore
162 by Lemma 1. On the other hand, since Ti>\\[ L \ui+i
——dz\—T-+O[—*-) z2 J ln2x Vina;/
,ui) J JUl
and
f'-1 Ja-i
( 0z \z + l V2 + l / 22
^ ^
\ fUl+1z J JUl
( 0u, V«« + l
+ l, z2
(
1 \ \Vh^J
we have
V
Z^
1
(X-b^:r r-
P
^"1 I \
±
ll
\y«-i
„
V^ + i
n
> °>Pni-l I
\
J
Pni
/
z
yin2x
Vin 2 - 5 xy
and therefore
by Lemma 1. (27) follows and the theorem is proved. Remark.
Since PU{X;W)
< P*{x;t;,v) < Pu,(x;Z,z), e < v,
the functions X(u,v) = X(v,v) = X(v) and A(u,i>) = A(u,u) — A(u) clearly satisfied the requirements in Theorem 3 (see [4-7]). 4. Estimation of Pu(x, q; £) We use the notations in Sees. 2 and 3. The aim of this section is to prove the following: Theorem 4.
The inequality
(39) holds uniformly in (to), where q =
0{x*).
163 Proof.
1) It follows by Selberg's method that the estimation _ / p
f2x\*\
t/»c(q)x A i d )
»{*>«{7) ) ^
/zlnlnaA
^
+0
(40)
{-^x-)'
holds uniformly in (u>) and q = 0{x*), where
*"[W-AM]' Md)={
,
r
(41)
[(rf-l)2-flnf + < 5 ( | ) - ^ '
-
-
in which r) is a preassigned positive number and
tf(d)=lim-^-
^ln2-L,
Y £
d>2.
(42)
pp' < id
A(d) has a similar expression for d > 6, and we may take A(d) — A(2) for 0 < d < 2. We refer to Wang Yuan [4-7] for the proof of these formulas but we should take £=(^)i/log5a;here. 2) By Brun's method, we know that the estimation
(43) holds in (us) and q = O(x*). We refer to Buchstab [10, 11] for the proof, but we should make the following changes: pr is the largest prime number <( —) T5 ,p r3 = Pr2 = Pn = Pr, and Prfc(4 < k
pJx,q-,(^)")<mM22^L+o(-f r-), \ \q J ]
\qj
J
gin 2 1
\qln3xj
and
P,L;p)A)<101,j«i+o( • ) .
164 3) Suppose that a, (3 are two real numbers satisfying 2 < a < j3 < 15 and that there are two non-negative and non-decreasing functions \(z) and A(z) with the following properties: They have at most finite number of discontinuities and the formulas (2x\1\
_ (
P»[™{J) and _. / Pu[x,q;[
^/a;lnlnx\
. c(q)x
J^^^Ul^j
/2z\*\ , c(q)x ^fxln\nx\ —\ < A ( z ) - ^ - +O 3
, ,
(44) 45
holds uniformly in (a>). Then X1(a)=maxi0,X(J3)-2
A(z)-^-dz\
(46)
and A 1 (a)=A(/3)-2 /
A(^)^d2
(47)
have the same properties as X(z) and A(z) respectively. We refer to Buchstab [10, 11] for the proof. Note that xi < | if ( ^ ) ^ < p < ( y ) ° , and so g = O(x3) = o ( ( f ) 3 ) - The existence of the functions X(z) and A(z) satisfying (44) and (45) follows from 1) and 2). 4) P r o m l ) - 3 ) , w e h a v e t h e following t a b l e : d \ 10 I 9 I 8 I 7 I ••• Ap(rf) 9 9 . 9 8 1 8 1 1 U 7 8 4 6 9 6 0 . 8 8 8 1 7 4 3 . 5 1 5 5 4 ~ ••• A 0 ( d ) I 1 0 0 . 0 2 0 7 3 | 8 2 . 7 2 0 7 | 6 8 . 5 2 5 1 1 | 54.39352 | •••
(48)
(see [7, 1 0 , 11]). T a k e ut+\ -ut= 0 . 0 1 . T h e n b y 3) a n d t h e f o r m u l a s
/
Ja — l
\{z)-^dz>Y,Kut) Z
t_0
Jut
-ydz
(49)
and
/
A( Z )^z<^A(« t + 1 )
-J-dz
(50)
for i t e r a t i o n s , w h e r e ^ p - = 0 . 0 1 , w e o b t a i n d \ 10 I 9 1 8 I ••• 1 4 . 3 1 1 ••• An(d) 99.98181 80.892035 63.59931 ••• 0 . 0 5 ••• A12(d) 100.02073 81.11841 64.403149 ~ 3 0 . 8 2 1 ••• T h e t h e o r e m is proved.
(51)
165 5. Estimation of Pu(x; $, rj) Theorem 5.
The inequality
(52)
P^(x;xKx^)>0m-^+o(-^w-) In i
\\n xj
holds uniformly in (u>). Proof.
Take (3 = 5 = 8 and a = 3.24 in Theorem 3 and set A(u,v) = A12(u)
and X(u,v) = An(v).
Then An(8)-/ J2.2A
A12 (-^-)Z-^dz> 0.01. \
z
+ 1/
(53)
z
The theorem follows. Theorem 6.
The estimation 1
1
CX
(
X
\
(54)
Pu{x;x^,x^)>0.29-T-+O[-Tr-) In i \ln xj holds uniformly in (u). Proof. Let
\0{u,v) = Xn{v) and A0(u,v) = Ai2(u). Let u t +i — ut = 0.01 and ut+i < /3. Then by Theorem 3 and the formulas \+1(P,ut)
= Xi+1((3,ut+1) - /
Ai(J^!Z)^dz
Jut-l
\Z + 1.
3{ut+l 1
> Xi+1(p,ut+l) - A, (< V
W
~ \ut+l
*+l
JZ
- l ) H*1'1ZA±dz(55) z
/ Jut-l
and yu t + i-i ( Bz \ z + 1 A i+ i(/3,u t ) = A i+1 (/3,u t +i) - / Ai I -^3-7,-2) —2-cte Jut-l \Z + I J Z
< Al+I(f3,ut+1) - A, f^"'"^,^ - l ) n+l~l \
U
t
J Jut-l
Z
A±dz (56) Z
we have (u,v) \i(u,v)
1 (8.05,8) I (8.05,7) I (8.05,6) I ••• 6 4 . 0 1 1 9 9 8 5 5 . 5 5 9 5 5 1 4 6 . 5 8 2 4 8 8 •••
(u,v) I (7.05,7) I (6.9,6) 1 (6.76,5) I ••• A i ( u , u ) I 5 0 . 8 3 3 6 8 5 | 4 3 . 8 6 9 9 8 5 | 3 9 . 9 9 4 5 3 7 | •••
(57)
166 and (u,v) I (10,9) I (10,8) I (10,7) I ••• I ( 1 0 , 2 . 8 4 ) I ••• X2(u,v) 89.912569 79.816805 69.635682 ••• 0.29 (u,v)
(8.89,8)
(8.75,7)
(8.58,6)
•••
(58)
A 2 ( u , v ) I 7 1 . 3 4 4 2 4 5 | 6 1 . 6 7 7 9 3 1 | 5 2 . 6 6 7 5 8 2 | ••• T h e theorem is proved.
6. Proof of Main Theorem Take e =
5 31 *5 24
— ^ . Then by Theorems 2,4 and 5, it follows that N(s) > S 5"i5.24-^ = S A ,
(59)
if s is sufficiently large. This is the main result announced in [1]. Then take e = 5 3ix4 84 ~ ^6' ^ y Theorems 2, 4 and 6, we obtain N(s) > S 5 3 iii.84- £
= SA!
(60)
if s is sufficiently large. Theorem 1 follows. Remark. If x~& is replaced by x^i or z A in Theorem 6, then it is possible to establish N(s) > s " by Theorem 3 with more complicated numerical calculations. Remark added on 7 April 1966. proved
We know recently that Kenneth Rogers has
N(s) > s~&,
s > so
(see Pacific J. Math. 14 (1964) 1395-1397). References [1] Wang Yuan, A note on the maximal number of pairwise orthogonal Latin squares of a given order, Sci. Sinica 8 (1964) 841-843. [2] R. C. Bose, S. S. Shrikhande and E. T. Parker, Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler's conjecture, Can. J. Math. 7 (1960) 189-203. [3] S. Chowla, P. Erdos and E. G. Straus, On the maximal number of pairwise orthogonal Latin squares of a given order, Can. J. Math. 7 (1960) 204-208. [4] Wang Yuan, On the respresentation of large even integer as a sum of a product of at most 3 primes and a product of atmost 4 primes, Ada Math. Sin. 6:3 (1956) 500-513. [5] Wang Yuan, On the representation of large even integer as a sum of a prime and a product of at most 4 primes, Ada Math. Sin. 6:4 (1956) 565-582. [6] Wang Yuan, On the representation of large even number as a sum of two almost primes, Sci. Rec. (New Series) 1:5 (1957) 15-19. [7] Wang Yuan, On sieve methods and some of their applications, Sci. Sinica 8 (1959) 357-381.
167 [8] H. F. MacNeish, Euler squares, Ann. Math. 22 (1922) 221-227. [9] A. A. Buchstab, Asymptotic estimation of a general number theoretic function, Math. Sb. 2 (1937) 1239-1245. [10] A. A. Buchstab, New improvements on Eratosthenes sieve method, Math. Sb. 4 (1938) 375-387. [11] A. A. Buchstab, On representation of even number as a sum of two integers with bounded number of divisors, Dokl. Akad. Nauk SSSR 29 (1940) 544-548.
168
Vol. XVIH No. 5
SCIENTIA
S1NICA
Sept. - Oet. 1975
Science Articles
ON THE REPRESENTATION OF EVERY LARGE EVEN INTEGER AS A SUM OF A PRIME AND AN ALMOST PRIME PAN CHENG-DONG
(SSJl)
(Department of Mathematics, Shandong University) DING XIA-XI (TX&t)
WANG YUAN (5E 5C)
(.Institute of Mathematics, Academia Sinioa) Eeeeived Oct. 21, 1974.
ABSTRACT
In this paper, we give a modified proof of Chen's theorem "every sufficiently large even integer is a sum of a prime and a product of at most 2 primes". 1.
INTRODUCTION
For brevity, we denote the following proposition by (1, a): Every sufficiently large even integer is a sum of a prime and an almost prime of at most 2 prime factors. The proposition was studied by T. EstermannL4), A. Eenyi[18!, Wang Yuan19'101, M. B. Bap6aHul-I2J, Pan Cheng-dong16-171, B. B. JleBHH»61, A. A. Byxurra6[13'14], A. H. BHHOrpaflOB1151, H. E. Bichert[8] and Chen Jing-runt2>31 successively by means of sieve method and large sieve method. The best result is due to Chen. He proved
Theorem 1. (1, 2) Chen gave the previous method an important improvement. duced and estimated the
£=
S
Especially, he intro-
o-i)
i,
(
where p, pt, p2, p3 are primes and CPi,2) denotes the condition x10 < pl^ x* ^ p2^ (—) . The aim of the present paper is to give a modified proof of (1, 2). First, we shall show that the estimation of Q may be derived from the following mean value theorem. Let 2 ^ y ^ x. Let it(y, a, q,l) =
^]
a
»
169 600
SCIENTIA SINTCA
Vol. XVHI
where |-1, for prime n, l
Theorem 2. estimation
1=
0, otherwise.
For any given positive constant A and positive number e ( < 1), the
,JL*. - -- JL KaYy>a'q> ° - ict) - KrV)
ci.2)
A
\log a;/ holds for log28^ < Ay ^ A2 < yl~*, where | / ( a ) | ^ 1, B = A + 7 amd ffee constant implied by the symbol "0" depends only on e and A. Next, we use the comparatively elementary sieve method given in the previous work of one of the authors (Cf. [10]) to replace the Eichert's method. We have by the same way that Theorem 3. There exist infinitely many primes p such that p + 2k is a product of at most 2 primes, where k is a given positive integer. Theorem 4. Every sufficiently large odd integer N can be represented as N = p + 2PC2), where p is a prime and P t2) is an almost prime of not more than 2 prime divisors. The other famous problems concerning the distribution of almost primes can be treated similarly. II.
THE PROOF OF THEOREM 2
To prove Theorem 2, we shall need Theorem A (large sieve). Let 1 < P < Q. Let M and N be positive integers and let b'ns be any complex numbers. Then *
M+N
2
/
TLT\
M+N
(21)
*
where 2
denotes that the sum is over the primitive characters mod q.
For brevity, we omit the index q of Xq. Cf. P. X. Gallagher [5] and Chen Jing-Eun [3]. The Proof of Theorem 2. 1)
Denote Z>x = logB x,
D= zhog- B a\
(2.2)
170 No. 5
PAN et al.: A SIMPLE PROOF OF CHEN'S THEOREM (1, 2)
601
Then we have
*(V, 0,3,0-
2
8
-
=
TTS
! ( I ) X W
S
( I
.
J :
W.
(2.3)
fl«=/(mod 0)
where (a, q) = (Z, g) = 1 and 2 denotes a sum in which Xq runs over all characx, ters mod q. Hence
*(y, a,ftZ)
i - 2
?>(«)
" = -TT 2 *(0*(«) 2 «»*(M)
a
«
9 ( 2 ) x3-¥X0
^ ^ ^ S 2 ^(0X(o) 9(2)
9,14 X,
2
»
»
«nX(«).
(2.4)
t», «/«,)= 1
It follows from prime number theorem that
£
a. - *(^-) + 0 ( 2 l) = H - + 0 U- e-"**) + 0(f).
(2.5)
Therefore we have from (2.3), (2.4) and (2.5) that
Z< S ^ - T 2 - ^ maxS*
S /(«)««) 2 ««*(») (a,« 2 )=l
+ ° ( T 4 - )<
max max
t».9 2 )=l
ks* • 7».» + ° (TZT-)>
V-V
S ff(«)Z(«) S *.Z(n) ,
(2.7)
where
J».«» S -TT E * «
^,<«<^2
"<Wo
in which fff(«) = / ( w ) , <^» = «», for (w, m ) = 1, g(n) — da = 0, otherwise.
2) Let
Iy.m = 4 % + /i2,'-,
(2.8) (2-9)
where
(2.10) 7
"- = 2
-TT S *
2 ff(o)^Cfl) 2 d.Z(») .
(2.11)
Then it follows by Siegel-Walfisz theorem (Cf. K. Prachar [ 7 ] ) that there exists a positive number s 1 = e^e) such that
171 602
SCIENTIA SINICA
Wm < S "TV 2 * I S +
2J
- 7 - T ZJ
«
ZJ
X,
fKQJ
Vol. XVHI
«-*(»)
«•
ZJ
^ O s ! ^ »
= OC-D^e-6^1"*7 logx) + OCfl^jm 6 ) (2 12)
-
"^T^H-)l
3)
\ log* x / Obviously, we have
I{H < S S
Z
^ ( i , *),
(2-13)
where 2]D1 < D < 2I+1Z>1, 2"% < J.2 < 2K+lA! and
Jft.G,*)~
2
-TT 2 *
Z'D^^O'+'D, "PW;
for A; < K and the 2
k+1
X,
2
2kAl<*<2k+'Al
»(«)««) 2*.^»).(2.]4) »
A! in (2.14) should be replaced by A2 when k = K. Let
/(«, z) = 2 ^
.
^1
Take
n'
(•'-it- -r 1 -)\
^2-15>
log a;/
T = e*""''.
(2.16)
Then from Perron's formula (Cf. K. Prachar I 7 | ) , we have
2
«U00 = - i : rTfl^X)
I1-)1 ds + 0(*-) + 6(.a),
(2.17)
where 6(a) Suppose that H
,0(1), for a | y,
(2.18)
l
O, for a | » .
Denote
/,(«,*) = 2 - - ^ .
/^ s >« -
2
^
•
(2-19)
Then we have immediatelj' that
f(s, Z) = /,(«, X) + /2(s, Z) + O(*- 2 ).
(2.20)
Let
ff*(«,Z)=
2
g(<0 (fl)
?
( 2 - 21 )
and
P}fl («) =
2
^ T 2 * I »*(«» ^)/'( s ' *) I' 0 - 1» 2).
(2.22)
172 No. 5
PAN et al.: A SIMPLE PROOF OF CHEN'S THEOREM (1, 2)
603
Since fir*O, *)/.(«, X) X-ds a-IT
S
gic(s,
*)/.(«, X) O-ds
j\-iT
S
= 0 (f (S •-* ) ( S •-*)) = ° ( ^ ) - OCO ,
(2.23)
therefore from (2.14)—(2.23), we have
2it J 1-/T
+ 0
|S|
2sr J o-iT
\S\
(2 24)
(r^r-)-
-
^ log^^ x J 4)
By Schwarz inequality, we have
pj!l(0<(
V
x(
- f r S * lflfc&*)l2V
2 2'Dt<«<2'+'C1
S
Take
<
P ( 3 ) X,
'
- f - S * !/«(«.*> I'V 0 = 1.2).
ff=(2'Z>,)2.
(2.25) (2.26)
Then we have from Theorem A that
« (fl" + 2 fc 4,)^ logx « a;^log-fl+1 x for s = — + &(— T < i < T). Let 2RH < I7 < 2R+1JT and
2 r)
/l (s, Z) =
2y&£, for 0
2'H<»<2'+'H
y.
Then E « log2^ and
Hence, it follows by Theorem A that
W
'
d.X(«) ;
for r = B
(2.27)
173 604
SCIENTIA SINICA
2>DJ V E
\ A,
Vol. XVm
&Dj
« (— + —V log2 x « log"* y • log2 x
(2.28)
for s = a + « ( — 2" < tf < T ) . Substituting (2.27) and (2.28) into (2.24), we have max ItflU, &) « * log~a+4 a; « a; log--4"3 a;.
(2.29)
Hence the theorem follows from (2.6), (2.12), (2.13) and (2.29). We deduce from Theorem 2 immediately (Cf. Pan Cheng-dong [6]) Corollary.
Under the conditions of Theorem 2, we have
(o,«)=l
= 0(-^-Y
(2.30)
M)Ae»*e v(g') denotes the number of prime divisors of q and B = 2A + 24.
Remark. Starting from the function
A
2
(w) 1( «* —»
oB=/(mod q)
we may prove the similar result too. HI.
1. Let rj = 2
THE SIEVE METHODS
Mean Value Theorem of Bombieri-BuHoapadoe
e, where s I < —\ is any given positive number. V 4J
Let q be a
positive integer and § > 0. Let £(x, q, £) denote the set of all integers with the form k = qm, where m ^ — and the largest prime divisor of m is not exceeding f. Further let
«(*,«, s) -
3 (fc)
" I <"<>) I m a x
S
*«fl(*.«.« where jr(j/ } fe, I) = 3r(y, 1, A, 0 .
Theorem B.
v<
*
(/
max '* ) = 1
*c* *, o - ^ r ,
(3.D
For any positive constant A, we have Biz, 1, as*) = 0 ( y ^ ) ,
(3.2)
174 No. 5
PAN et al.; A SIMPLE PROOF OF CHEN'S THEOREM (1, 2)
605
where the constant implied by the symbol "0" depends only on e and A Cf. E. Bombieri [1] and A. H. BHHorpaflOB [15]. 2. Brun's Method Let
2 s=S y ^ x be two integers. Let a,q;dit
( 1 < i < r)
O)
be a sequence of integers satisfying O,2) = l, «,-^0( mod p,.), ( l < t < r ) ,
ff<
(3.3)
where 2
(3.4)
Denote
(3 5)
*« - •- n ^ 4 n (i - r-hv)' Pi«y P — 2 p> 2 \
CP — 1 )
-
y
where y is the Buler constant. Theorem C. Suppose that C is a positive constant. Then there exist two nonnegative and non-decreasing functions A(ff) and A(«) (0 < a ^ C), each of which has only finite discontinuities, such that
V
<
(5!^ <
PM (X) g ,
V
V
V W ;
l0gX
}
A («)
C
V
' ^lia:
«p(2)logf
\
\
(1 + 0 (Ml2£3£)\ V
V l0gX
)}
(3.6)
+ O(log>x-R(x,q,(^yyj holds uniformly in a and (a>). In fact, we may take
Ml-± A(a)-{ V ^
^Cft+DO™), for«>7, (2& + 4 ) ! ^
(3.7)
If), for a < 7, and A(a)=
V
A
(2*+ 5)!
z1
(38)
lA(8), for a < 8, where A = 1.5 + £ and r = log 1.5 + e, in which s is any given positive number.
175 606
SCIBNTIA SINIGA
Vol. XVHI
3. Selberg's Upper Bound Method r
Let c > 0,P = TT pi and £* < —.
Kd)
^
for d\P, where /(ft) = (p(ft) TJ
Let
/<*) '
u
&<* /(O
(3 9
- >
p
~ 2 . Further let P— 1
PI*
(3.10) d2\P
d^P
where [dl!d2] denotes the least common multiple of ^ and d2. Theorem D. The estimation P.C.*. Q, I ) < QCx, Q, i) + OOog2 x - B(x, q, £))
(3.11)
and
(3.12) hold uniformly in a(0 < a < 6) and (w), where 4er,
for 0 < a < 2, ?25I
A(a) =
a
,
for 2 < a < 4 ,
~' ~f ^ f
(3-13)
2 ^ ^ /Or4<«<6, a - 1 - — log — + *(«) 2 2 in which *(o) = f f
— 4.
^
dfcfa, (a > 4).
(3.14)
Byxuira6 Method.
Theorem E. Let X(a) awi A(a) fee two functions with the properties as those stated in Theorem C. Then the functions defined by [max V(x(a), A(/?) - ('"' ^&A,(o) = i J«-i 2 U(a), /or
0 < a < 1 + 6,
dz\ /
for l + e < « < / 9 < C ,
(3.15)
176
No. 5
PAN et al.: A SIMPLE PROOF OF CHEN'S THEOREM (1, 2)
607
and
fmin V(A(a), A(/?) - Lf""l1 ^z A^a)-] ~ lA(a),
dg\ /w 1 + s < « < 0 < C, ' (3.16)
/wO
/tave Wie same properties as those of the functions X(a~) and A(_a) respectively. We refer A. A. ByxiiiTafi1141 and Wang Yuan1101 for the proof of Theorems C and B and Wang Yuan wl for Theorem D. IV.
THE PROOF OF THEOREM 1
1. The Estimation of Fw(a, /?). Let /? > a > 1. Let
r*(«,£)= where p denotes prime.
2
(4-D
P.(x,m{jf),
Then from Theorem B, we have
+ O(—I—V 2. Let P =
JJ
(4.2) The Estimation of Q.
p. Let a; = i/ be even integer.
Further let a = 1, q = 2 and
_l_e_ 2<*4 ?
dt = a;(t = 1, 2, • • •) • Then we have
£< S
2
«. + oc*i)<2
+ O(*i) = 2 2
^K
S «-( 2
2 ^-P.P2, [di,*],*) + OGB*).
Obviously, ([d,, d 2 ], ptp2) = 1 and ld = O(log«).
£ <
2
2
O2
Hence we have
2 X*M (pctd,,1^])
+ O| log1*-
2
lA*(d)|3*'>
2
*(*,ftft,d,*)--^-
177
608
Vol. XVHI
8CIENTIA 8INICA
+ o(**) < 2 S S v * „([££]) + O( log's- 2 \ \
d<*i U,*)=l
IfOOlS"™
Z
fW<*> *>*>*>-~Jk\
» 2 *3°
WK
\ \
' / /
/ /
(«,<»= X
+ OQxi), where
/(a)-|
i i /x\± fl, for a = PiP2 and a;10 < pt < a;3 < p 2 < (— I ,
W
1.0, otherwise.
Therefore it follows from the Corollary of Theorem 2 (for A = 5) and Theorem D that fi<(8er
2
"
V (6 M 2 log-^-
+ 8)-°*2-(l + 0(-±-))
^
^
W ^
P1P2
10,(2 -
i)
< (»«' j" ^ n ^ * + ») ^ 0 + 0 (-V)) < (7.01474 + 2<J) - ^ - ( l + 0 ——")), log2 a; \ logix"
(4.3)
for s sufficiently small, where 5 = o ( l ) as s —> 0. 3) We have A0(7) = 13.95578 and /l o (8) = 16.00624 by (3.7) and (3.8) and A,(«)(0 < a < 6) is given by (3.13). Take /? — « = 0.01. Then we have ^ 0 (3 + 0.01i)(0 < i < 400) by Theorem E. For examples, X0(6) = 11.90332, A0(5) = 9.77058, Ao(4) = 7.41296 and Ao(3) = 4.44824. Take also /J — a = 0.01. Then we have Xi(.a) and A ^ a ) from A0(ce) and A0(a) by Theorem B, for example, ^ ( 5 ) = 9.87844. 4) Let x = y be even integer. Further let
M = -i2
2 1
1.
Let o = 1, q = 2 and di = x(ji = 1, 2, • • • )•
P.(a;, 2 R a;*) + — + 0 (*»).
(4.4)
2
Then from Theorem B, (4.2), (4.3) and 3), we have
P,Qc, 2, xh -M> 2^(5) - |-[], \
y ° d« - 3.50737 - *) -^~ 2
2 Jfff s(5 — z)
/ log a;
X ( l + 0 (—-—•")) = 2 (9.87844 - 10er (2; — — ^ Mog^a;^ ^ J.(5-2ZX2z-l-«logs)
178 No. 5
PAN et al.: A SIMPLE PROOF OF CHEN'S THEOEEM (1, 2)
- 2er log — 3.50737 - d) X - ^ _ (l + 0 (-^—)) 1.5 / log2 a; \ Hog* a:" Cx > ' (l + o (—^—^\) > 1 (for sufficiently large x). 10 log2 x \ Hog* a:"
609
(4.5)
Let p denote prime. 11 p'\x and p' | (a: — p), then p = p. Since the number I 2. of prime divisors of x is 0(a; 6 ), hence Pw(.x, 2, a;10) + O(x10) is equal to the number of primes p satisfying 2 < p < x, if p' | (a; — p), then p > xr°.
(4.6)
Since the number of primes p satisfying (4.6), such that x — p has a prime divisor l
1
p > xw, is P<£.x, 1p, a;10) and the number of primes p ( < a;), such that x — p is I
divided by p'Kp > »10)> is at most f>*
10
hence the number of primes p satisfying (4.6), such that x — p has at most 2 prime xx
XX
factors in xw < p' ^ a;3 or 1 prime divisor in s;10 < p' < a;3 and 2 prime divisors > Xs, is not exceeding M. Consequently, it follows from (4.5) that there exists a prime p such that x—p has at most 2 prime factors for sufficiently large x. The theorem is proved. REFERENCES [ 1] [ 2] [ 3] [i ] [5] t 6] [ 7] [ 8] [ 9] [10] [11]
Bombieri, E.: On the large sieve, Mathematilca, 12 (1965), 201—225. Chen, Jing-run: On the representation of a large even integer as the sum of a prime and the product of at most 2 primes, Kexue Tonglao, 17 (1966), 385—386. Chen, Jing-run: On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Set. Sin,, 16 (1973), 157—176. Estennann, T.: Eine neue Darstellung und neue Anwendung der Viggo-Brunschen Metode, J. Sei. und Ang. Math., 168 (1932), 106—116. Gallagher, P. X.: Bombieri's mean value theorem, Mathematika, 15 (1968), 1—6. Pan, Cheng-dong: On the representation of large even integer as a sum of a prime and an almost prime, Ada Math. Sin., 12 (1962), 95—106. Praehar, K.: Primzahlverteilung, Spr, Ver, (1957). Biehert, H. E.: Selberg's sieve with weights, Mathematika, 16 (1969), 1—22. Wang, Yuan: On the representation of large even integer as a sum of a prime and a product of at most 4 primes, Aata Math. Sin., 6 (1956), 565—582. Wang, Yuan: On the representation of large integer as a sum of a prime and an almost prime, Set. Sin., 11 (1962), 1033—1054. Bap6aH, M. B.: HoBbie npHMeHemm 6ojibuioro pemera KD. B. JlHHHHKa, Tpy. HH. Mai. u.u. B. M.
[13]
PoManoecKoeo, 22, 1961. Bap6aH, M . B.: IlnoTHOCTb Hyjieii L - p w o B .iwpHxpe H 3 a « a q a o cjioweHHH npocTbix H n o i r a npocrbix •mean, Mar. c6, 61, (1963), 419—425. ByxiuTafi, A. A.: HoBbie pe3yjibTaTH B HccjieAOBaHHH npofijieMH rojibA6axa-3ftnepa n npoSjieinbi
[14]
n p o c m x iHcan 6jiH3HeiWB, RAH CCCP, 162 (1965), 739—742. ByxuiTa6, A. A.: KoM6HpHaropHoe ycnjieHHe Merafla spaToccjjeHOBa peuieTa, VMH
[12]
(1967), 199—226.
CCCP,
22,
179 610 [15] [16] [17] [18]
SCIBNTIA
SINICA
BHHorpaAOB, A. H: O IMOTHOCTHOH ranoTe3e unit L-paacm flHpnxpe, HAH
Vol. X V I H CCCP,
cep.
Mar.,
23,
(1965), 903—934. JleBHH, B. B.: Pacnpe^ejieHHe "noMTH npocTbix" qnceji B neji03Ha>!Hbix noJTHHOMHajibHux, nocjieflOBaTejibHOCTHX, Mar. c6., 61 (1963), 401—419. FlaH, MSH-AYH: O npeAcraBJieHMH qeTHbix qaceji B Bane cyMMbi npocToro H HenpeBOcxo^iimero 4 npoCTbix npoH3BeAeHHd, Soi. Sin., 12 (1963), 455—474. PeHbH, A.: O npeACTaBjieHHH qeraux iHceJi B BHfle cyMMH npocToro a noTra npocToro <mcen, HAH CCCP, 2 (1948), 57—78.
180
REMARKS ON A THEOREM OF DAVENPORT* WANG YUAN Institute of Mathematics, Academia Sinica Received 6 January 1975
We use x, a, c to denote the n-dimensional real vectors, \x\ = \{xu...,xn)\
= {x\ +
---+x2n)1'2
the modulus of x, and A an n-dimensional lattice, i.e. a set formed by the vectors
where a i , . . . , a n are a given linearly independent vectors in n-dimensional Euclidean space and u\,...,un are any integers, { a i , . . . , a ^ } is called a basis of A. In this paper we shall use Brun's method to prove the following: Theorem 1.
Let A be an n-dimensional lattice and c»(l
be any given n - 1 vectors. Then there exists a basis { a i , . . . , a n } of A such that ai
- iVcil = O(log 3 N),
1 < % < n - 1,
holds for any number N(> 2), where the constant in " 0 " depends only on A andci's. This gives an improvement of a result due to Davenport. In his original result, the error term should be replaced by O(Ne), where e is any preassigned positive number and the constant in "O" depends on e,A and c,'s. Besides its interest itself, Davenport's theorem is also useful in the study of diophantine approximations and geometry of numbers (see H. Davenport [1], J. W. S. Cassels [2] and C. G. Lekkerkerker [3]). We may also obtain some improvements on generalised Davenport's theorems by similar method. L e m m a 1. (Brun) Let Q be a positive integer and pi < p 2 < • • • < Pr be a set of prime numbers not dividing Q. Let P(Q;pi, • • • ,pr) be the sum of non-negative real *Acta Mathematica Sinica, 18:4 (1975) 286-289.
181 numbers an's, where n satisfies the following conditions: n > 1, n = 0(mod<2), n ^ 0 (modpj),
1 < i < r.
(1)
In particular, P(Q) is the sum of an's with n>l,
n = 0 (modQ).
(2)
If there are M and i? depending only on an's such that P(Q) = ^ + O(fl),
(3)
hereafter the constants in "O" are absolute constant, then there exists an absolute constant a (>0) such that
(4)
P(Q;Pu:.,Pr)>^-f[(i-pj+o(P^R).
We refer to G. Ricci [4] and Wang Yuan [5] for the proof, but the ind n in [5] should now be replaced by n. Lemma 2. Let q be an integer >2 and s,t be integers such that (t,q) = 1. Then there exists an absolute constant K such that there is a positive integer u satisfying (tu + s, q) = 1 in any interval with length not less than i^log q. Proof.
1) Let G be a real number, H = Klog q and q\ = \\
p, where p
p^q
V < log5 q
denotes prime number, 5 = Then
2
£=
Q99 and K is a constant which will be defined later. Y,
l>Si-E 2 ,
(5)
G
where Ex =
Y,
1 and S 2 =
G
^
^
1.
p\q G log6 q tu + s = O(mod p)
Since the number of prime divisors of q is at most 2 log q and
(6) G
m holds for (t,q) = 1, hereafter 6 denotes a number satisfying \6\ < 1 but not always the same in different occurrences, we have
V (x+e)<-^r- Vi + E 1
s2=
p > log6 g
< 2 F l o g 1 " % + 21o gg <4ii'log 4 -' 5 g.
(7)
Set ( l , if n = tu + s, an = < (^0, otherwise,
G
+ H;
in Lemma 1. Let pi < P2 < ••• < Pr be all prime numbers satisfying p|g and P < log"5 ?. Then Ei = P ( l ; p i , . . . , p r ) > ( 7 i J
JJ
A-^+O(p2-99),
P < log5 g
by (6) and Lemma 1, and therefore
lo 2 1 si > ^ 7 r r - fi + ofr^—)) +°( g "^) ^ ^f^ ' (8) lo lo
<51oglogg \ \ g g9// loglogg by Mertens theorem, where 7 is Euler constant and r is an absolute constant. By (5), (7) and (8), we know that there exists an absolute constant qo such that log log q 2) Suppose 5 < qo. By Eratosthenes sieve, we obtain
s
=
£
1= E
G < u < G + ff (tu + s,q) = 1
G
=2 ^ d|?
E Md) d\{tu+s,q)
E
G<«
= H]\(l--\+6q>vK
i = "E^r+0Ei
ff
d|9
d|?
log5"1 g + 6q,
where v is an absolute constant. Since q < qo, we may choose K sufficiently large such that E > 1. The lemma follows from 1) and 2).
183 Lemma 3.
Let b j , . . . , bn be a basis of A and n a, = _ ^ Vijbj,
1< i <m < n
j=i
be a set of vectors in A. Then a necessary and sufficient condition that a i , . . . , a m can be extended to a basis a i , . . . , a^ of A is that the determinants of all m x m submatrices of (vij),
1 < j
1
have no common factor. See J. W. S. Cassels [2]. Proof of Theorem 1.
Let t>i,..., bn be a basis of A and n
Ci = YirVhi>
1 i n
(9)
^^ '
i=i
where r^'s are real numbers. Now we proceed to show that we may choose a basis n ai = _ _ U y b j ,
l < i < j
(10)
of A such t h a t
v^ =Nrij+O(\og3N),
1 < j < n,
\
(11)
where A^ is any given number > 2 and the constant in "O" depends only on n and rjj's.
We can choose Vij's with the property: For any integer I < n, the integers Ri = det(vij),
1
1 <j
and SI = d e t ( v i j ) ,
l < i < I ,
2 < j < I + l
are all nonzero and coprime. In fact, we may prove it by induction. Suppose that / = 1. Let v\\ be the integer which is nonzero and has the least distance with Nru and let W12 be the nonzero integer which is coprime with vu and nearest to Nr\2- If j > 2, we choose t>y to be the integer nearest to Nrij. Hence (11) holds for i = 1, j ^ 2. Suppose i = 1 and j = 2. If v\\ = ±1, then v-12 clearly satisfied the above conditions and (11). Otherwise, take t = l,s = 0,q = \v\\\ and G = Nr^, it follows from Lemma 2 that we may choose V12 such that the above conditions and (11) are satisfied. Hence •Ri = vu and Si = v\2 satisfy our requirements.
184 Now suppose that I > 1 and that v^s are well defined for i < I. Choose vij to be the integer nearest to Nrjj if j ^ / , / + 1. Expanding Rj and 5 / according to their last columns, we obtain
Ri = ±vnRi-i
+ A and
5/ = ±vIJ+1Si-i
+ vuB + C,
where A, B, C are definite integers. By the assumption of induction, i?/_i and S7_i are nonzero, (-R/_i,5/_i) = 1 and 5/_i = O(NI~1), since 5/_i is a sum of (I — 1)! products of I - 1 Vij's. If S/-1 ¥" =tl> then by Lemma 2 with £ = ±.R/_i, S = A, q = |S/_i| and G = iVr// we may choose VJI such that Rj ^ 0, (i?/, 5/_i) = 1 and w/7 " ^ r / 7 = O(log 3 15/-!|) - O(log 3 N). If 5/_i = ± 1 , we may clearly choose VJJ satisfying the above requirements. If Rii 7^ ± 1 with this vn, then by Lemma 2 with t = ± 5 / _ i , 5 = vuB + C, q = |i?/| and G = Nrjj+i, there exists w/,/+i such that Si ^ 0, (SI,RJ) = 1 and w/,7+1 - NrItI+1
= O(log3 |i?i|) = O(log 3 N).
Otherwise, if Ru ^ ± 1 , we may choose evidently w/,/+i satisfying the above conditions, (11) follows by induction. Since {Rn-i, SVi-i) = 1> it follows from Lemma 3 that we may choose an such that a i , . . . , a n is a basis of A, and finally ai-Ncil
=O(log3N),
l < i < n - l
by (10) and (11). The theorem is proved. References H. Davenport, On a theorem of Furtwangler, JLMS 30 (1955) 186-195. J. W. S. Cassels, An Introduction to the Geometry of Numbers (Springer, 1959). C. G. Lekkerkerker, Geometry of Numbers (North-Holland, 1969). G. Ricci, Sur la Congettura di Goldbach e la costante di Schnirelmann, Ann. del. R. Scu. Nor. Sup. di Pisa 6 (1937) 70-115. [5] Wang Yuan, On the least primitive root of a prime. Ada Math. Sin. 9:4 (1959) 432-441.
[1] [2] [3] [4]
185
Vol. xx No. i
S C IE N T I A
j.™.. Feb. 1977
S IN I C A
ON JIHHHMK'S METHOD CONCERNING THE GOLDBACH NUMBER WANG YUAN (3E
7t)
{Institute of Mathematics, Academia SiniccO Received February 5, 1976.
ABSTRACT
In this paper, some conditional results concerning the Goldbaeh number are proved. ple, assuming that the density hypothesis of §(s) is true, the inequality "i
(ln.AO ' 3
,.
For exam-
\N — p — p'\ =£ C(E)
always has solutions; here e is any pre-assigned positive number and p, p are primes.
It seems that there is a gap in JIHHHHK'S original proof of the similar theorem.
I.
INTRODUCTION
The even number that can be represented as a sum of two odd primes is called the Goldbaeh number. The binary Goldbaeh conjecture may be stated that every even integer > 4 is a Goldbaeh number. Let 0 < v < 1/2 and T > 0. Let N(T, v) denote the number of zeros of Riemann's zeta function f(s) in the rectangle -• + v < < r < l ,
U| < T .
(1.1)
Now, let us state the Riemann hypothesis ( R ) and the density hypothesis ( D ) as follows:
(R)
N(T,v) = o, (o
r>oY
and (D)
N(T, v) = OCT^-^Cln (T + 2 ) ) O ,
(o < v < — + e, \ 37
T > 0), /
where r and c ( > 0) are constants, g is any pre-assigned positive number and the constant implied by the symbol "0" depends on e only. Since
ivT0(r)>cJv'(r,o), ( T > T 0 ) ,
(1.2)
where c and To are positive constants and N0(T) denotes the number of zeros of f(s)
186 No. 1
ON JIHHHHK'S METHOD CONCERNING GOLDBACH NUMBER
17
satisfying
|*| < T ,
(1.3)
(cf. A. Selberg [4] and N. Levinson [ 2 ] ) , there is no need for us to consider the case v = 0 in (D). Since £(s) has no zero near the line a = 1 (cf. H. M. BiraorpaaoB [5] and H. M. Kopo6oB [ 6 ] ) and N(T, v) is comparatively small for — + e < v < — 2 37 (cf. M. N. Huxley [ 1 ] ) , we need not consider the case 37
h £ < u < —• either. 2
We use N to denote even integer, p, p prime numbers, c positive constant, e any pre-assigned positive number and c(e) positive constant depending on e only. However, they are not always equal to the same values. Theorem 1.
Suppose that
N(T, vXc(e)2 71 - 2 »ln(r + 2),
(o < v < — + eV v 37 *
(1.4)
Then for any given N, there exist p, p' such that 148,,
\N — p — p'\
(1.5)
Suppose that
F(r,v)
(o
37
(1.6)
/
Then for any given N, there exist p, p, such that \N — p — p'\ < c ( e ) ( l n i V ) 3 + 5 .
(1.7)
In general, we may prove Theorem 3.
Suppose that \cTl~2v (M(T + 2))r,
forO
#(2»<J
1
Lya-^/ci+e,)
for
(1-8)
a < v <
where a, r, e t are constants 0 < a < 1/2, r ^ — 6a and Sj > 0. iV, f/iere exist p, p such that
T/iew /or <ww/ grwew
\N - p - p'\ < c(e) (ln^)C3+r)o-2«)-'+e_
(L9)
We have also Theorem 4. Let q and N he two given positive integers such that q
e(s), where h satisfies 0
187 18
Vol. XX
SOIENTIA SINICA
The proofs of these theorems are based on JIUHHHK'S method (ef. KD. B. JIHHHHK [7]). Theorems 2 and 4 give some modifications of the corresponding theorems in [7], where JIHHHHK also stated a result that under the truth of (1.4) with 0 < v ^ 1/2, the right-hand side of (1.5) might be replaced by 0((lniV) 7 ). The present author, however, is unable to understand his arguments. It may have existed a gap in his proof. (See Remark in V.) II.
SEVERAL LEMMAS
We use the notations x = N~r + 2itiS, ( 0 O < l ) , <W~
2
lnplnp',
(2.1) (2.2)
(2.2)
«,(&,#)= S ^ T C O ,
(2.4)
where 2 j denotes a sum over all zeros pk = fik + itk of f (s) in the critical domain 0 < C T < 1.
S2(S,i\O= 2
x-^rCft).
(2-5)
flf,(d,^)= 2
^TCft).
(2.6)
Lemma 2.1. For — 1 < cr < 2, we Ziave
r(«)
d9 = ^ 1 e" w « /w .
a;
e
Jc (jv-i
+
2^3)2
^g
'
where C denotes the closed contour which contains a segment [ — B, R ] and a semicircle jBe'^O <
^
A(n)A(n)
= Q(N) + O(*/~N(.lnNy').
Lemma 2.4. Suppose that Nx = JV + H, where H denotes an integer satisfying H = O(VN (In A^)2). Then
188 No. 1
ON JIHHHHK'S METHOD CONCERNING GOLDBACH NTTMBE3
19
Q(NO = e \ \ 8(9, Nye2'iN^d& + O(ViVCln^)2).
(2.7)
Lemma 2.5. We have tf)2)-
8(9, 2V) = x-1 - 8,(9, N) + O((ln
(2.8)
Proof. From Mellin's transform, we have Ks)x-!jr
flf(d, A") = - ~—\
(2.9)
(s)ds.
Obviously, we have
#"~*
==
\\x\s
e-'°(?-
e*'-'""*^
First, suppose that 9~^Q.
arct
*sy,
for
9 > 0, C2 10")
Shifting the line of integration in (2.9) to the line
a = — —, we have
8(9, N) = x"1 - 8,(9, iV)
— [~*+'°° T(s)x~' -£ (*)<&•
2nri J - { - • • »
Q
(2.11)
Since - f ' ( - — + **) = O(ln(|*| + 2 ) ) Q \
2
(2.12)
/
(cf. Prachar [3]), we have by (2.10), (2.11) and Lemma 2.1 that
rT+'" T(s~)x-S -£-' («)<& « r*"1 e~'""g^~\ntdt J-|-i»
^
J2
« f" r : e"' " c t K ^ In tdt « 1,
(2.13)
for the ease N9 < 1. Now, suppose that iVa ^ 1. Then —^— <arctg —^— <—^—, ixN9 2xN9 2TTN9
and J-i-.oo
r(s)x~' -^ (s)ds « Q
+ f"
J2
(2.14)
e "»N» r 1 In trf*
e'l^TF r 1 In *d* « (In N)2.
(2.15)
Substituting (2.15) into (2.11), we have the lemma. The ease 9 < 0 may be treated similarly. The lemma is proved. Lemma 2.6. (JIMHHHK [7]). Suppose that -r\ satisfies — > 77 ^ 4iV-1. T&ew we have 4
189
20
SCIENTIA 8INTCA
P' | 8&,ff)| 2d9 « S
E (**. + 1)'* "*
X (t fc + I)"*,"* i?1-**,-"*.
^
I h1 — hj + 1
and
f2' i«3(&, ^ ) i 2 ^ « S X
Vol. XX
S
CI**,I
e-
(
'^+%'arctg7^r;
(2.16)
+ D"**-*ci**,i +1)**.-*
1 ^"^"^(-""-sW). I *»,-**, I + 1
(217)
We need also gome results concerning the critical zeros of £ ( s ) . Lemma 2.7. In any interval it, t + 1), the number of critical zeros of f ( s ) does not exceed c l n ( | t | + 2 ) ( c / . Prachar [ 3 ] . ) Lemma 2.8.
(BHHorpaAOB [5]-Kopo6oB [ 6 ] . ) f ( s ) Tias MO sero in the domain
(2.18) Lemma 2.9. (Huxley [ 1 ] . ) We to« 37 iV(r, v) «
3Cl-2i>) 2.2(1+2.)
+ r f
/
w
8 . _
21 (2.19) <
v
<
1
)
where the constants implied by the symbol "<3C" depend on s only. From Lemmas 2.1 and 2.8, we have Lemma 2.10. Suppose that 0 < 9 < 8JV""1.
Tfeew we 7iaw
^ ( 3 , iV) = O(JVe- f
(2.20)
Lemma 2.11. We have f \8(9, N)\2d9 = —N]nN+ 0(Nlnln3N). Jo 2 HI.
(2.21)
JIHHHHK'S METHOD
Lemma A . Suppose that N~i < K = X(iV) < — . If 4 (" \81(9, N)\2d9 Jo
« JV(lniV)- £ / I 0 ,
(3.1)
hereafter the constants implied by the symbol "
190 No. 1
ON JIHHHHK'8 METHOD CONCERNING
GOLDBACH NUMBER
21
\N — p — p'\ « K"1 O-nNY'2.
(3.2)
J ( # i ) = \* S(.S,Nye2*iNSdd. J-« Then by Lemma 2.5 and the Schwarz inequality, we have
(3.3)
Proof. Let
(3.4)
d9 + B, -*
2
x
where |B | < 2 ( " |ifif^d, iV>|\te + OCBO,
(3.5)
Jo
in which
«, = J\*-^\* iS^NWdS + A/«r/^(lniV)2 V J» |*| 2 Jo
\
Jo \ x \ 2
+ y/» j * | ^ ( d , i V ) | 2 ^ ( l n i V ) 2 + (lnJV)4. From Lemma 2.2, we have rs j s * " ^ ^ = J-« a;2 Since we have
N
-*,/„ + 0([°°d9\a Nie-N>/N + 0 ( K - 0 2 \J* d /
Jo | x | 2
(3.6)
(3.7)
Jo N-2 Jw-« »>
(3.8)
it follows by (3.1), (3.4), (3.5), (3.6), (3.7) and (3.8) that J(2V,) = iV16-N-/N + O(iV (In iV)- 6/M ).
(3.9)
H = K-'OnAO 672 ,
(3.10)
Denote and
r(9)= S
e2""9.
(3.11)
0
Then we have
|T(9)| < |»|-l
(3.12)
for d 6 [K, 1/2] and 9 € [ - 1 / 2 , - K ] . Let r = [4/e] + 1 and a; = xy + • • • + xr, where 0 < Xj < if, (1 < j < r ) . Further let ^ denote the integers of the form N + x. Then we have from (3.9), (3.12) and Lemmas 2.4 and 2.11 that
2 9(^0 = « S f, S(d, #)2e2'"7'.9
+ O(-v/¥ffr(ln^)2) = e S Ntf-W + (KNH'Qn iV)-£/20)
191_ 22
SCIENTIA SINICA
Vol. X X
+ 0 (fl"r(ln N)~"/2 [* 18(9, 2V) 12d») = eN 2 e-'(l + 0(2\r».ff)) + O(i\rfl"r(ln #)- 6/2 °) = NHr(l + O(ln i^)-E/2°) > 2, for TV > e(s).
(3.13)
The lemma follows. IV.
T H E ESTIMATION OF INTEGRAL
Denote M= [21nNl
and Mt= [inN - cQnN~)i] + 1.
(4.1)
In this section, the constants implied by the symbol "
(o < v < — ) ,
(4.2)
where r is a constant and ^(v) is a non-increasing function satisfying — ^ /t(v) ^ 3. Then \8l(9,N)\2dS
t
~ s ~ A -S-^"^-M-A 1 ^(i n _^)2+r. (4.3)
T-=2Nn.
(4.4)
^]-^"^
1) Let
Then we have #^2^8,
-J-^arctg^— >—.
(4.5)
Hence it follows by Lemma 2.6 that P' \82(9,N)\2dd«Z1
+ S2!
(4.6)
where
'B 2 >0
X
"u ! 1^ , e-^,+^'^, I **, — hj + 1
(4.7)
and S2 is obtained if the condition pki < /Sfcj in It is replaced by /Jfcj > /9fcj. Clearly 2:, < ! ; „ + 212 + Z13,
(4.8)
192 No. 1
ON /IHHHHK'S METHOD CONCERNING GOLDBACH NUMBER
23
where
OO^OVr
0
X
ttcfi'NT
X
i-
(49)
e-(%+tK)M*T
0<«t 2 « t l
1 e-<'*1+<*2V4«rj I *Jc, — tkt I + 1
(4.10)
and J o is obtained if the condition concerning tki and t/Ci in S12 is replaced by 0 < **,<**, and tk2>NT. 2) Obviously, we have «
tr+l)T
^•12« 2 J
2 J ^Si>?
2_I
s=N
PKl
2J
|, _
%
1=1
I'fc,
f
, ,
1
»fc,l + -
e
1
Hence it follows by Lemma 2.7 that
s a « n~lT S
" S i5 ^ L se~'/411 « 9?"12'2 S ^ln sr > 3 e~sM"
S
« JV2 (In iV)3 e-N/8» 2
s l n s 3e
) "'/8" « ! •
(
(4-n>
Since 2 13 satisfies the same relation as ^ 12 , we have by (4.8) that (4.12)
Sl«In + l. 3) Let Im denote the interval we have
—+ L2
m
~ , M
2
1- — . Then from Lemma 2.8, if-I
2n=i>u(m),
(4.13)
».= i
where
o« B l <wr
X
^
o<«B2
e-»*, + '*, Wrf .
(4-14)
Z u (m) « 2ffl(ffl) + Zn 2 (m),
(4.15)
I**,-'*,! + 1
Obviously, we have
193 24
Vol. XX
SCIENTIA SINTCA
where
2u.w = 2 X
and
2
2
<(Cj<(/+2)r
17 ? I o - i «" ( '*' + l *' V 4 r f . I »*, — ffc21 + 1
2i»(m) = 2 X
(**. + D'* "* (**, + D^-V-'*-'*•
2
2 e-
I**,-**,! + 1
(4-16)
(**, +1)'* "* (**, + D"*.-*^*.-**.
(t +
K 'K)M«T.
(4.17)
4) Suppose that m > 2. Since 1 « (iVr)1/M « 1, we have
2n,(m) « §
((s + l)Z0»*-y-w*. ^
2
ln(s+ 2 ) y
e-'^
sT
« ^ V " * 0 n ^ ) ! 2 0 + i)WMe-'/4"iv ((* + i)y,
TO
)
« r ^ " + " ^ ^ ~ ^ 7j-2WM (In ^) 2+r 2 ] (s + 1)"^ + "^X 1 '"^ e-'/4" « I «
v
«A
M'V^MA
"''(lni^)2^.
(4.18)
For the case m = 1, it follows by Lemma 2.7 that N-l
2m(l) = 2
/ =o
(In (s + l)r)2e-'/4*iVr((s + l)r, 0) N-l
« T(ln iV)3 2
(s + l)e- 5/4 " « iVij (In N)3.
5) From Lemma 2.7, we have
J,u(m) « I7"1 2 x
2
2
*=[(j+2>n
p«,
« r- 2
/ =o
((« + l)r)"*-* i,l-w*, e-'/te
2
2
(**1 + i)"*.-ie-*'/4"T
Pkt,pkle'm sT
(Cs + i)T)'*r*,'-»». «-'/*•
(4.19)
194 No. 1
ON JIHHHHK'S METHOD CONCERNING GOLDBACH NUMBER
25
00
X
(x + 2yh~i
2
e~*MltT In (a; + 2)
N-l
« T~l 2
( 0 + l)Z7)**i-H1-J/i*.e-'/4"
2
a;**.-* e-'Mln (x + 2)dx
X f°° J [CJ+2)T]
N-l
« 2
((« + l)27)w*r1»71-2''*«e~"'4'
S
(4.20) X f°° /fc,->e-» f <
/4lt
2m . fra-lV,
ln(j/ + 2)dt/
2m\
I ^•^"("B"X
7m-]V, Zm\
"="; ,^-B-A'-ir; ( i n Ny+rf
tor m>2,
(4_2Q)
2
[^(lniV) , for m = 1. Trom (4.12), (4.13), (4.15), (4.18), (4.19) and (4.20), we have S1 « NrjQnNy + 2 iV^" + " ( 2 ^ i ) ( l "^ ) ^"(^X 1 -^-) ( l n iy)2+r> m=2
Obviously, X2 satisfies the same relation as Si.
(421)
Therefore we have by (4.6) that
f2"1 |S2(d, N)\2dd « ^7?(ln Ny + 2 i^"S" + " ( "^ X '-B-J^I-B-X l -^r) ( l n Ar)2+r_ n>=2
(4.22) (4-22)
Similarly, we may prove that (4.22) holds also, if 82(9, N) is replaced by $ 3 ($, iV)' Since f" 18^9, JO 12^9 < 2 f2" |flf,(d,iV) 12d9 + 2 (2" | &(», ^ ) 1 2 ^ , (4.23) Ji)
Ji
Jij
hence we have the lemma. V.
THE PROOF OF THEOREM 1
Let _148_£
N = K(iV) = (ln iV) 13 2 _
Let £l
= l(T3e and M2 == [2 f— + g^ lnivl + 1 . L \ 37 / -I
(5-1)
(5.2)
Then it follows from Lemma 2.9 and the supposition of the theorem that there exists a positive constant e2 = e 2 (s) such that N(T,v)«\
fT 1 - 2 "ln(T + 2), for v € J m ( K m < M 2 ) , x (1_2U) } , for v e / m (ilf2<m
(5.3)
195
26
Vol. XX
SCIENTIA SINICA
hereafter the constants implied, by the symbol "<SC" depend on e only. Let L be an integer such that
(5.4) Then
« Ji£_ fii\
( ln Ny « ^(in ^)-e/io2 V ,
(5.5)
for 1 < i < L and ^ J\fM + T+i^ m=M2+l 1
«^i+^
M > ( l n JV)2 <<; jyr M + TfT^
M-l/il)l + e 2 V ^2 / +
S
2
2Mi
I+H' « (lnJV)3«iV6
_ S 2 ,flnNl' / 3
2
M ^ (ln
«i\r(lni^)- 2 .
Ny
(5.6)
Hence by Lemma B, we have
(
S/2'""1
_ 13 ,
I ^ C ^ i V ) ! 2 ^ « iV«(ln Ny 2~' + iV(ln iV)-£/1° 2 38
,
«/2'
+ #(ln NT2 « JV(ln NTe/l° 2~^'
+ iV(ln N)~2.
(5.7)
Obviously, we have by Lemma 2.10 that (•«/2L Jo
I^CS,^)!2^^
Ci/N Jo
,„
|-S' 1 (d ) JV)| 2 ^«iVe- £fI ' w/ .
(5.8)
Consequently, it follows by (5.7) and (5.8) that C*
2
J^
CW2'-1
j o |^(d,iV)i (Z&= 2 J s / 2 , +
f»/2L
Jo
|Si(3,^)|2cZ9
I £,(3, JV) 12 da « iV(ln Ar)-e/io _
( 5 9
)
Hence, Theorem 1 follows from Lemma A. We omit the proofs of Theorems 2 and 3. They are parallel to the proof of Theorem 1. Remark. In his original proof of a theorem which is similar to Theorem 1, JIHHHHK used the following arguments: "Hence, it is sufficient for us, if the inequality
jyi+wo (ikY~^ « \ 2' ) or
ik.« or
1
7
N
(lnN)7
1__
2v-
196 No. 1
ON JJHHHHK'S METHOD CONCERNING GOLDBACH NUMBEB
holds for 0 < v < — —
27
^—^" (cf. p. 515 of [7]). If we take r = 1,
and HN = (lnJV)~7, then the above inequality is not satisfied with v =
. 2 (In N)wln Hence it seems to me that the conclusion O( (In iV)7) should be replaced by
VI.
THE PROOF OF THEOREM 4
1) Denote
ffGO-f1'
i0TX
(•0,
Tx=
S
=X
°
((U
>
otherwise,
ZCOe2""'*,
S(9, iV, Z) = 2 A(w)Z(w) e~",
(6-2) (6.3)
n-2
8^9, N, X) = 2 «~ Px r(^),
(6.4)
•where ^ j denotes a sum over all critical zeros of L(s, X). Then similar to Lemma "x 2.5, we have SO, N, X) = ^ x
where
2 - 8&d, N, X) - S2(9, N, X~),
S2(.9,N,X) = 0«lnNy).
(6.5) (6.6)
Since S(— V
+ 9,N)-~
2
/
- ~ J] fxZ(o)flf(d, i^, X) + O((ln iV)2), qp(g) X(J
(6.7)
for (a, q) = 1, where ^ j denotes a sum over all character mod q, and since r%0 = (i(q) and | r z | < \ / 2 for all X mod g (cf. Prachar [3]), we have 8 (— + 9, N\ = ( - f - 2 fzZ(o)(flf1(», tf, X) + »,(», ^ , ^))) 2 + 0(N0n N?) =
'f9!* + - ^ T 72 ( S ^ ( a ) ( S 1 ( 9 , N, X) + 8£9, N, Z))Y qo(2)2x2
f3 2 ^Ca)(i8i(», ^ ; *) + &(3, ^ , ^)) + 0(iV(ln^)2). (6.8) 2
2l q
(' iS^N^^l'dd^NvO-nNy,
J—1
(6.9)
197
SCIENTIA SINICA
28
for — ^
JJ
Vol. XX
^ 2N'1.
2) Suppose that iV1 is an integer satisfying N/2 ^Nt ^N. Let
UN,, ,) = S [I ' «(», A^V"".'**, where ~ > 2q
n
(6.10)
Then it follows by (6.8) and (6.9) that
^ — . N
JoCNi, V) = 2
2 ] e*"™t'« f* #(— + 9, N\ e2'iN^d9
*,!* (o,9,)=l
J~1
\
'
+ 0 ( ^ 0 + O(B2) + O ( ^ ( l n JV)3),
where
(6.11)
*i = S - r - y S j ^ S TXz(o)(/8f1(a, iv, JQ +flf2(»,^, z)) [ ds, (6.12) and B
(' k~' S ^(«)('8 f i(», ^> *) +ft(»,^ ^)) &>• (6-13)
' = S -7^7 2
By (6.6) and (6.9), we have R
i = 2 ^ - ^ r (' E S f ^ ' T, zZ'CoX&Ca, ^, z) +flf,(s,y, x)) «,i«
4
X (|S1(^iV,X)| + O((lnJV) ))^« 2 * ^ (
ln
x?1
J
-i
^)%
(6-14)
and ?1I«
x
(S \T~\
/
S rxZ(6)(Sl(9, JV, X) + S2(9, 2V, Z)) f T d9 1
""" 9,19
X (f'
J-'\M,)=I
/fi 2
S
X"1
^ J - 1 (a.^jij
V'' 2 /
2 fzAf(6)(SI(d, JV, Z) + S2(9, N,X))\ daY2
< 2 ^ - T T (
<
9,19
S^SrvclnW
(O5)
198 No. 1
ON JIHHHHK'S METHOD CONCERNING GOLDBACH NTJMBEB
29
Hence from (6.11), (6.14) and (6.15), we have
+ OCf1))
e^'N^(Nte-N^
UNly , ) = 2 - 4 ^ 7 S
+ 0 ( 2 ^ ( l n t f ) 3 ) + 0 fe /,g\,/2 Nrj«KhiNy»)
= iv-N'/N II Ci + ^ . ( P ) ) + o ( 2 —rV) + 0 ( S 8.^7 (In ^) 3 ) + O ( S
/ J/ ! 1/2 ^ I / 2 (In ^)V2)»
(6-16)
where
ffw.Cp) = '
P
(6-17)
— ( p _ xy;
for pliVi.
3) Let K=(lnJV)~3"*,
Br=(lniV)3+e
and
r=[-^j+l.
(6.18)
Let 2 ] e-2"'9*9.
r(S) =
(6.19)
2r
Let 9Jt be the sum of intervals complementary set of 9JI in
— — —, — + — , 1
.
(0 < o < q — 1) and m be the
Further let Nx denote the integers of
the form Nx = N-q(.hl+
where 0 < h < - ^ - . 2r
J
(6.20)
••• +hr),
Then — < Nt < iV and 2
= 2 f / -»(». ^ye2-'"'^ = I ] /„(#„ 4) + S J>(^» T)'
where
Afc-Wt \
Since
q/
i8f(d,^)2eJ-"''»d9.
y j (jf i i \ = f g(9 NyT(&ye1'"»d9 « «-' P \8(9, N)\2d9 qJ
Jm
(6.22) (6.23)
for 9Cm, therefore by (6.18) and Lemma 2.11, we have
V
-
Jm T(9) « K"1,
i^
(6 21)
J"
199
30
SCIENTIA SINICA
Vol. XX
« Hr (In N)~Er/2 Mn N « ^ - .
(6.24)
Consequently, it follows by (6.16) (notice that WNi(p) = 'P'iv(jp) for p\q), (6.21) and (6.24) that
J = n (i + ^(p)) 2 ^~Ni/N + o ( - ^ 2 -fr-)
+
° (-??-) ~ Ip|I ^ + ^ P ) ) 2 ^i«~"'/w + OCNHranN)-"). (6.25) \ lniv / 9
Nj
Since i\T is even if q is even, hence
II (l + 9 r N (p))>c>0, where c is an absolute constant.
Consequently, we have
J > c Y ) Nle~N^/lf + O(i^ffr ( I n ^ ) " E / 4 ) > », for TV > c ( e ) .
(6.26)
JVfl"',
(6.27)
3e 2 (2r) r
The theorem follows. REFERENCES
[ 1]
Huxley, M. N.: Large values of Dirichlet polynomials, Ada Arith., 24 (1973), 329—346; n , Ada Arith., 27 (1975), 159—169; m , Ada Arith. (to appear.) [ 2 ] Levinson, N.: More than one-third of zeros of Biemann's zeta function are on 0 = 1/2, Adv. in Math; 4 (1974), 383—436. [ 3 ] Praehar, K.: Primzahlverteilung, (1957) Spr.-Ver. [ 4 ] Selberg, A.: On the zeros of Biemann's zeta function, Skr. Nor. Vid. Akad. Oslo; 10 (1942), 1—59. [ 5 ] BHHorpaflOB, H. M.: HoBaa oueHKa (JjyHKUHH §(1 + it), HAH CCCP, cep. Mar, 22(1958), 161—164 [ 6 ] KopoSoB, H. M.: OneHKH TpHroHOMeTpHieCKHX cyMM H HX nptuioxeHHR, VMH CCCP, 4 (1958) 185-192. [ 7 ] JIHHHHK, K>. B.: HeKOTopwe ycTOBHbie TeopeMbi, KacaroniHecH SHHapHbix 3a«aq c npocraMH qncjiaMH, UAH CCCP, 77 (1951), 15—18; HeKOTopue ycjioBHue TeopeMbi, KacaioniHeca 6HHapHwx npo6jieMbi roJibfldax, HAH CCCP, cep. Mar; 16 (1952), 503—520.
200
REMARKS CONCERNING A TRANSFERENCE THEOREM OF LINEAR FORMS* WANG YUAN AND YU KUN-RUI Institute of Mathematics, Academia Sinica ZHU YAO-CHENG Institute of Applied Mathematics, Academia Sinica Received 1 June 1977 Revised 22 July 1977
Let n > 2 and 9\,...,9n be a set of real numbers such that l,9i,...,9n independent over rational number field Q. For a real number x, let ||x|| = min(a; - [x], [x] + 1 — x)
and
are linearly
x — max(l, \x\).
We use £ to denote a preassigned positive number and c\, c2 the positive constants depending only on e, n, 9\,..., 9n.
Proposition A.
There exists C\ such that (£i---x n ) 1+e ||:ri6>i + - - - + : E r A | | > 1
holds for any given set of integers x\,...,xn Proposition B.
satisfying x = maxi<j< n \xi\ > c\.
There exists c2 such that for any integer yn+\ > c 2 , we have y1nX\\\yn+i9l\\---\\yn+len\\>\.
In this paper, we shall use Mahler-Baker's method to prove the following transference theorem: Theorem 1.
Propositions A and B are equivalent.
A similar result may also be established if the e in Propositions A and B is replaced respectively by certain functions E\(x) and £2(2/71+1) which take positive values and tend to zero decreasingly. *Acta Mathematica Sinica, 22:2 (1979) 237-240.
201 Proof of Theorem 1. 1) Assume first the truth of Proposition A. We proceed to show that Proposition B holds. We may assume that yn+\ > 2. Let hn+iOiW = IVn+iQi ~ Vi\,
(1)
l
Since 1,6i,..., 9n are linearly independent over Q, we have
\\yn+i9i\\ > 0,
1 < i < n.
Let ^ = (yn+l||2/n+l^l||---||yrl+l^n||)"||2/n+l^ir1,
1 < t < n.
(2)
Without loss of generality we may assume fi
>
Since
(3)
it is impossible that
1,
1 <j
(4)
\Xk\ < fk • • • Vn,
\yixi -\
\-ykXk+yn+ixn+i\ < 1-
Since the absolute value of the determinant of linear forms on the left-hand sides of (4) is y-n+i and the product of right-hand sides of (4) is also yn+i by (3), it follows by Minkowski's linear form theorem [3] that there are integers x\,..., xk, xn+i not all zero and satisfying (4). The last inequality in (4) yields that yixi H
VykXk +yn+ixn+i
= 0.
(5)
Hence x±,... ,xk are not all zero. By the definition of k, we have
• • • xk
(6)
< ifX • • •
by (3) and (4). Therefore by the identity j/ia;i -\
h VkXk + 2/n+i^n+i = (zi#i H k
1- xkdk + xn+i)yn+i
202 and (1), (2), (4), (5), we obtain k
< ] T Nllyn+i^ll
\xi6x + ••• + xk9k + xn+1\yn+1
»=i k t=l
(7)
(7)
To prove Proposition B, it is sufficient to assume e < 1. Suppose that y n +i > 2 and y1n++£1\\yn+i0i\\---\\yn+i9n\\
e, n, 6\,..., 6n such that
(A^---A^) 1 + !fe||A ; r i 0 1 + --- + Azn0Ti|| > i
(9)
holds for any not all zero integers x\,... ,xn, in particular, for the yn+\ with (8) and those x\,... ,Xk,Xk+i = • • • = xn = 0, not all zero and satisfying (4). By the combination of (6)-(8) and e < 1, we derive that 1 < (AiT• • • \x~k)l+^HAzifli
+ •••+
\xk9k\\
< A f e ( 1 + ^ ) + 1 (5i • • • z f c ) 1 + ^ Hn^i + • • • + xk6k\\ < A^Hzifli + • • • + < A
n+2
xkek\\yn+1yf+l
fc(y n+1 ||y n+1 fli|| • • •
\\yn+19n\\)-nyt+l
= A"+2fc(^.||j/n+1^|| • • •
\\vn+i6n\\)±y:£
n+2
<\
ny-^.
This means that if (8) holds, then yn+i < n ^ k / \ 2 " ( ^ + 2 ) , The right-hand side of the above formula may be regarded as c 2 in Proposition B. Therefore Proposition B follows. 2) Assume now that Proposition B holds. We proceed to prove Proposition A. Let x = max \xA, l
where x\,...
,xn are integers. Suppose that x > 1 and (^•••£ n ||a;i0i + ---+a; n 0 n ||)« <
L
(10)
Otherwise Proposition A is obviously true. Suppose
\ X l 6 i + --- + x n e n + x n + 1 \ = \\x1e1 + --- + x n e n \ \ .
(n)
2(B Set i>i = ( x 1 - - - x n \ \ x 1 e 1 + --- + x n 6 n \ \ ) ± x - \
\ < i < n .
(12)
Since 1,9\,..., 6n are linearly independent over Q, we have ipi > 0,
1 < i < n.
Consider t h e system of linear inequalities oiyi,...,yn, \y% - yn+i6i\ < ip%, \xiyi H
yn+\:
i < i < n,
h xnyn + xn+iyn+i\
(13)
< 1-
Since the absolute value of the determinant of linear forms on the left-hand sides of (13) is ||a;i0i + • • • + £ n (? n || by (11) which is equal to the product of the righthand sides of (13) by (12), it follows by Minkowski's linear form theorem that there are integers yi,...,yn,yn+i not all zero and satisfying (13). If yn+\ — 0, then ipi < 1,1 < i < n by (10), and therefore we have j/j > 0, (1 < i < n) by the first n formulas in (13). This leads to a contradiction, and so yn+i ^ 0. We may assume that yn+i > 0. The last formula in (13) yields xiyi H
h xnyn + xn+iyn+i
= 0.
(14)
Therefore from the identity xiyi +
h xnyn
+ xn+1yn+i
= (xi6i + •••+ xn9n + n -^2xi(yn+i6i -y{)
xn+1)yn+1
i=l
and (10)-(14), we obtain n
\\xi9i H
hx n 0 n ||i/ n + i < ^_Xi|y n + i0i - j/j| < *=1
n
yixjipj i=l
< n ( x 1 - - - x n | | a ; i 0 i - | - - - - - | - x n 0 n | | ) i < n.
(15)
By (12) and (13), we derive IVn+lOl — 2/11 - - - IVn+lQn ~ Vn\ < 1>1 • • • fa = ll^l^l + • • • + Xn9n\\.
(16)
Now suppose that Proposition A is not true then there exists e with 0 < e < 1 and a sequence of integer sets ( x i , . . . , xn) with x = maxi<j< n |XJ| —* co such that {xx • • • £ „ ) £ * ||a:i0i + • • • + xn9n\\ < 1.
(17)
It follows from Proposition B that there exists a positive integer r such that (T^+i)1+*||r2/n+101||--.||Tyn+10n|| > 1
(18)
204
holds for any positive integer yn+i- Let xi,... ,xn be a set of integers such that (17) « and let y±,... ,yn,yn+i (yn+i > 0) be a set of integers holds and x>ns?r satisfying (13) which is induced by x\,... ,xn as above. Then from (15)—(17) and e < 1, we obtain •n/n+i||Ty n +i0i|| • • • ||-n/n+i0n|| < Tn+1yn+1 \yn+i6i - j/i| • • • \yn+i6n
n+1
< r
yn+l\\xlel
n+1
+
---+xn6n\\
n ( x i • • -x n ||a;ifli + • • • +
= nTn+1((Xl
• • • xn)^\\xtf!
X (£i • • • £ „ ) - " ||zi01 -\ <
- yn
xn9n\\)±
+ •••+
x
n
6
n
\ \ ) ^
hXn0n||"
nr^x-iniy^ 2n
<{nsT
n(n+2)
'
_ i . x /
x
)n(Tyn+i)
N _ S.
,
» < (ryn+i)
s_i
".
This gives a contradiction with (18), and thus Proposition A follows. References [1] K. Mahler, Ein Ubertragungsprinzip fur lineare Ungleichungen, Gas. Pest. Mat. 68 (1939) 85-92. [2] A. Baker, On some Diophantine inequalities involving the exponential function, Can. J. Math. 17 (1965) 616-626. [3] Hua Loo-keng, Introduction to Number Theory (Science Press, 1957).
205
Vol. XXII No. 3
SCIENTIA
SINICA
March 1979
A NOTE ON A TRANSFERENCE THEOREM OF LINEAR FORMS WOLFGANG M.
SCHMIDT
AND
(.Department of Mathematics, University of Colorado, USA)
WANG YUAN (EE
(Institute
x)
of Mathematics, Academia Sinica)
Eeeeived September 22, 1978.
ABSTRACT
In this paper, a transference theorem of linear forms in diophantine approximation The proof of the theorem depends on a theorem of Mahler.
is proved.
1. Let m and n be integers ^ 1 and [0,j (1 <J i ^ m, 1 ^ j ^ n) be a given set of mn real numbers. Let ||a;|| denote the distance from a real number x to the nearest integer and x= max(l, \x\~). Property A.
For any given e > 0, the inequality
( n n*«*. + • • • + 6i.*.\\) (*•• • -^) i + e < i N
/
<=i
CD
has only finitely many integral solutions G"i>"""»*OProperty B. For any given e > 0, the inequality ( f [ \\6uVi +•••
+ dmiym\\) (ft- • -gmy+° < 1
(2)
has only finitely many integral solutions (j/i, • • •, ym~). In this note, we prove the Theorem.
The properties A and B are equivalent.
For the case m = 1 and 1, 6n, • • •, 0,B are linearly independent over the rational field Q, we have the result of [1]. 2.
The proof of the Theorem depends on the following:
Lemma 1 ( M a h l e r ) . Let /fc(x) ( 1 < J < O be I Unearly independent homogeneous Unear forms in the I variables x = ( # , , • • • , £ / ) and let S^Cy) ( 1 < k < 0 be I Unearly independent homogeneous linear forms in the I variables y = ( ^ , • • •, y,~) of determinant d. Suppose that all the products cc,% ( 1 < i, j < 0 in
have integer coefficients.
If the
inequalities
206 No. 3
A TRANSFERENCE THEOREM OF LINEAR FORMS
|/»(x)|<X,
277
1<*<*
are soluble with integral x ^? o, then the inequalities lff*(y)l
l
ore soluble with integral y ^= o. See K. Mahler [2] and also J. W. S. Cassels [3]. 3. Proof of the Theorem. If there is a 6i} € Q, then Properties A and B are obviously both false. So we may suppose that all the d'^s are irrational numbers. First, we shall prove that if there exists an i with 1 ^ *' ^ m such that 1, 8n, • • •, 6,-n are linearly independent over Q, then Properties A and B are both false. Since there exist integers au- • •, an not all zero and an integer a0 saah. that a0 + afin + • • • + and!n = 0,
(3)
it follows that (1) has infinitely many integral solutions (Jcalt • • •,fca,)(fc = 1, 2, • • •)• Hence Property A is false. Since 0,-,- 6 Q (1 < j < n), it is necessarily n > 2 by (3). Without loss of generality, we may assume that an > 0. Take e = — —. Then it follows by Dirich2(« — 1) let's theorem that for any given integer t > 1, there exists an integer q such that 1 < g < <"-> and ||0;Jg|| < f1, Take y = aog and c = n max I a< I. Hd^H < ct~\
1 < j < n — 1.
(4)
Then 1 < j < « - 1,
and further by (3) ||0,.y|| = Pma.q\\ = HCfliflrt + • • • + afl-.0,-.n-.)«ll S Iki0,-,3ll + • • • + I k - A n - i d l < c*"1Consequently, we have
IIMl • • • Weij/W*' < (cro-Cc*-1)^8 < c+2r^ < l if t is sufficiently large. Since ||0,-jdl < f 1 and 6a I Q, there are infinitely many integers q satisfying (4) when t runs over all the integers > 1. Hence we have infinitely many integral solutions Qyu'",ym) of (2), where yi = anq and yf = 0 ( j =N» »)• So Property B is false. Similarly, we can prove that if there exists a j with 1 ^ j < n such that 1, By,-••,6mi are linearly dependent over Q, then the Properties A and B are false also. So in the following, we may assume that for l ^ i ^ m , 1, diu •••,din are linearly independent over Q and for K j < n, l,dl}, ••-, 0mj are linearly independent over Q. By the symmetry of the two Properties, it is sufficient to prove the following assertion. If Property A is false, then Properly B is false also. (5)
_ _ _ _ ^ _ ^
207
278
SCIENTIA SINICA
Vol. XXTT
Suppose that there exists e > 0 such that (1) has infinitely many integral solutions (jd, • • •, xn~). Write (6)
Z = xl---xn. For any integral solution of (1) with X ^ 2, we define Au • • •, Am, A by 0<\\0atCi
+ • • • + 0inxn\\
= X~*i (1 < i < m) and Bx,---,Bn
and A, + • • • + Am = A,
(7)
by
x, = XB>,
(1 < j < « ) .
(8)
Then A,- > 0 ( 1 < t < m), Bt > 0 ( 1 < j < r») and Bi + • • • + Bn = 1. The expo•nents A I; • • •, Am and B1; • • •, Bn depend on X. By (1), we have A > 1 + e.
(9)
Let liaT
mia(i
"'"'i"-:) =2P A
(aaX^oo).
(10)
Denote by Xn+i (1 ^ i ^ m) the integers such that 10ft«i + • • • + fl/A + as.+, | = ||0fta!, + • • • + fl,.a;.||, 1 < » < m.
(11)
1) Suppose that p > 0. Then there are infinitely many integral solutions of (1) such that m i n U , , - - - , ^ ) >p>0. A
Write
(12)
17 = max(l, A — (n + m — 1) m i n ^ , • • •, 4 m )).
(13)
Then from (9) and (12), we have — <max(—-—, A M + g
l-(fi + m - l U < L /
(14)
Let x = (a;,, • • •, xn+m~) and y = (j/ I; • • •, yn+m~), and let
(15) and fX-^j/,-,
l
ff,(y) = jz^-/-((? 1 .^y l + • • • + 0m,,-m2/ra + j/,), Since m+n
S
m
/.•(x)fi'.(y) = S
n a;
«+<j/< — S
K
^™+i»
(16)
208 No. 3
A TBANSFEBENCE THEOEEM OF LINEAB FOBMS
279
and since there exists an integral solution of (1) satisfying (12), it follows by Lemma 1 (with I = 1 and d = XV~A') that there exists an integral point y ^ o such that |»i I < ( « + » - 1) x '
1
^
r + A
>,
1 < i < m,
(17)
and \8nyi + ••• +omjym
+ ym+i\
<(ti + m - l ) I 1 ^ r " ' \
l
(18)
By (9) and (14), we have \8iflh + ••• + enjym + ym+l\
< (n + m - l ) ! " ^ "
<(« + m-l)X"+"-1 < („ +
m
< 1,
_
u
;
1 ) z -5^=r-••(irr.c-+-t)p)
(19)
1 < i <«,
if X is sufficiently large. Hence (jylt—, ym~) ^ o. Since the right-hand sides of (17) are all ^ 1 by (13), we may replace the y{ by j?,- (1 < i < m) in (17). Take 8 > 0 such that 1 - in - 1)3 1 + m8
= max
/_j__; V1 + e
1
_
( w+ w
_
1 )
\+ /
5>
( 2 0 )
Then by (14), (17) and (18), we have
( ] I llflitfi + • • • + emlyj) (&• • -gmy+s <
(w
+
m
_
1 ).^i+.) X iTsr T «>+-«>
+ <
' t->+f- l W)
= (« + m — i)»+»(i+*)2:~ "+"1-1 *> 1+'"*
A
>
< (n + m - l)«+»«+«z" " + m " 1 < 1,
(21)
if X is sufficiently large. Since 1, Qxi, •••,Qmi are linearly independent over Q for 1 < j < n, it follows by (9) and (19) that there are infinitely many integral points (i/i, • • •, i/m) satisfying (21) when (x : , •••,«„) runs over all the integral solutions of (1) satisfying (12). In other words, Property B is false. Thus the assertion ( 5 ) is proved. 2) We now proceed to prove the assertion (5) by induction on m. Since p = 1/2 for m = 1, the assertion holds for m — 1. Suppose that m > 2 and that the assertion holds for m — 1. If p > 0, then the assertion (5) is true by 1). In the following, we consider the case p = 0. Take
209
280
SCIENTIA SINICA
Vol. * X T T
(22) Without loss of generality, we may assume that (1) has infinitely many integral solutions (#!, • • •, xn~), such that (23)
~-
•*.)l+J/i
< ( ( n H0"^ + • • • + 0,.s,||) Gv • •^)1+£)1~" < 1 by (7) and (22). Hence it follows by the hypothesis of the induction that there exists a t > 0 such that
(]& lift/* + • • • + em-i,}ym-i\\) (fr • •gm-ly+t < 1 has infinitely many integral solutions (,yu • • •, ym-C). many integral solutions Ctfn" '> Vm-u 0) for s = t. hence the Theorem is proved.
Consequently, (2) has infinitely Thus we have the assertion (5),
REFERENCES
[ 1]
Wang Yuan, Yn Ktm-rui & Zhn Yao-oheng: A note on a transference theorem concerning the linear forms, Ada Math. Sinica, 22 (1979). [ 2 ] Mahler, K.: Ein tjbertragnngsprinzip fiir lineare XIngleichnngen, das. Pest Mat., 68 (1939), 85—92. [ 3 ] Cassels, J. W. S.: An Introduction to Diophantine Approximation, Camb. University Press, (1957).
210 Chin. Ann. of Math. S (1) 1981
A NOTE ON SOME METRICAL THEOREMS IN DIOPHANTINE APPROXIMATION WANG Y U A N
Y U KTTNRUI
(Institute of Mathematics, Academia Sinica)
§ 1. Introduction. Suppose that A is a set in w-dimensional Euolidean space Bn. If (xi, •••, xn) £A implies that (x'ly •••, x'n) £A for any 0
«
(l;
Further let
»"(A) = g«A(?)
and
Q(h)=±+(q)q-\
Then for almost all (9t, •••, 6n) £Bn, we have
N(h, olr ..., en)=w(h)+o{w(hyQ(hy{\ogyr(h)y+°).
(2)
r
Letg =(g'i, •••, qm) and r = (ri, •••, rm) denote the lattice points in Bm, where #i's a n d r f ' s are positive integers, 6= (#i, •••, 0m) a point i n Rm, q8 = qx9^
the soalar product of q and 8 and d(q) — 2
\-qm8m
!• We also use q<Ji to denote
^ = max(^1, •••, qm) *^h. Similarly, we may define Q
where 0t= (0a, —, 9im) (l^i
(3)
%{h) - 2 $(q) d(q). Q
Then N(h,
0i, - , 6n) -!P-(A) +O(x(h)*(logx(h))*+>)
h o l d s f o r a l m o s t a l l (&!, ••, $ n ) = ( 9 l l t ••-, 9 l m , ••-, 0nl, •-, Manuscript received May 28, 1980.
0nm)€Rnm.
(4)
211 CHIN. ANN. OF MATH.
2
VOL. 1
Theorem 1 gives a modification of a theorem of Gallaghera\ Take Aq to be the set n
of points (xt,---,xn) satisfying 0<,Xi
(K«<w)
and
/t
vxf-scn
Then it is easily proved by mathematical induction that f2~", forT<2"
^ ^7r(l0g-f)' Take Aq = E(q(logq)n).
0th6rwise
-
Then q(logq)n>2" and
^q) - n.i - (ff(kgff)^^g ^.(^(JLQawll))' for g>8. Hence
^(A) = 2 7 T 2 (?(log?)•)-1 (log( q a J n q ) n )) s +0(1)
Obviously Q(h)=O(l). Let N(h, 9t, •••, 6n) denote the number of integers satisfying K j < A and the inequalities and
0<{q6^<^(l
(ft {^,})g(log ? )"
Then it follows by Theorem 1 that N(h, 0U - . , 6n) = (
w
^
log log^+O((log log A) * (log log log A) a+«)
;
(5)
holds for almost all (8lt •••, 9n) £Rn. Let |g|| denote the distance from the real number x to the nearest integer. Then from (6) with some simple combinatorial considerations, we may derive Theorem 3. Let N*(h, 6it •••, 0n) be the number of integers q satisfying ( i l Mill )
Kq
(6)
Then for almost all (6lt •••, #„) fzRn, we have
N*(h, 6it ..., 9n) = (n j " 1 }
f
loglogA+O((loglogA)*(logloglog/02+8).
(7)
Similarly, we may derive from Theorem 1 that for almost all (9x, —, 9n~) £Rn, the inequality
( n ll^.fl )g(log ? )" + '
(8)
212 NO. 1 A NOTE ON SOME METBICAL THEOBEMS IN DIOPHANTINE APPROXIMATION
3
L e t / ( I ) - 1 and f(k) =/fc(logyfc)" or k(logk)n+'(k>l). Put i = max(l, \x\). Take AQ = .E(f(q1)'--f(qm)) in Theorem 2. Then we may obtain for almost all (0n, •••, flM) €.Rnm the asymptotio formula of the number of lattice points q satisfying max | q{ \ *^h and
ni^?i+-+M-iin/(?y)
(9)
From this formula we may derive Theorem 4. The inequality W\eaqi + -+9imqm\\ f i f e (log ^)" r s ) < 1
(10)
has only finitely many integral solutions for almost all ( # u , •••, 0nm) €zRrm, but
n\\Oaqi + -+0imqm\\ niglUmqT") < 1
(H)
has infinitely many integral solutions for almost all (On, •••, O 6-B. Especially, it follows from Theorem 4 that property A in Schmidt and Wang's"3 transference theorem holds for almost all (6llt •••, 0nm) £RnmThe proofs of Theorems 1 and 2 are based on the method of Schmidt^ and we may also treat the similar problems in non-linear diophantine approximation (Cf. Schmidt [3]). Remark. We propose two conjectures. 1° Suppose that On, •••, anm are n real algebraic numbers such that 1, an, •••, Onm are linearly independent over rational field Q. Then n l|a,igi+-+o, m gj| ng} + «»i. 2° Suppose that
/3IJi=er",
where r(j(l
Put n=l or m=l. Then 1° and 2° are Theorems of W. M. SehmidtW ana A. Baker^ respectively.
§ 2. Proof of Theorem 1. If 2 i K g ) < ° ° , t t e theorem can be proved by BorelCantelli's lemma (Cf. Gallagher [1]). Now we suppose that 2X90 diverges. Evidently, we may confine ourselves to the case (01} ••-, 9n) £Gn. Let
y(q,0i,-,0J=
P(q, qOi-pi, - , q0n-fn),
pi, l < i < n
y(h q, 0i, - , ^n)=
S
Kq, q0i-pi, - , qOn-p»),
l(q)=\1ii-\\(q,6u l(k, q) = \1-\\(k, Jo Jo
»>,0.)Mi-M» q, K - , Ot)del-d0n,
213 4
CHIN. ANN. OF MATH.
I(K q, r)-[-[y(k, Jo
q, 9lr - , 9n)y{k, r, 9U - , 9n)d91-d9n,
Jo
U, «, 9X, - , 0n) -
N(k,
and notice
VOL. 1
± y(k, q, 9X, ••; 9n) a=u+i
N(v, 9lt - , 9n) = ±y(q, 9U - , 9n).
Lemma 1. I(q)=^(q), I(h,q)=$(q)
(12) (13)
K?)=£-£r(g, ot, - , 9n)d91-d9n = [Q-\"q~ny(g, g'^u - , q-i9n)d9t-d9n Jo
Jo
-?-
2 Pi, l
{"-{"Pig, 91-p1> - , 9n-pn)d91-d9* J0
JO
=
iQ°,q) = q-"Pi, (PI,2qXhJO (*-fV(7, 91-p1, -, 9n-Pn)d91-d9n JO 1-SJ.sn
= xjj(q)
q)nq~".
Now we proceed to prove (13). Divide the sum l(k, q,r)= 2 \1--\10(q, q91-p1, - , q9n-pn) i>(, (Pi,a)
Jo
X/8(r, r^-sa, •-, r9n-sn)d91--d0n
into n+1 parts
!(*, g, r)=I0+- + In, where /^ is the sum of all the terms with exactly j indices %, •••, if satisfying
(14)
qpi — rsi = 0.
We first estimate JoIo<
2
Pi, »I
asi-rpntO
arrtl+o
P-fiSfa, qBx-pi, - , q9n-pn)p(r, r9t-su - , r9n~sn)d9r-d9n
JO
r
JO
-^i. 8
- \
.1-Js.
-
8 }
y 8 ( ? , grfll, .-., qffH)
~^
x/3(r,r9'1-^rPL,
..., &„-«*>-**' )dff1-d9\.
Write (g, r) =d, ^ =rfg'and r = dr'. Then we have gs ( -rpi = M . For given ^=£0, p,
214 NO. 1 A NOTE ON SOME METEICAL THBOEBMS IN DIOPHANTINE APPEOXIMATION
5
is determined uniquely modulo q', so we have /o
-,
rOn--^)d8r~den. (15)
Put
J(Xt, .-., »,)-[" -f~ £(?, ff0lf - , ff^0/8(r, r91--^-,
-,
r6n-^f)
xd0r ••<$,,. We proceed to prove that J decreases as a function, of kt when X,f>0 and increases when Xj<0 for « = l, •••, n. Without loss of generality, we may suppose 4 = 1. In faot, for fixed $a, —, 0n and A,a, •-, K, P(q, q&i, —, q6») and /g(V, r ^ — ^ - , —, r ^ n —2—) are the characteristic functions of two intervals on ^i-axis which have fixed lengths and start from 0 and —^— respectively. The measure of the overlap of the two intervals is \"_J(q, qOi, .-, q9.)fi(r, rO,-^-,
-,
r9%-M-)dBu
It is a decreasing function of Xi for A,iX) and an increasing function for Xi<0. Hence J decreases for X^O and increases for Xi<0. From this fact and (15) we have X dBi- • -dejXf -dXn = i/> (q) iji (r). As for I)(j>l), since 2 2 2 [-i1 Kq, qOi-pi,-, qOn-Pn) (Pi,Q)
(16)
JO
x/8(r, rOt-h., —, r^-« B )^i-<^» 1—
= 2
2
2
(Pi, Q)<1C qs,-rp,*O
qsi-rp,=O
Pi
^
[ " -f
J _^i_
Vn
" i8(j, gd, -, qff.)
J -jgn.
x ^s^, r g' t - gfr-fPi., ..., rff^-3h=r^L,
x /8(r, r ^ - - ^ - , - , r ^ - f e ,
re>n_i+1,
..., r ^ ) ^ i - d ^
r^- i+1 , - , r9n)d9r-dSm
- , r9%-,-^s=L, r6^j+1> - ,
r9^dJ91-M%dii'-dK-i
q, r)>q~>,
215 6
CHIN. ANN. OF MATH.
VOL. 1
hence
W
"WM(*,
(17)
q,r)q~\
(13) follows by combining (14), (16) and (17). The lemma is proved.
Lemma 2.
P - f V f a 0i, —, en)d6r~d9n = WX>v),
Jo
(18)
Jo
P - T ^ C * , «, «, ^, - , 9n)d6r~den^ ± *Kq)
Jo
?-[ JO
JO
Jo
(19)
g=a+l a
N{h, u, v, 9U •-, enyd9v-d9n<W(u, ^) + 2(2"-l) 2 tf,(q)dk(g), S=ii+1
(20)
where dk(q) = 2 1. d|(j,
Proof (18) and (19) follow from (12) immedeately. (20) follows from (13). The rest part of the proof of Theorem 1 is similar to the proof of Theorem 1 in Schmidt [2], and we omit it here. § 3 . Proof of Theorem 2. If E'K<7)< I =°, ^e theorem can be proved by BorelOantelli's lemma (cf. Gallagher [1]). Now we suppose that 2,ip(q) diverges. We may also confine ourselves to the case (fix, •••, &n) £(?„»,. Let w(0) = 0 and co(h), h~>l, be an increasing integral-valued function which tends to infinity. Set #' = {0} \}{h>0\
a>(h-l)
S" = {h>0\co(h)
we define intervals of order t to be (y2*+v1} (u+l)2*+va'], where u, i>i, Vz are non-negative integers such that i>i<2* and Vi, Vi are the smallest non-negative integers satisfying w2* + i>x € S, (u+1) 2* +1)2 € 8. Lemma 3. Every interval (0, x] with %£S can be expressed as union of intervals U /j of the type described above, where no two of intervals It are of the same order. Proof (cf. [2]). Put L 0, otherwise. y(Q, 6i, - , 0,,)=
2
0(g, Q0I-PI,
-,
Q0»-P«),
Pi, 1«£»
I(g, r) = f - f y(s, 6U -, 6n)y(r, 8X, -, 0n)d01-d0a, &(«,*)=
jGm
J Um
2
U
N(u, v, 0lt •», 0n)= 2 v(g, 0i, "-, 0«), U
216 NO. 1 A NOTE ON SOME METRICAL THEOEEMS IN DIOPHANTINE APPROXIMATION
and notice
7
N(v, 01; - , 6n) = ^y(q, 0u - , 6n).
Lemma 4. (21) I(ff)=«Kff). If q and r are linearly independent (this fact is abbreviated to q, r, I. i.), then Kff, r)=./r(ff)iKr). Jf q and r are linearly dependent (this fact is abbreviated to q, r, I. d.), then
(22)
(23) I (9, r)<$ (q) <£ (r) + (2" -1)A{qu n)
n r3
^ 0 . Let
(
qi
qf-qm\
r-L r 2 - - - r m 1 0 I(m"2> /
where Iw is the ZxZidentity matrix. Obviously, det T = q1r2 — q2r1. Write T0f = & = (in, -', ilm)(Ki
andM(?m={(a;a, ••-, xm)\0<x{<M, Ki<m}. -(
y(q, &„ - , 6n)y(r, 6U - ,
= Jlf-"m|g1r2-g2r1|-" 2 f
-f
Then
en)det-d0n
&(q, £u-p1; - , U~fn)
x/3(r, it»-H, - , L2~s«)d£i-d£». (24) Put T(MGm)x-xT(MGm)=D(M). Let Pj and P 2 (P1<=:D(M)c:P2) be two nmdimensional parallelepipeds whose surfaces are parallel to the corresponding surfaces of D (M) with distance *Jnrn. Since
f - ( fi(Q, in, - , Li)/3(r, fM, - , L*)d&-dL = <]>(Q)
•"(
P(Q> tv-fu
- » fi-3V>
x/3(r, €iS-si, ••-, £n2-sn)di!f-dgn
Let M-*o°, then we have (22).
217 OHIN. ANN. OF MATH.
8
VOL. 1
2) T a k e r = ( l , 0, •••, 0) and Ar = Gm, then i/»(r) = 1 . Since all the components of q are positive integers, q and r are linearly independent. Hence (21) follows from (22). (Notice that the proof of (22) depends only on the linear independence of q and r.) 3) Suppose q, r, l.d.. Put /l W =\ \
_22- ... g» \ g* g1 , 1 0 I*"- ' /
£ = TF0 ( (l«Kw).
Evidently, deW = l. We have J>(, S( J TF(Om)
J ITWm)
x/8(r, n f u - « i , - , r1|nl-sn)rf?1---dfn = I 0 + - + I n , (28) where It is the sum of all the terms with exactly j indioes iu •••, if having qxS( — ript = 0. For real a1; •••, an, by the method similar to the proof of (13), we have 2
Ft. si
J Oi
<
2
•"
P(q,qiVi-pi,-",qiV«-pn)0(r,r1r}i-s1,---,r1'on-sn)dr]1--dr]n
J an
%
' -
P(Q, qir/i, - , qiVn)
x /8(r, r W x - ^ ^ i . , - , r^-gA~r^)drA-«H<^(g)
/o
JO
2
f1+O1-
V,, si ^a, «iS(-rij(-*.O ktoi
(29)
2J
•••
0(9, qivi-pu —, gi^n-pj
BiSi-tiPi*O aiSi-riP(=0
x/3(r, rnTi-Si, •••, riife-Sn)^—^
=
2
2
Qt'i-riP&O ais,-rip ( =0 l<«
I
"1
•••[ „ *'
0(a, q*k.-, qi->0
" " "37Ql
n)ql\
218 NO. 1 A NOTE ON SOME METRICAL, THEOREMS IN DIOPHANTINE APPROXIMATION
9
m
Taking ai = gi1 2 f t f « ( K * < « ) , we have 2
2
[
•••[
p(.«< ix, ii Jw(.am) l<»«;»-j »—J
-••• «&.-#.. 2 JO
JO
/3(tf> ?ifn-3>i, —, giLi-Pn)
Jir«j«)
2 f -
p !( .si Pi, s» Jax Oisj-ripi+O g ^ i - r ^ ^ O
Jo,
Hence Ii<( n )«A(ff)^(?i, OST 1 .
(30)
(23) follows from (28), (29) and (30). The lemma is proved. Lemma 5. f -f f -f
N(u, v, 0t, »., ejMi-M.-Wiu,
N(u, v, eu •», e^dffi-de^wiu,
v),
(31)
vy+2(2»-i) 2 +(g)d(q). (32)
Proo/ (31) follows from (21) immediately. Let r
f •••( y(Q, K - , en)y(r, ex, - , en)d0v~Mn
u
-
2 HQ,r)+ 2 I(Q,r)
r
r
U
U
a
<W(u, i)) + 2(2»-l)
2 d.r.l
WtiMqu rdq?
(33)
a.
by Lemma 4. If r is linearly dependent of q and satisfies u
-Q* ——?—P
(a 6)=1
'
'
then 6|g,(K»<m), — < - | - < l . Conversely, if (a, 6)=1, b\qi(l
—<-y
••-, -v-<7m) is a lattice point which is linearly dependent of q and
satisfies w
2
2 ^(fc,-f^W 1
Wo)Mqi, rdgi1- 2 «A(«) 2
Q.r.li.
- 2 *(«) 2
UK*
(0,
6)=1
2 -f-^r1^ 2 Hv) 2 i&= 2 +
219 10
VOL.1
CHIN. ANN OF MATH.
Combining (33) and the above inequality, we obtain the lemma. Takeea(A) = [ z (A)]. Let L,= {(u, v) |«€»S', i>£#', (oi(u), w(i))] is an interval of any order t, w('u)<2'} and let A* = A*(s) be the greatest integer satisfying w(A) <2 S . L e m m a 6.
2 Proof
f •••( (N(u, v, $lt •-, 6n)~W{u,
v)yd0v-d0n^O(s2').
By Lemma 5, every term in the above sum has upper bound
2(2»-l) 2 4>(Q)d(q). U
We first sum over all the (u, 1>) £L, for which (G>(M), G>0»)] is an interval of fixed order t. Since intervals of order t cover the positive axis exaotly once, we have the upper bound
2(2n-l)X(h)=O(2°).
Summing over t, the lemma follows. Lemma 7. There exists a sequence of subsets o\, ov-- in (?„,„ with measure* J
such that holds for anyhGS', Proof
N(h, 0U —, 6n) =W(h) +0(2^s^ + *) a>(h)<2s and (fiu —, 0«) 6GBm\o-s.
W e define a, to be the set of (0lt •••, 0n) £Gnm for which
2 CM, vtaL,
(JV(«, v, 0U ••; 0n)-W{u,
v)y<s*+>2>
f34)
does not hold. By Lemma 6, we have M. = O(s- 1 -'). co(h) <2', then by Lemma 3, (0, w(A)] can be expressed as union of at Ifh€8', most s intervals («(«), w(v)], where (w, v) GL,. For (filt —, 0n) £Gnm\o-,, summing over these (M, V) , we obtain
(N(h, 0X,.-, 0n) -w(h)y=cE
(u.v)
(N(u,«, eu '-, 0n) -wbi,
v))y
(u,u)6l,
by (34) and Oauchy's inequality. The lemma follows. Proof of Theorem 2. Since 2 s " 1 - 8 < ° ° , there exists for almost all (0lt ••-, 0n) £Gnm a s o = So(^i, •••, 0n) such that (0t, •••, 0n) £cr s for s>s0. Suppose that (0±, ••-, 0n) has such a So and h is so large that co(A) >2 So . Pick s satisfying 2*~1
h£S".
220 NO. 1 A NOTE ON SOME METRICAL THEOREMS IN DIOPHANTINE APPROXIMATION
11
Then we have Similarly Since N(h', 6U the theorem follows.
\W(h) -W(h') \<\x(h) -x(h') | <1 | W (h) - W (h") | < 1 - , 0J
References [ 1 ] Gallagher, P . X., Metric simultaneous Diophantine approximation, JLMS, 37 (1962), 387—390. 12 ] Schmidt, W. M., A metrical theorem in Diophantine approximation, Can. J. of Math., 12 (1960), 619—631. [ 3 ] Schmidt, W. M., Metrical theorems on fractional parts of sequences, TAMS, HO (1964), 493—518. C 4 ] Schmidt W. M., and Wang Yuan, A note on a transference theorem of linear forms, Set. Sin., 22: 3 (1979), 276—280. [ 5 ] Schmidt, W. M., Simultaneous Diophantine approximation to algebraic numbers by rationals, Acta Math., 12S (1970), 189—201. £ 6 ] Baker, A., On some Diophantine inequalities involving the exponential function, Can. J. Math., 17: 4 (1965), 616—626.
221 12
CHIN. ANN. OF MATH.
i
7C
m
VOL. 1
^«
*
£ P. X. Gallagher £3«Jgfcia, # & £ W. M. Schmidt W i l f t ^ a . &nm^tfl, MM: 1°. MfJl^m^^i (Pi, - , 0«) €Rn, &4tf
(w5i),
loglogA+O((loglog/>)v»+«),
e -^» «,. 2° W. M. Schmidt i5i7Clft$S&£S*l#£J! ^ ^ T A ^ ^ W M (On, - , 9J)
222
ACTA ARITHMETICA XLVIII (1987)
Bounds for solutions of additive equations in an algebraic number field I by
WANG YUAN *
(Beijing, China)
1. Introduction. Let k be a rational integer ^ 1. Similar to Waring's problem, one can show by the Hardy-Littlewood's method that an equation a2 x\ + ... + as xj = 0, where at, ..., as are given rational integers but not all of the same sign, has a nontrivial solution in nonnegative rational integers xlt ..., xs, provided only that s^Ci(fe). (See, e.g., H. Davenport [3]). Here we use c{f,...,g) to denote a positive constant depending on f, ..., g. As for a bound of these solutions, it was shown by J. Pitman [10] that if s ^ c2{k), then there exists a nontrivial solution in nonnegative integers such that (1)
maxxj
where c2 and c 4 are explicit. Under suitable conditions and if s is very large, the estimation can be considerably improved. (See, B. J. Birch [2] and W. M. Schmidt [11], [12].) In particular, Schmidt proved that if s^c 5 (/c, e), the equation fli^+
•••+asxks =b1y\+
...+bsyks
with positive rational integer coefficients has a nontrivial solution in nonnegative rational integers x l 5 ..., xs, yx, ..., >'s such that (2)
max(xh
yj)
^ max(ah
b/lk+c.
We use hereafter e, e l5 ... to denote arbitrary preassigned positive numbers < 1. The number l/k in (2) is best possible. Although the circle method is still used in the proof of (2), the treatment of the minor arcs is completely distinct from that in Waring's problem. * Supported by the Institute for Advanced Study, Princeton, N. J. 08540.
223
118
Wang Yuan
It was Siegel ([13], [14]) who succeeded in dealing with Waring's problem in an arbitrary algebraic number field by his generalized circle method, and he obtained the result corresponding to Hardy-Littlewood's estimation on G{k). Siegel's result was improved by R. G. Ayoub [1], Y. Eda [4], O. Kdrner [8], R. M. Stemmler [15] and T. Tatuzawa [16], [17] in various aspects. By the combination of the methods of Schmidt and Siegel, we can generalize Schmidt's theorem to an arbitrary algebraic number field. Let K be an algebraic number field of degree n. Let Kip) (1 < p ^ r t ) be +r2) denote the real conjugates of K and let K{q) and K.(q + ri) (rt + l ^q^rt the complex conjugates of K, where rx +2r2 = n. Throughout this paper, the indices p and q are over the sets of integers cited above. For yeK, we denote n
by }'w (1 < / < « ) the conjugates of y and by N(y) = Y\ 7li) the norm of y. Let ;= I
7j (1 < i ^ n) be numbers of K and x ; (1 ^ j < n) be real numbers. We set ^ n
= Y, xjlj
an
n
d define £(l) = £ x}yf
j= i
(1 ^ i ^ n). We use the notations
i= l
lid = max |
S(^)=fe>
and
Eg) = exp (2TUS(£)),
i= 1
where exp(x) = ex. A number y of K is called totally nonnegative if y(p) > 0. Let a1, ..., OLS, / ? ! , . . . , & be 2s nonzero totally nonnegative integers of K. Consider the equation of the type
(3)
<*!%+...+«,% = 0!^+
...+?,£.
A set of numbers Ax, ..., A,, fit, ..., fis satisfying (3) is called a nontrivial solution o/(3) if Ai, ..., A,, / i l 5 ..., fis are totally nonnegative integers of K, not all zero. Set (4)
m = max(N(ai),N{Pj)).
In this paper, we shall prove the following THEOREM. Suppose s ^ c6(k, n, E). Then the equation (3) has a nontrivial solution such that (5)
max(JV(Af), N(nj)) < m1/([+£.
Here and below the constants implicit in
224 119
Bounds for solutions of additive equations
this investigation we will show that if s ^ c7(k, n, s), the equations a1 al 2\ + ... + as as Xks = 0 has a solution in a1, ..., as, / l 5 ..., ks, where each a, is 1 or — 1 and where Xl (1 < i ; ^ s) are totally nonnegative integers, not all zero, with max N&) < m a x ( l , |N(a,)|, ..., |iV(as)|)c. i
2. Several lemmas. L E M M A 1. Lef fl5 ..., f Pl+r2 ^ e a sef of real numbers r,
(6)
satisfying
rj+r2
I
p=l
X f, = 0-
« = rj + l
Then there exists a totally nonnegative unit a of K such that c8" l e" < a{p) < c8 e",
c8" 1 e'q < \(T(\ < c8 e'9,
where c8 = c8 (K). See, e.g. Lemma 1.1 in Hua Loo Keng and Wang Yuan [6]. (Put a 2. There exists a rational integer c9 = c9 (K) such that for any integers a, /? of K, where fi # 0, there exist a rational integer I and an integer co of K such that 1 ^ / ^ c 9 and \N(k-ctip)\ < \N(P)\. See, e.g., K. Ireland and M. Rosen [7], p. 178. L E M M A 3. For any t integers ) \ , ..., y, of K, not all zero, let y be a nonzero element of the integral ideal a = (ylf ..., yt) with the least norm in absolute value. Then LEMMA
£9! 1',/y,
1< i
are integers. Proof. Set a = y, and P = y in Lemma 2. Then there exist a rational integer /,- and an integer oii such that |W(U-o>,-y)l<|JV(y)|,
K/,
Since /,• yt — &),-yea, it follows that JV(/,-ft—<»,•?) = 0. Therefore /,ft —
7
h \7 J
are integers. The lemma is proved. 2 LEMMA 4. For any t integral vectors (ft, dt) (1 < / ^ t) of K , where y, / 0
225 120
Wang Yuan
(1 < i < 0, if 51
=
=
d,
then
(7i,^)=^yZi(y^),
Ui
where y is defined in Lemma 3, and where 5 and Xi (1 < i < t) are integers.
Proof. By Lemma 3, X\ = c^-lih (1 < ' < 0 are integers. Let yi
it
Then <5; = ay; (1 < / < t). Since ((5j, ..., dt) = <x(y1, ..., yt) = <xa is an integral ideal and yea, 3 = ay is an integer. Therefore c9!
c9!
c 9 ! y;
c9!
\c9\J
The lemma is proved. LEMMA 5. For any nonzero integer
c io = 4max £ \cof\ *
j= I
and
JVP = — - c 1 0 , l
\
a
I
Kp^n-
Since the matrix (a}Jp>) (1 ^ p < r1? 1 ^ j < n) has rank r 1 ; we may suppose det(a»jp)) ^ 0 (1 =$ p, j' ^ rj). The system of linear equations
has a unique solution. Set aj = [xj] (1 ^ j ^ r j , where [x] denotes the r
i
integral part of x. Then we have an integer y = £ a,- a>, satisfying y(p)
j=i
3. Reductions. PROPOSITION 1. Suppose that x ^ 1/fc and s ^ c u (/c, n, x, e). Then (3) /jas a nontrivial solution with m a x (JV(A,), N(/i,.)) «
OT
X+
'.
226 Bounds for solutions of additive equations
121
The case x = \/k is the theorem. One can prove by Siegel's method that if s ^ c12{k, n), then the equation of the type
a, A5+ ... +«,A*-«, + 1 ^ + 1 - ... - M * = 0 has a nontrivial solution in totally nonnegative integers Als ..., As such that max JV(Aj) « max N(a,)Cl3<*'1*,
(7)
i
where a ! , . . . , a s are given nonzero totally nonnegative integers and l ^ f <s-l. It will suffice to prove Proposition 1 when m is large, say m ^ c14(fc, K, x, e). In fact, if m < c 1 4 and s^ c12, then it follows by (7) that (3) has a nontrivial solution such that
ma.\(N(X,), N(pj)) <§ mCl3 « ctf < w ?+E . Let X be the set of x such that Proposition 1 holds. Then (7) shows that X is not empty. It is clear that X is a closed set. Hence the proof of Proposition 1 is reduced to proving that if x > l/k and xeX, then there exists an x' e X, where x' < x. For 1 < 7 < s, set t, = k-' (log N(ot,)1'" + log |aj°| ~l),
1 < i < n.
Then (6) holds, and therefore there exists a set of totally nonnegative units a} (1 ^ j =% s) such that
eg l Nia/'+iaf)-llk
< of ^ c8 N{aj)1'* {af)" x / \
eg J N (<Xj)llnk | a f |" 1/(t < |
< c| N(tXj)lln,
eg * iV (a7)1/n ^ |aj«> fff| < c| AT (a,-)1/n,
1 < j ^ s.
Similarly, there exists a set of units T,- (1 <_/<s) such that c^Ntfj)1*
< |j8j«>TJ**| < 4 A/^) 1 '",
1 < ; < s.
Let Then (3) becomes
(3)'
k a'lX'l +
...+a'sKk
= ft < + ••• + f t ^ -
227
122
Wang Yuan
If Proposition 1 holds for x' and for the particular equation (3)', then we have a nontrivial solution of (3)' such that max(NW), N(^)
< max(N(«;), N(P'j)Y + <.
Since N(ty = AT (A,), N(^ = Nfji,), N(aj) = N(af), N(ft) = AT (ft) (1 < i < s), we have a nontrivial solution of (3) with max (A/(A,), N(fij)) 4mx+c. u i.e., Proposition 1 holds for x' and for (3). Hence we may suppose without loss of generality that a, and ft satisfy
cfs1 A/fa)1'" < a{" < c 15 A/fo)1'",
cfs1 ATfa)1'" < |aj«| < c l s iV(a,)1/n,
CiV ^(A) 1 '" < # " < c 1 5 ^(A) 1 '",
^r, 1 A7(ft)1'" < |j8}«| < c 1 5 A/(ft) 1/n ,
(8)
1 < i < s, where c 15 = c15(/c, K). In what follows, x will be a fixed number > 1/fc for which Proposition 1 holds. Take y sufficiently small such that 1/fc + 6c 13 ny + 20ny < x
(9)
and
22fcny < 1,
and put (10)
x' = max (x (1 - \ y) + y/2kn, 1/fc +6c13ny + 20ny),
so that x' < x. We proceed to prove that Proposition 1 holds for x'. Let e1 — min(e/8x', e/4) and divide the interval [0, 1] into a finite number of intervals (/}• of length < e t . If s is large, one of these intervals / will be such that many of the coefficients a l 5 ..., as are of the type
NioLd^rn',
me I.
We may therefore suppose without loss of generality that N(«,)
, ^ . .^
.x
Similarly, we may suppose !%<+•
'<»<••
Let a* = m1 max A7 (a,) and fc" = ml max N(ft). Let p, and q, be the largest i
rational integers such that N (a,) rf" ^ a"
i
and
N (ft) ^f" < b",
1 < i < s.
228
123
Bounds for solutions of additive equations
Since m ^ c 14 , an/N(ti^ ^ ml and bn/N(fi^ ^ mx, we may suppose
Hence JV(a,)/jf">io"
Nifid^" > \bn,
and
1 < i < s.
Set a| = a, pi, ft = ft «?, A, = p, AJ, ^ = 9 j /<; (1 < i < s). Then (3) becomes (3)', and by (8), a| and ft- satisfy (2c 15 )" J a < a?"> < c 15 a, (2c 1 5 )- 1 a
(2c 15 )" J a < |a,-^>| < c 15 a,
(2c 1 5 )- 1 fe<|ft ( « ) |
l
Suppose that Proposition 1 holds for x' and for the particular equation (3)'. Then there exists a nontrivial solution of (3)' with max (AT (A;), Ntf)) < max (a", bf+tl* < w ( 1 + £ i ) ( J C + £ / 4 ) <^ m*'+«/2. Since /V(af) = ml iV(aj)maxiV(aj)/mEl maxN(a,) = a"m~£liV(a,)/maxN(a,) ^ a B m" 2 e i ,
1 < i < s,
i
we have pf < rf" ^ fl7N(a,-) < m2£l < mE/2,
1 < i < s,
and therefore N(Aj) « p? JV(A0 < w x ' +£ ,
1 ^ i ^ s.
Similarly N(ni)<^mx'+£,
Ki^s,
i.e., Proposition 1 holds for x' and for (3). Thus in proving Proposition 1 for x', we may suppose that (11)
c16a
c16a < |aj«>| < c 1 7 a,
c 16 b < j8lp)
c 1 6 6 < | ^ 4 ) |
l^/<s
for certain positive numbers a, b, where c16 = c16(/c, X) and c 17 = c17(fc, X). 4. Continaation. In what follows, A will be the integer occurring in Proposition 1, and s > h. Set (12)
z = y/2kn2.
cu(k,n,x,e)
^ _ _ _
229
124
Wang Yuan
We distinguish two cases. A. There is a subset of h elements among a l 5 . . . , a s , say a l 5 . . . , <xh and there is a subset of h elements among /? l s . . . , / ? „ say /? t , . . . , j5A, and there are totally nonnegative integers at, ..., ah, i j , . . . , rfc such that (13)
0 < ||T,|| < mz,
0 < ||CT,.|| < nf,
1 < i ^ /i
and
\N{a)\ > m», where a is a nonzero element in the integral ideal {oL-^a^, ..., a.hah, PiTi, ...,phrh) with the least norm in absolute value. By Lemma 5, we may choose a nonzero integer y such that ||y|| ^ c 10 and ya is totally nonnegative. By Lemma 3, a,- =
and
p; =
-,
1 < i < fc
are all nonzero totally nonnegative integers. Therefore it follows from the case x of the Proposition 1 that the equation
a', A?+...+«U? = ft !*?+•••+&/*? has a nontrivial solution satisfying max(JV(A;), JV(/*J)) « max(JV(aO, N(P'j))x+e « „,(!+*«-«(*+«). Let Aj =ff;A,', /I, = T; /i; (1 < i < /z) and Af = ju,- = 0 (Ji < i: < s). Then by (10) and (12), the equation (3) has a nontrivial solution with max(N(Ai), N(nj)) ^ m u+*«-)»<*+*> + « ^
u
m d-w2)(x+ e )
+« ^
m*+£
We are thus reduced to case B. For any h elements, say a 1 ; ..., ah, among a.l, ..., as, and for any h elements, say f}lt ..., f}h, among jS^ ..., /3S, and given any totally nonnegative integers ox, ..., ah, xl, ..., zh satisfying (13), the integer a defined as in the case A satisfies JiV(er)| <my. Condition B depends on k, n, h, m, y, and it is denoted by B(k, n, h, m, y). PROPOSITION 2. Let q = 1 or — l. Let
(14)
m = max (a", b")
and let a l 5 ..., as, plt ..., /Js be nonzero totally nonnegative integers satisfying (11) and B(k, n, h, m, y). Then if s ^ cls(k, n, h, y), the equation
M i + • • • +«,#-& M - • • • -P./4 = «
230
Bounds for solutions of additive equations
125
has a solution in totally nonnegative integers Xx, ..., Xs, fiy, ..., /xs, x, not all zero, with max(JV(A,.), N{jij)) 4 m 1 ' " 2 0 " , i.j
|| z || < m6>'.
Now we proceed to show that Proposition 2 implies that Proposition 1 is true for x'. Let x, x', y, z, h be as above. Suppose that c 12 and c 18 are integers. Let s = uv, where u = c 18 and v = c12. Replace the indices 1 < / < s by double indices 1 ^ i < v, 1
(15)
£ (<xfl ^ + ... +a? u 4-ft, rf, - ... - A ^ J = 0.
i= 1
For each i, 1 < i < u, the coefficients a a , . . . , a,-,,, Pn,...,Piu satisfy the conditions in Proposition 2. Hence there are totally nonnegative integers X'n, ..., A;u, rt'ls ..., / C X.-. not all zero, such that «n ^ ' i + • • • + «.u %* ~ A i A*** — - - - — A« ^ : * = «3i X; with
max (AT W,.), iV(/i;,)) « m 1 ' ^ 2 0 ^ ,
||Zj|| < m6^.
We may suppose that Xi ^ 0 (1 ^ i ^ u). Otherwise we get a small solution straightaway. Take qx = ... = qv-x = 1 and qv= — 1. Then by (7), the equation
has a nontrivial solution satisfying maxNfaMin* 1 3 1 *. i
Let Ay = 7(Ay, /iiy- = JiUlj (1 ^ i < «, 1 ^ j ^ «). Then we have a nontrivial solution of (15) having
max(JV(^), Nfad) < mllk+6ci3"y+2Ony Uj,t,l
< m*'.
Thus Proposition 1 holds for x'. be an integral basis of K, h the 5. Weyl's inequality. Let oii,...,(on different and D the absolute value of the discriminant of K. We can choose a basis gl, ..., Qn of b~ ! such that
S
^
= \0,
otherwise.
m 126
Wang Yuan
Set £, =X1Q1+
a n d n = y l a ) 1 + ... + yncon,
...+xHgn
where x; and yt (1 < / ^ n) are real numbers. We denote by P(T) the set of (>>!, ...,yn) satisfying 0 ^ tfp) < T,
|i/(*>| ^ T;
a sum where /I runs over all integers such that 0 ^ Aip> «$ T, \ki9\ < T,
£ AsP(T)
and
^] a sum of integers /z satisfying \\n\\ ^ T U| 6 P(r)
LEMMA 6 (Siegel). Le? h ^ 1. Then for any £, ?fere ex/s? an integer a and a number fi of t)~ 1 such that
0<||a||
M-f}\\
(0
max(^|ot ^-^"l, |a |) ^ D~m,
1^ i < n
am/ N((a, i?b))
h>2
5
7 (Mitsui). Let A, B, h be positive numbers satisfying A ^ 1,
+ r2
D and 1 ^ B < 2'4~r2 D'^h.
X
min(A,
|l-£(c>JJ.)r
1
Then for any £
(1 < ; < n))
|/i|eP(B)
here and also in Lemma 8, a denotes an integer satisfying the conditions in Lemma 6. See Theorem 3.1 in Mitsui [9]. Notice that in the proof of his formula (3.42), we may use the estimation |JV(a)| > c||a|| instead of |iV(<x)| > cT with c = c(K). LEMMA 8 (Weyl's inequality). Let G = 2kl
and L(f) = X E(#t),
w
»ere
XeP(T)
Let h be a number satisfying +k+r
kl2*
Then
2DTk-i
k
T > k!
25+k+r2D.
232
127
Bounds for solutions of additive equations
Proof. By Holder's inequality X
X1
X1
X
^E£(u 1 r i ^+...f- 2
<2t 2 ii
" - j;E£(u 1 r i {+ ...)|2^2
2k 2 1
xi x
x2 x
4... A,
where (16)
x2
xk^1 x
Ai = / c ! A t . . . V i .
|Ai|eP(2r)
(l^/
and A runs over all solutions of the conditions
X+ X h + ...+XigeP(T)
(1 ^ ( l < . . . < i g ^ k - l ,
C
Let ,4(^) denote the number of solutions of (16). Then by the well-known properties of the divisor function, we have W
lO(T' 2 ),
otherwise.
Hence m
r
^ T"^-^+Tn(G-k)+£2YJ\IdE(fiXO\,
where the summation is extended over all fi, X satisfying HeP(k\2k~l I*-1)
and
XeP(T).
Since £
£(/MC) = ^(T" 1 " 1 min(r, |1 -E(najjC)\~l
(1 ^ 7 ^ «)))
XeP(T)
(cf. Siegel [14], p. 332), we have \L(Zf
4 T*G-»+T"iG-k)+e2ZjT'<-lmin(T,
Let /I = 7 and B = k\2k~l
|L(C)I
Tk~\
(1 < ; < » ) ) .
Then by Lemma 7, we have
^7 ^T
\1 -E(/uojS)\~*
I N I + 7 ^ T + ^ ^ ~ + ~7^J llNI + T + 7*/
233
128
Wang Yuan
The lemma follows.
=*
6. Schmidt's lemma. In this section, we shall generalize Schmidt's lemma to an arbitrary algebraic number field. £3 LEMMA 9 (Schmidt). Suppose that T > cig{k, K, e3), C ^ T" and |JL(£)I ^ C. Then there exist a totally nonnegative integer a and an integer f} such that
and 0
fT"\G
("c) r''
where a = a'y and fi = /?'y in which y is an integer satisfying \\y\\ ^ c2o(K) and a', P' satisfy the conditions of Lemma 6 with h = 7Jc~E3(C/T'I)G.
Proof. We have / r \G
/T''-1/C+£3\G
and
Let
Then /i satisfies the condition of Lemma 8 for T^ c19. By Lemma 6, there exist an integer a' and a number /T of b " J satisfying
Ha'S-Zni*:*- 1 ,
0<||«'||
and the other conclusions in Lemma 6. Take E2 = £3/2G. Since
and T""1/G+£2^C7^2"£3
we have by Lemma 8 that
c^iL(ai«r" +t2 na'ir 1/G ,
234
129
Bounds for solutions of additive equations
i.e.,
There exists a nonzero integer yy such that ||yx||
(o, b) = l.
We write •• -> a. Let t > 1 and F(t) be the set consisting of y = x, QX + ... ... +xngn satisfying (xj, ..., xn)eGn,
x, (1 < i'^ n) rational numbers, and
y->a
N(a) < t".
Let h = abm20ky-yln
(17)
and
t = mytn.
For any ye.T(t), subject to y -* a, we define the basic domain B r by {(xu ..., xn): (xj, ..., x n )eG n , ^ = x ^ ! + ... +x (I e (1
(18)
such that ft ||^ — yo|| < 1 for some y0 = y(mod b" 1 )). We may prove that if yt ^ y2, then J5yi nBy2 = 0. In fact, suppose there is & £eBn
r\By2, i.e., ^||^-y o ,|| < 1, where yOi = y,(mod b" 1 ) 0 = 1, 2). For
simplicity, we set yOl- = y, (i = 1, 2). Write max W > - y j " | , r J ) =
ffj0,
1 ^ i < «, 1 < ; < 2.
Then
f\of<\,
maxiafy^t,
j=l,2,
i= 1
and thus |y<«-y<0| < |^(0 _ y (0| + 1^(0 _ y 0 ) | ^ ^ - 1 ^ ( 0 + ^ ) )
= h-'af
af^r1+(
^ 2h-lofo2»t,
Suppose yt -* a,- (i = 1, 2). We have JV(01a2)|JV(y1-y2)|<(2r1f3r
Ada Arithmetica XLVIII.2
1 < i < n.
J235
130
Wang Yuan
since m Js c 1 4 . On the other hand, ax a2{y1 — y2)i> is an integral ideal, and thus N(a1a2)\N(y1-y2)\^\N(b-1)\=D-1. This gives a contradiction, and therefore the assertion follows. We define the supplementary domain E by (19)
E = G n-
U B y.
yefW
We use the notations £,
A=b (20)
$(£)=
....+xnQn,
=X1Q1+ ilk
20
m *,
X Ei^XH),
dx^dx^.-dx^
B =a
1/k
20
H = m6",
m \
T,(Z)= X £ ( - A / a
Ki<s,
S(^=nS,(a T(Q=f\Tt(Z) >=1
i= 1
and
F(«= X S(c)T(O£(-^), where q = 1 or — 1. Let Z denote the number of solutions of the equation in totally nonnegative integers At, ..., Xs, nlt ..., fis, x satisfying i,eP(A),
ft,eP(B)
(1 < i < s),
zeP(«).
Then (21)
Z = j ; jF«)dx+fF«)dx.
We shall show that under the assumption made in Proposition 2, Z > 1. 8. Supplementary domain. Take £3 such that (22)
£ 3 <1/2G,
e3(l- + 20y)
and s so large that
(23)
s>l^+fc. z
LEMMA
(24)
10. Suppose that (x 1 ; ..., xn)eGn and
\F(a>H"(ABrm-A.
236 131
Bounds for solutions of additive equations
Then £ lies in a basic domain. Proof. We may suppose that Then F(^)^HnA''(h~1)B'ls\Sh{^rh+1, and thus by (24) and m ^ c 14 , we have |S,(£)| > |S»(0| > Anm-5'is-"+1)
= C,
say for 1 < i < h.
By (20), (22) and (23), we have m S/(.-*+l)
^ ^1/4^.-11+1) ^ ^1/20 <
1 A ^-'3t
and therefore It follows by Lemma 9 that there are totally nonnegative integers
= mz
and ||^ ; CT ( .-
5G
«s-'1+1M£3"'t<m^-k,
1
since m^ cx^.. After a recording of fi1, ..., f}s, we may also suppose that
iT1(£)i>...>iT,(a Similarly, there are totally nonnegative integers T,- (1 < i < /J) and integers ^, (1 < i; ^ h) having 0 < ||T,|| < nf
and
||<SAT,-^,|| < m z B " \
1 ^ i < h.
Hence by (11), (12), (20) and m > c 14 , we have 11% /Sj Tj - IAJ a,ff.ll= 11% 0; T; - ^a, o-,. j3; T ; + {«,ff,-/?j T ; - ^ a, tr,\\ < H/Jj Tj|| ||^a, tr, - Vl\\ + ||a,
+
am2zB"k^m2z-2Oky<\,
and thus
Since ^i/J/Tj —^a,-ff,- is an integer, we have Thus by Lemma 4, the 2/i integral vectors c9!(a, o1,, (/>,-) and c9!(/y,T,, i^,)
237
132
Wang Yuan
(1 < / ^ h) are all integral multiples of an integral vector (a, x), where a is a nonzero element of the integral ideal (ai<7j, ...,a.hah, ^ T X , . . . , f}hTh) with the least norm in absolute value. Therefore the condition in case B yields that 0 <\N(a)\
<my.
Let ff~1Tl) = b/a>
(a, b) = l.
Then a\a, and thus
N(a) <\N(a)\ <my = tn. Since Hc^H < mz, we have
and by (11), (12), (17), (20) and m > c 1 4 , Ir(') _ C T ( ' ) \ - 1 T0')| _
|^(0.-(') ^(0
, n (i)|
^ " ' m ^ - ' = a-lb-lm~20ky = a-1b-lm-2Oky+yl2nk
+ nz
1
1
Therefore <^eBv, where y = a~ x(mod b" )- The lemma is proved. 9. Basic domain. We use the notations C - y = C,
rj = y1a>1+
G,(y) = N(a)- 1 £
...+yna>n,
£(M*y),
dy =
//,(y) = N(a)- 1 ^
•i(moda)
/, (C, /I) =
J £ (a,- ^* 0 ^ ,
G(y) = f l G,(y), '= 1
E(-p,tfy),
/j(moda)
•// = (C, B) =
P(/»)
(25)
dyx...dyn,
J £ ( - A /7k 0 ^ ,
1 < i < *,
P(B)
H(y) = fj «,(y), i=i
/(C, >D = f[ hiC, A) and i=
I
1=1
where y -* o. LEMMA 11. Let a be an integral ideal. Let N(a, T) be the number of elements v of a satisfying 0 ^ v(p) ^ T.
\v(q>\ ^ T.
238 133
Bounds for solutions of additive equations
Then
where To = max(iV(a)1/n, T). See, e.g., Lemma 3.2 in Mitsui [9]. Notice that the conclusion is still true for the number of v satisfying v + nea, 0 =$ v(p) + n(p) < T and \v{q) + i/q)\ ^ T, where /^ is a given number in a residue class mod a. Now we can prove the following lemma by the Siegel argument (see [14], pp. 328-330). LEMMA 12. Suppose that £eB y . Then (26)
St(Q = Gi(y)Ii(i;,A) + O{t2A"-1)
and (27)
Ti(Z) = Hi(y)Ji(t;,B) + O(t2B"-1),
1 < i < s.
Proof. Determine positive numbers 0(O (1 < i < «), with 9W = 0iq + r2\ such that 0(Omax(/iiC(O|, r'A^a) 1 '") n
= D 1/2n f l max(/i|C0)|, r l N(a)l/nyinN(a)1/n,
1 ^ i < n,
Then f ] 0W =
Dll2N(a),
i= 1
and it follows by Minkowski's linear form theorem that there exists ctea such that 0 < |a(l)| < 0(l), 1 < i < n. Hence aa" 1 = b is an integral ideal and
N(b) = |JV(a)| JV(o)-1 ^ (fl tf'Wa)-1 = v ^ ; i=l
hence b belongs to a finite set depending on K only. Let ax, ....,
1^ i^ n
satisfying ||Tl.||=O(||a||) = O(max0 w ) = O(f). i
Let n run over a complete residue system modulo a, and k over all numbers
239 134
Wang Yuan
in a such that X + fieP(A). Then S,(£)=
(28)
£
E(ociMky)
£
£(a,(A + ^ C ) .
/I + / J S P M )
Expressing A in terms of T,- (1 < i ^ n), we obtain ^ = 01*1+
where gi,...,gn
•••+9n^n,
are rational integers. Let G(A) denote the cube
(S,, . . . , Sn): ff = S 1 T 1 + . . . + S B T B ,
gfj^Sj < 3 , - + l
(1 < / < « ) .
Then ||
ii^+^c-^+^cii^iiff-AiiiiciKiiff+Aiir^p+^ir1)^^-1^-1, and therefore by (11), £(a,(A + /i)*C)= f £(a,(a + /i)kC)rfs + O ( a f / i - 1 ^ - 1 ) ) where ds = ds!...dsn. Since iV(a) ^ f" ^ ^l by (17) and (20), it follows by Lemma 11 that the number of X with a\k and A + /ieP(/t) is (^(^(a)" 1 /I"). Therefore X a|/
£(a,.(A + /i)kO =
X
J £(a,(
a|;. G(X) X + fteP(A)
Let F denote the domain in the s-space defined by 0 < a(p) + n(p) ^ A,
\a{q) + ^(«»| < A.
Then the volume of the area belonging to exactly one of
(J
G(-i) and
oi;.
F is dominated by 0{N{a)-1 tA"-1). (See, Siegel [14], p. 329.) Therefore by (17) and (20), we have X
£(a,.(A + /i)*C)= j£(a i (a + M f C ) ^ + O(iV(Q)- 1 f 2 ^- 1 ).
Let
240
_ _ ^ 135
Bounds for solutions of additive equations
we have
E(«i(A + tift) = N(ariIi(t,A)
X
+ O(N(a)-it2A»-1).
A + pePM)
Substituting into (28), we have (26). The proof of (27) is similar. 10. Continuation. We use En to denote the whole n-dimensional Euclidean space. LEMMA 13. We have (29)
7,.(C, A) « f ] nan(A, i= 1
fl"1''^0!"1'*)
am/
J,(f, B)« f[ min(B, ft"1/k|C('V1/k)-
(30)
i= 1
See, Siegel [14], p. 335. The only difference between the proofs of (29) and the corresponding formula of Siegel is that we use <x{p) x(p) and |aj" T(9)| instead of his T(P) and IT*^. LEMMA 14.
= G(y)H{y)E{-qxy)
O((AB)ns(ab)-"m-2Ok'll'-11}').
J /(C, A) J(£, B)dx +
Proof. By Lemmas 12 and 13, we have S(0T(0
= G(y)if(y)/(C, A)J(C, B) + O{{ABTt2mzx{A~*,
B" 1 )).
Let C(P) = « P ,
(31)
Cq) =
uqe^.
The Jacobian of xt, ..., xn with respect to up, uq, q>q is equal to the product of the Jacobian of xit ..., xn with respect to C
2 r2 D 1 ' 2 n«V It follows by (17) and (20) that
J d x ^ n ( f d«p)n(j" I M,du < ,^,)«r n = (a6)-"m-20'""'^
By
P
0
4
-it
0
241_ 136
Wang Yuan
and Therefore (32)
\S(Q)T(0E(-qX0dx By
= G(y)H(y)E(-qXy)jI(C,A)J(C,B)E{-qxQdx + By
+
0{{AB)ns{ab)-nm-20kny-11?).
In the integral in the right-hand side of (32) we replace E{ — q%Q by 1. Then by (20) and Lemma 13, the error is (AB)m \\\xQ\dx <^(AB)mHh'n-1
^(AB)ns(ab)-"m~2Okn:>-lly.
By
Hence (33)
f S(c) T(Z)E(-qX{)dx
= G(y)H(y)E(-qxy)
By
j I(C, A)J(C, B)dx + By
+ 0({AB)ns{ab)~nm-20kny-'Lly). If (*!,..., xn) is a point in En-By, then /i|C(l)| 5= 1 is true for at least one index i. By Lemma 13 and (31), we have
J
En-B
I(C,A)J(C,B)dx
{f\mm(A,a-1/k\C{i)\-llk)Umin(B,b-1/k\i:U)\-llk)Ydx
<$ J En-By
i = l
j=\
<$( f (^)" s/ ''«" 2s/k ^M)(jmin(/l 2 , a-s'kv-s/k)min(Bs, x( } ]min(A2s,
a'2s/kw~2slk)min(B2s,
b-^v-^dvf1'1
b~2s/kw-2slk)wdwd(p)r2
x +
-n 0
+ (|min(^ s , a- s/k u- s/lt )min(B s , b-slku-slk)du)ri o
x( J J (abr2s/kv~4s/k+1dvd(p)( -n
i,- 1
x
] JminO4 2s , a~ 2s/ * W - 2 ^) x -it 0
xmin(B2s,
b'2slkw'2slk)wdwd(pf2~l.
242 137
Bounds for solutions of additive equations
Since \min{As,
a-
s k
i u~stk)mm{Bs,
b~s/kiTs/k)du
0 1
4 A* Jmin(B\ b~ *'k u~ s'k) du o <^/ls(
J
Bsd« +
b-slku-s'kdu)^AsBs-kb-1
f
and jmin(/l 2 s , a- 2s/k w- 2s/ *)min(jB 2s , b~2s/kw-2slk)wdw
<s
A2sB2is~k)b~2,
0
we have by (9), (12), (17), (20) and (23), J
I(C,A)J(t;,B)dx
2s k 1
l - (abrslkAlri'1)sb~("i~1)B{ri~1Hs~k)A2r2Sb~2r2B2r2(s~k)
+ Ari'b~ri
Brils~k) h*slk~ 2 (ab)~ 2s/k A2(r2~l)s
^h2slk~i(ab)~s'kb~n+1 +
h
4slk 2
A{n~1)sB{"~1)is~k) 2slk
~ (aby
b~
n+2 (
<(ABr(abrnm-2Okny(m~i^l+y»
B "~
+
b~ 2{'2~ U B2{r2~
m k )
+
2Hs k)
-
+ rrn^l+~^)
<{AB)m{abynm-20kny-lly. The lemma follows by substitution into (33). 11. The singular integral. Let q' = y\a>1+ ... +y'na>n, C = X\Q1+ ... + x'nQn,dy' = dy\...dy'n, dx' = dx[ ...dx'n, t, = Ar,' and C = a^b'1 m-2OkyC The Jacobians of yv, ..., yn and x l 5 ..., xn with respect to y\, ..., y'n and x\,...,x'n are An and (a~* b~* m~20ky)n respectively. Set % = ixja (1 ^ i ^ s). Then where by (11), y, (1 < i < s) satisfy (34)
c 1 6
c16<|y|«)|
1 < i < s.
Let us write rj' and £' as ?/ and C again and let p
where P = P(l). Then /,(C,yl) = /r/,(O,
Ki<s.
243
138
Wang Yuan
Similarly, we have Jt(Z,B)
l
= B"MQ,
where J i ( Q = $E(-ys+ir,kOdy,
l^i^s,
p
and ys+1 = ft/ft (1 ^ i ^ s) which satisfy c16
(35)
1 < i < s.
c16<\M
Set
/(0 = f l /,-(0, i=l
J(Q=f[
i=l
Ji(0
and
Then we have j /(C, i4) J(C, B)dx = (/lB)" s (afe)- n m- 20ltn) '
(36)
Now we shall treat the integral
then F is said to be monotonic over I. LEMMA 15 (Tatuzawa). Let F(xlf ..., x,) be a finite product of positive bounded monotonic functions over the rectangle {(Xl,...,xt):
(Hi
0^x,^c,
/ / we write ,
x
sin27i-lx nx
then lim
^ • o
+ O, ..., +0).
f ... $F(xl,...,xt)xx1(x1)...Xx,(xt)dxl...dxt=&'F(
0
0
See Tatuzawa [17], pp. 47-49. LEMMA 16.
0
= D(
1 - 2s),2 k - 2ns N
(y i
?s)
- 1/k J J Fp Y[ Hq, P
9
244
139
Bounds for solutions of additive equations
where
in which Up denotes the domain 0<w,
l
w2s = w t + . . . + w s - w s + 1 - . . . - w 2 s _ 1 ;
and where 2s
H
q
Uwi'k~ldwi-dw2s-id
= [ K, 1 = 1
in whicn ^ denotes the domain 0 < w, < lyj"!2,
1 < i < 2s,
-re< ^ < 7i, 1 < ; ^ 2 s - 1 ,
Proof. By Lemma 13, we have
/.•(O^riminClJ^r 1 '*),
1<'"<*,
and J,(0 (1 < i < s) satisfy the same inequality. Then by the transformation (31), we have
\\I{QJ(C)\dx
En
P
0
q
-it 6
which converges for s > k. Therefore <*>=
lim
4>(Q),
*(O) = J / ( 0 J ( 0 d x ,
where (2 denotes the closed region of x defined by
Ki < AP,
|»,| < A,,
I«;I ^ A;
,
^
vq
in which ^P-C
vq-
,
^_.
Consider 2ns real variables ytJ (1 < i < 2s, 1 < j < n). Let It = yaOil+ ... +yin(on,
dYt = dyn ...dyin
and let P, be the domain 0^n|p)^l,
l^'Kl,
lKi^2s.
.
245
140
Wang Yuan
Let and Z
« + Z« + r2
Z
« ~ Z 9 + '2
V2
V2'
Since r(«) = v" +
iv
r ( « + r *' = " ' " ' ^
"
7 = "'"'"«
7
= ^
+ >
'"«
we have i=l
p
q
q
The Jacobian of x l 5 ..., xn with respect to fp, t>,, ^ is equal to |det(CJ»)|-! | -if 2 = D J ' 2 . Set du = d^!...dr r i dvn + !...dv r i +,2dv'ri +!...dv'fl
+rj.
We have *(fl)= J ... f (/y1...dy2ljexp(2jui t px
Pzjldx
j=i
n
P2S
= D''2 J . . . f dy 1 ... < /y 2 l jexp(2ni(5;ii p » p +!«,»,+X«i»i))^
=^>1/2 j - 1 n^p(«p)n(^K)^K))^...^2Sf
l
P
2J P
4
Let zj = f,
1 =$ i ^ n.
Then ,,(i) M (0d _ -
/.,(0 M (0d i
i ,,(0 M(0
-,(0
M (0k
_
_
v(0
M (0k
\
T h e J a c o b i a n o f ) ' 2 v , i , ..., >' 2 s,n w i t h r e s p e c t t o t r , . . . , t n i s e q u a l t o |det(/cy« ,K>*-1 a>y>)rJ |det(^ ; >)| = JV(fc"»lyj,1 /;L"kl), and the Jacobian of f1; . . . , tn with respect to u p , u,, w^ is
|det((o(r'))r1|'r2 = O- 1/2 .
246 141
Bounds for solutions of additive equations
Therefore *(O) = J UXxp(up)U(Xxq(uq)Xx'q(u'q))du Q
P
x
q
x J...
J
N(k-l\y;.1r,lrk\)dY1...dY2a-1,
where Q is a closed region containing the origin of u in its interior and du = YlduPY\(duqdu'q). p
«
Let
ttf> = yncoV+ ...+yjM?
-utf'e1**"1,
1
The Jacobian of y ^ , . . . , y jn with respect to ujp, ujq, u'jq is Idettwi'Or 1 N(k~l |J7J-"|) = D" l'2(k~l \nj ~k\). Then we have «
OP
2s-1
x [ ] {dun...dujiri+r2dil/jtri
+
1...d\)iln+r2),
where /? denotes the region OsSu^sSl — 7t < «Art =^ 7T
(1 < j < 2 s - l , 1 < / < r , + r 2 ) , (1 < r ^ 2 s - l , r i + l < r < r i + r 2 )
- y ^ ^ " = ^ - ( y ( i p ) « 1 P + . . . -ylfi'-i «2.-i.p), l7a^-k,-(^«ii 2 e I '* l € +-..-y!& ) -i«Ji 2 -i.,^ a '- 1 '')l,
Therefore $=
lim
where
F;= lim k-2syfs'1 lxx(up)dup\ i J e W , . . . ^ . , ^p"""
Qp
-Upi=l
247
142
Wang Yuan
in which wj is used instead of uip, Qp denotes the range of up in Q and U'p the domain 0 < w| < 1 ( 1 < i < 2s),
y
and where
H'q = lim fc"4'|yg|-2 Jx,,K)^K)d«,^;x V^""10
Qq
in which wj stands for uiq, i/>,- for i//,-9, g , denotes the region of uq and u'q in Q and Vq the domain 0 ^ w; ^ 1 (1 ^ i < 2s),
(1 ^ j ^ 2s-1),
-K^II/J^K
lyKl2 wai = ^f - W < / 2 e1*1 + • • • - i&-1 < - . «*2*~ Ml2-
By Lemma 15 and the transformations y\p) w[ = wt (1 ^ i ^ 2s) for the integral F'p and \^\2w't = w, (1 < i < 2s), tpj = 0; + ^ (1 < ; < s), p, = 0, + ^, + 7t (s+ 1 ^ / < 2 s - 1) for H'q, where 0, = arg y^ (1 ^ f < 2 s - 1), we have
F'p = /c"2s f] y!" 1 " 1 '"^, i=\
H'q = /c"4s fl l y i T ^ w , , i=l
and the lemma follows. 12. The proof of theorem. By Lemma 11, we have (37)
X
1 = ^ 7 - H " + 0(//n-1).
Therefore by (9), (14), (21) and Lemma 10, we have z
Z
=
O(Hn(AB)ns{ab)-nm~2Okny-17y).
jF(Qdx +
yer(r) B y
For a given a, the number of y in T, subject to y->a, is O(iV(a)). By Theorems 35 and 76 in Hecke [5], it follows that the number of a with AT (a) = disO( X l) = 0 ( / 4 ) . Therefore dx...dn=d
E1*
£
iV fl
( ) ^ Z d2
and by (20), (36) and Lemmas 14 and 16, we have Z = J0 2(f, H)(AB)ns{ab)-nm-20kny
+
O(Hn(AB)ns(ab)-"m-20kny-14y),
where
Jo =
D^-2^k-^N(yl...y2srllkYlFPUHq p
i
248 143
Bounds for solutions of additive equations
and S=®(r,H)= £
I G{y)H{y)E{-qXy).
XeP(H) yelXt)
Let £ * denote a sum, where y runs over a reduced residue system of (ob)"1, y
mod b" 1 . Thus ye(ab)" 1 , (y, b" 1 ) = (a, b" 1 ), and we take only one y in each class modulo b" 1 . Then
®= I
I*G(y)Htv) I £(-«zy)+
= SJ + S2,
I
I*C(y)H(y) X £(-«zy)
say.
By (37) we have ®i = I
1 = ^ ~ - H " + O(/f"- 1 ).
If TV (a) > 1, then X(moda)
(See, e.g., Hecke [5], p. 197.) For any given integer n, it follows by (17), (20) and Lemma 11 that the number of v e o and v + fieP(H) is V^iV(o)
VV(a) 1 - 1 " 1 /
Hence if the domain ^e.P(H) is split up into a union of complete residue sets (mod a), plus a few others, remaining elements, say R elements, then
and therefore by (17) and (20)
&2< I ^H"-1
Y.*R
X!
rf3
^H'-'r^^H"™"2'.
Consequently, we have S^c22(X)H". It follows by (23), (34) and (35) that Jo *s c23(k, K, h, y). Therefore Z > c24(k, K, h, y)Hn{AB)ns{ab)-nm-20kn» if m $5 clt(k, K, x', e). The theorem is proved.
> 1
249 144
Wang Yuan References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
R. G. Ayoub, On the Waring-Siegel theorem, Canad. J. Math. 5(1953), pp. 439-450. B. J. Birch, Small zeros of diagonal forms of odd degree in many variables, Proc. London Math. Soc. 21 (1970), pp. 12-18. H. D a v e n p o r t , Analytic methods for diophantitie equations and diophanfine inequalities, Lecture Notes, Univ. of Michigan, 1962. Y. Ed a, On Waring's problem in algebraic number field, Revista Colombiana de Mat., 1975, pp. 29-72. E. Hecke, Lectures on Theory of Algebraic Numbers, Springer-Verlag, 1980. H u a Loo Keng and Wang Yuan, Applications of Number Theory to Numerical Analysis, Springer-Verlag and Science Press (Beijing), 1981. K. I r e l a n d and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1982. O. KSrner, Vber das Waringsche Problem inalgebraischen Zahlkorpern, Math. Ann. 144 (1961), pp. 224-238. T. Mitsui, On the Goldbach problem in an algebraic number field I, J. Math. Soc. Japan 12 (1960), pp. 290-324. J. Pitman, Bounds for solutions of diagonal equations, Acta Arith. 19 (1971), pp. 223-247. W. M. Schmidt, Small zeros of additive forms in many variables, Trans. Amer. Math. Soc. 248 (1) (1979), pp. 121-133. - Small zeros of additive forms in many variables II, Acta Math. 143 (1979), pp. 219-232. C. L. Siegel, Generalization of Waring's problem to algebraic number fields, Amer. J. Math. 66 (1944), pp. 122-136. - Sums of m-th powers of algebraic integers, Ann. of Math. 46 (1945), pp. 313-339. R. M. Stemmler, The easier Waring problem in algebraic number fields, Acfa Arith. 6 (1961), pp. 447-468. T. T a t u z a w a , On the Waring problem in an algebraic number field, J. Math. Soc. Japan 10 (1958), pp. 322-341. - On the Waring's problem in algebraic number fields, Acta Arith. 24 (1973), pp. 37-60. Wang Yuan, Bounds for solutions of additive equations in an algebraic number field II (to appear).
INSTITUTE O F MATHEMATICS ACADEMIA SINICA Beijing, China
Received on 30. 5. 1984 and in revised form on 18. 11. 1985
(1428)
250 ACTA ARITHMETICA XLVIII (1987)
Bounds for solutions of additive equations in an algebraic number field II by
WANG YUAN*
(Beijing, China)
1. Introduction. We use the conventions and notation introduced in [4] throughout this paper. Let au ..., as be a set of integers in K. Consider the additive form A(a, ^ = a 1 a 1 t f + . . . + a I a I # .
(1)
where a = (a1, ..., as) and A = {Xx, ..., As) are vectors. A set of numbers a, k is called a nontrivial solution of the equation (2)
A (a, A) = 0
if each a, is 1 or — 1, and the Aj (1 < i < s) are totally nonnegative integers of K, not all zero. Write
PI=max(P1||,...,PJ|)
and \A\ = max(l, ||ai||, .... W ) .
In this paper, we shall prove the following theorem by the combination of the methods of Schmidt [1] and Siegel [3]. THEOREM. Suppose s^ cx {k, n, e). Then the equation (2) has a nontrivial solution with (3)
M\<\A\:
This gives a generalization of a theorem due to Schmidt [1]. He first established the estimation (3) for the case of rational field. If a, X is a nontrivial solution of (2) with (3) for the case of k = 2, then a, A =(Ai, ..., As), with Ai = k} (1 < i < s), is a nontrivial solution of the linear equation <x1a1A1+ ...+asasAs
= 0,
having
PII«WI2. * Supported by the Institute for Advanced Study, Princeton, NJ 08540.
251^ 308
Wang Yuan
Therefore we may suppose k > 1 throughout this paper. If A is not identically zero, put A' = ^A, a where a is a nonzero element in the integral ideal (cc1> ..., as) with the least norm in absolute value and c2 = c2(K) is a rational integer such that
(4)
Let X be the set of x such that if s^cs(k, a nontrivial solution with
n, x), then (2) has
U\\<\A\X. (4) shows that X is not empty. Let x be the greatest lower bound of X. The conclusion of the theorem is x = 0. We will suppose that x > 0 and we will reach a contradiction. We may choose y such that (5)
0
and
x + kx2-kxy-k2x2y
< y.
(See Schmidt [1], p. 222.) Take z so small that (6)
y+12xz <x,
z < y/10,
z < 1/10.
Then pick x' with (7)
max (y +12xz, x — xz/(2n)) < x' < x.
We proceed to prove that x'eX. It will suffice to prove the assertion when |i4| is large, say \A\ ^ c6{k, K, x'). (See Wang Yuan [4].) And we may suppose clearly that cct =£ 0 (1 < i < s). Finally, pick x" with (8)
max(y +12xz, x-xz/(2n)) <x" <x'
and choose ^ such that (9)
(l+e 1 )x" + 4e 1 /fc<x / .
252
_ ^ _ _ _ Bounds for solutions of additive equations... II
309
Since l<|N(aj)|=|«J 1 >...«i'*|<|ay ) ||4-- 1 we have min\ctf>\>\A\-"+1. Divide the interval [ —n + 1, 1] into a finite number of intervals {/} of length ^ et. One of these intervals Jt will be such that there are not less than sx numbers among <X;'s satisfying
\«P\=\A\'1, where S j ^ s / l —£
/ VL i J
e1el1,
+ 1 ) . We may suppose without loss of generality that
/
a l 5 ..., aSJ satisfy the above relation. Similarly, there exists / 2 e U } such that there are at least s2 numbers in a x , ..., ccs satisfying
\af\=\A\e\
e2el2,
where
We may suppose that a t , ..., aS2 have the above property. Continuing this process, we obtain ( numbers among a,'s which we may suppose to be a l 5 ..., a, such that (10)
\of/«j»| < l^lf1,
1 < y, / < t , l < i < n,
where ^ S / ( [ ^ ] + l J . Suppose that if a 1 ; ..., a, satisfy (10) and t ^ c7(k, n, x1), the equation a 1 a' 1 Ai*+...+a,a;A;' t = 0 has a nontrivial solution satisfying max|A} (l) |«|i4| x '. Take cs (k, n, x') = ( — + 1 J c7 (ft, n, x') and set a,- = a[, A, = A,' (1 < i < t) fl.
= 1, A, = 0 (t < i < s). We have a nontrivial solution of (2) with
ii AIM u r . Hence we may suppose that the coefficients of (2) satisfy (11)
layyotj0! < I 4 ' 1 ,
Kj,Ks,
l^i^n.
253
310
Wang Yuan
Let at (1 ^ j ^ s) be a set of totally nonnegative units such that (12)
\N(aj)\lln
(See, e.g., Lemma 1 in [4].) Let a" = \A\ rational integer such that
|iV(a,)|P?n
1
1
Since \A\ > c6 and a /|N(a,-)| > Ml"* , we may suppose
Hence Set (13)
a; = a, o-J1 P?
A, = a\l at Pt k[,
and
1 < j < s.
Then Ax (a, X) = ai a, ^ k + . . . +
= a\A(a,
A),
where X = (X'lt ..., X's). By (11) and (12), we have
1< 7 < s , 1 < i < n
a<\cx.f'\
(i)lk^(0~(i)-l 1/* n 4)
(1
'
\a® ~' ff(0l -
'
x
J
J
J
J
'~ l^aWa?-1
«|iV( aj )| 1/(t ">|JV( ai )|- 1/(t " ) |^r i/ ' [ « M| 2 E l / \ 1 < ; < s , 1 < i < n.
Since |N(«()| = \A\mi |JV(af)| max|JV(a7.)l/Mri max|iV(a,)| '
j
= a-|iV(a l )||Ar"Vmax|JV(a I )| ^
fl-14"2"1,
we have
(15
»
**im<w'~t-
Suppose that the equation A^a, X) = 0 has a nontrivial solution such that \\X\\<\Axr
<\A\h+H)x".
Then it follows by (9), (13), (14) and (15) that (2) has a nontrivial solution satisfying U\\<\A\(i+ei)x"+4eilk<$\A\*\
254 Bounds for solutions of additive equations... II
311
Thus it will suffice to prove that if s ^ c8 and if a,- (1 ^ i ^ s) satisfy (16)
c9a< \af\
1 < ; ^ s, 1 < i < n,
where c9 = cg(k, K), c10 = clo(k, K) and a > c6(k, K, x"), then (2) has a nontrivial solution with (17)
IIAM a*".
Of course c8 depends on k, n, x", but since k, n, x, y, z, x', x" are fixed, we will not indicate the dependency of c 8 (and of subsequent constants) on these parameters. We shall first prove that this assertion can be derived by the following Proposition 1, and then give the proof of the proposition. PROPOSITION 1. Suppose that a,- (1 < i < s) satisfy (16). If s^ clt, then either (2) has a nontrivial solution with (17) or there is a nonzero integer x such that (18)
A{a,$ = X,
XeP(a6z),
^eP(a%
1 < i < s,
where each a, is 1 or —1 and Xj (1 < j < s) are totally nonnegative integers of K, not all zero. We may suppose that c5(k, n, 2x) and c n are integers. Denote v = c5, u = c l x and s = uv. Replace the indices 1 < / < s by double indices 1 ^ i ^ v, 1 < X u. Then (1) becomes A(a,X)= where ^ = (an, ..., OjJ, kt = (kn,...,
£ 4 ( 0 , , A,-), kiu) and
u
A(Oi,
A,) = X <Xy Oijk\},
1 < I < 0.
If there is an equation, say Ai(ah Af) = 0, which has a nontrivial solution having
P,-Max", then we have directly a nontrivial solution of (2) with (17). Otherwise it follows by Proposition 1 that there are nonzero integers Xi, • ••, Xv satisfying M*i, l,) = Xi,
XieP(a6%
V W ,
!<*<», K./<«,
where each atj is 1 or — 1. Since the equation
B(b, n) = xihiA+--- +XvKrf, = o has a nontrivial solution with ||/4I « max ||fc||2* « a a 2 ~ ,
255 312
Wang Yuan
the equation (2) has a nontrivial solution ft;**;;, ju,Ay (1 < i < i>, 1 < ; ' < « ) satisfying
maxIMyll ^||/i||maxPJI ^a>+12x*
i
by (8). Hence it remains only to prove Proposition 1. 3. The circle method. Let t = az/n and h = a1+ky~z/n. Let F(t) be the set consisting of y = xx gt+... + xngn satisfying (x 1; ..., xJeGn, x, (1 < i < n) rational numbers, y-> a and N(a) < t". For any y e T (f), subject to y —• a, we define the basic domain By by
(19)
{(xu ..., x j : (x 1; ..., x n )eG n , £ = x t e i + ... +xngn such that A||^ —Toll < 1 for some y0 = y(mod d"1)}. We may prove that if yt =£ y2, then B7l nB V 2 = 0. In fact, suppose there is a ^ e B y i nfl, 2 , i.e., /ill^-yo.11 < 1, where yOi = y;(mod ^" 1 ), i - 1, 2. For simplicity, we set yOi = yt (i = 1, 2). Denote
max(h\£m-yf\, r1) = af,
1 <j < 2, 1 < i < n.
Then
n oy> < 1,
maxaf-'^t,
i=i
j = 1, 2,
•
and thus
lri° - y?l < l^(i) - y'!0! +1£(0 - r'i'l
Suppose yt -* a,- (i = 1, 2). We have iV( ai a2) |JV( 7 l -y 2 )| < (2/T»t 3 )" < ZT \ since a ^ c 6 . On the other hand, a1a2(yi—y2)5 1
N(aia2)\N(y1-y2)\>N(S- )
is an integral ideal, and thus
= D-i.
This gives a contradiction, and therefore the assertion follows. We define the supplementary domain E by (20)
E = Gn- U By. yetXt)
We use the notations i = x, Qi+ . ..+xngn,
(21)
dx = dx1...dxn,
Sftf)= X ^(Or^iA AeP(B)
Ki<s,
B = a\
H = a6z,
S(Q=flSt(®, i= 1
256 Bounds for solutions of additive equations... II
313
and
F(0= £ Stf)E(-ft), XeP
where at (1 ^ i ^ s) are defined by a2p
(22)
l«%-il
-1="^T'
fl2p=
|a ( $
(1
"^"
*^ri)'
(2r1 + l < ; ^ s ) .
aj = l
Let Z denote the number of solutions of the equation
a 1 a 1 A5+...+a I a,A* = z in totally nonnegative integers Al5 ..., As, # satisfying *eP(tf),
1 < i < s.
l,eF(B),
Then Z= X jF(«^+|F(Od{.
(23)
yer\t) By
E
We shall«show that under the assumption made in Proposition 1, either (2) has a nontrivial solution with (17) or Z is > 1. 4. Supplementary domain. In this section a, = + 1 (1 ^ i ^ s) which are not restricted by (22). LEMMA 1 (Schmidt). Suppose that
T^cl2(k,K,s2),
OT1'116^2
and | £ E(£i*)| > C, XsP(T)
k 1
where G = 2 ~ . Then there exist a totally nonnegative integer a and an integer /? such that ^ )
r2
~k
and
0 < ||a|| < [^—J T*\ See, e.g., Wang Yuan [3]. LEMMA 2. Suppose that s^c13 and £eE. Then either (24)
\F(^\
or there is a nontrivial solution of (2) with (17). Proof. Take e2 such that (25)
0 < £2 < c 14 < 1/(2G),
257 314
Wang Yuan
where c 14 is a constant to be determined later. Set (26)
h = m2.
and
m = c5(k, n, x + e2)
Choose c 13 ( > Skn) sufficiently large such that if s ^ c 13 , then n(k + 2/y) ^ and by (21), we have (fl*a2yi/(s-*+l) _ gn(t+2/y)/(s-fc+l)
<
jfl
If (24) fails to hold, then ir\S1(Q...SM(Q\>F(®>H"B*-k>a-2: We may suppose without loss of generality that \S1(Q\>...>\Sg(®. Hence we have \Sh{Orh+1
B*h-l)
>
Bn{s'k)a~2n,
and thus by (25), B^s~h~k+1)l^~h+1)a~2nlis~h+1)
|S-(£)|">
> B"(Bka2)-n^-h+1)
> B"-'2 > Bn~UG+e2,
1 < i < h.
Take C = B" °2 in Lemma 1. Then it follows that there are totally nonnegative integers ec; (1 < i < s) and integers pj (1 < j < s) such that 2
llfo*i-AII
(27)
0
2G£2
,
,
i
1
Denote T, = fttr?" (1 < i ^ h). Then
iifoo?-T,ii ^ i k i r ' I I ^ ^ - A I I «i?-** 2 ^,
i ^ / ^ /,.
Therefore by (16), we have
Itotf T,-«X*III < iK^X-^a^ll + lK^.^-T,.)^.^!! «aB-t+4(tG£2,
U i ^ l i ,
i.e., the vectors / ; = ( a ^ , T;) (1 ^ i ^ /i) satisfy (28)
l|det(/;,/,)|| « aB- t + 4 < l G £ 2 ,
1 < i,j ^ h.
Let ft be a nonzero element in the integral idea (aio\, t t ) with the least norm in absolute value. Then fi\c2cx1(Tk1 and fi\c2ti. (See § 1.) Set c2v.la\=
Pa,
C2T!=J?T
and
f = {a,x).
258 315
Bounds for solutions of additive equations... II
Then
c2fi=fifWe may choose ft such that \N(o)\1"><$\\a\\<\N(a)\1">.
(29)
(See, e.g., Lemma 1 in [4].) We have also two integers & and T' such that /? = a 1 ff* 1 T'-T 1 (T',
therefore i.e., c 2 = ax' — to'.
(30) Set g = (a', T'). Then by (30),
=C219if+C21*l'i9,
fi where a.- of a'
cpt=
Tj
T
,
,
and
,
\j/i =
a af of T
T,-
,
l ^ i ^ h
are integers. By (27), we have 1<JV(^)^|4»|B2(""1)G£2, i.e., (31)
Itffl^JT 2 *"" 1 ^ 2 ,
U;<M
Since C j a ^ = fia, we have by (16), (27), (29) and (31) that (32)
a\\a\\-'B2kGt2 > \p*\ > a\\a\\-'S'2'^^2,
1 < i < n.
Therefore it follows by (28) that (33)
m
= ||det(/J,/)|| < ^ m a x l ^ r 1 lldetV,,/!)!! «|M|B"* + 6 k l l C £ 2 = M,
l^i^h.
1. Suppose M ^ 1. Replace the indices 1 ^ / < h by double indices 1 ^i,j^m. Define
where a,- = fo,lf ..., aim) and A, = (A,ls ..., kim) (1 < i < m). It follows by (26), (33) and the definition of the set X that the equation Al(al,ll) = 0
259 316
Wang Yuan
has a nontrivial solution having 114)1 4Mx+e\
(34)
l
Let m
m
c
1
ft = I ! j= i
j=
1 < i < m.
i
The first coordinate of # is m
At = C2 *ffS fl.v 4 9»y.
1 < i < m.
Therefore c2)3i/(T (1 < i ^ m) are integers. If ftlt ..., f}m are not all zero, then let x be a nonzero element in the integral ideal (/?j, . . . , /?m) with least norm in absolute value and satisfying
\N{x^llH<\\x\\<\N{x^lm. Then a\c2x and by (29),
IWNIIzll-
Consider the form
B(b,n)= £ptbtrf, i=l
where b = (bu ..., bm) and p = (JUJ, . . . , nJ. Since m
A = Z aij%j<*ij°kiv
1 < i < m,
we have by (16), (27) and (34), ||AIMal?2*Ge2M(*+£2)\
l
and therefore the equation B(b, fi) = 0 has a nontrivial solution satisfying /
MM max (l,
2kG *2 aB
M(x+°2)k\x+E2
^
J .
Consequently (2) has a nontrivial solution bi au,
m a{J Aij
a, = 1,
(1 ^ i, j ^ m),
4 = 0
(/i
satisfying 2G 2
+
/
nR2kGe2
AJ*-x + e2>k\x
||J|| < 5 * M- '» max ( l , ^ 5 - ^
)
+ e
2
260
Bounds for solutions of additive equations... II
max
2
317
r—w—)
2
, say.
I
Since c2a1o*1 =fio,by (27), (29) and (33), we have 2
||<x||«aB
(max|j8(l)|)
l
<£aB
2
,
2
,
i
M 4aB and therefore I
= aB2kGci M(x+E2)k + 1M-' jj-k+8knGe2f
k+6knG °2 B-
n~ k+ SknGE2-.(x + e2)k
Since a > c 6 , we have IIAII
a
x+kx2 kxy k2x2 +c
~ ~
y i5(k->i)s2
Hence if c 14 is sufficiently small, then by (5) and (8), we have the desired (17). 2. Suppose M < 1. We revert to the indices 1 < iI ^ h. By (33), we have \jjt = 0 (1 =$ i ^ /i), i.e., c 2 /j (1 ^i^h) are integral multiples of the integral vector / . Let
B(b, n)= I«,of6,^, wherefe= (6j, ..., bh) and fi = (fi1, ..., fih). Let / be a nonzero element in the integral ideal (a t a\, ..., <xhcrk) with least norm in absolute value and satisfying \N(x)\11" > \\x\\ > \N(x)\u". Then a\c2x, and thus ||a|| < || z ||. Hence the equation B(b, fi) = 0 has a nontrivial solution satisfying
||/,||<max(||ai«Tf||Hr1)X+t2 ^ a f l ^ l W I - 1 ) 1 * ' 2 . i
It derives a nontrivial solution of (2): Oi = bh ki =ViHi aj = l,
Xj = O
(1 ^ i ^ h ) , (h<j^s).
If c 14 is sufficiently small, then
UW^B^iaB^M-1)*^2 _ < a
x + t2 + 2Gyt2 + 2kGy(x + E2)e2 II | | ~ x ~ £ 2 x + xz/(2n)||ff||-X
261_
318
Wang Yuan
If |M| ^ a*1", then by (8), we have ||A|| ^ ax-xzH2n) ^
QX^
i.e., (17) is true. Now we suppose that |M| < tf/n. Then by (27) and (31), we have ||5-
if c 14 is sufficiently small. Let 0~1Td = b/a, (a, b) = 1. Then a\a, and thus iV(a)^|JV(cx)|
£ - y = C,
ri = y1o)1+ ...+yna>n,
dy = dy^.^dy,,,
0,(7) = JV(a)-1 X £(«<«;/?)
(1 ^ i ^ s),
fi(moda)
G(y)=t\G,(y), /,(C,B)= 1 E{a,a,rfQdy
(1 < i < s)
P(B)
and
/(C,B)=ri/,(C,£), i=l
where af (1 < i < s) are defined by (22). LEMMA 3. Suppose that £eBy. Then St (0 = G, (y) I, (C, B) + 0 {a2*'" Bn~J). See, e.g., Lemma 12 in Wang Yuan [4]. LEMMA 4.
/,(£, B) « fl m"»(B, a - ^ l ^ r 1 / k ) ,
1 < i < 5.
See Siegel [3], p. 335. LEMMA 5.
IS(§E{-X®dx = G(y)E(-Xy) J/(C, BJdx + O ^ - ^ a - - 7 ^ .
262 Bounds for solutions of additive equations... II
319
Proof. Let The Jacobian of xlt ..., xn with respect to up, uq,
Dll22r2Y\u9. 9
Suppose £eB y . Then by Lemmas 3 and 4, we have S(Z) = G (y) I(C, B) + 0 ( B - - l a2*"). Since by (19), ^dx^h'n
= a-n-kny+z,
By
and by (6), (l + 2/n)z-y < — Iz, we have (36)
\S(!;)E{-tf)dx By
= G(y)E(-Xy) J /(C, B ) £ ( - z O ^ + 0(B^- t »a- n - 7z ). By
In the integral on the right-hand side of (36) we replace E(~xO by 1. The error is
B" JllzCIM* ^B^Hh-"-1
By
by (19) and (21). Hence (37)
= G(y)E(-XY) \ I{L B)dx + O{B«*-*a'"-1*).
$S(OE(-xOdx By
By
If (x 1; ..., xj is a point of En — By, then the inequality /i|C(0| > 1 is true for at least one index i. Since s ^ c 13 , it follows by Lemma 4 that
J I{C,B)dx4 £n-By
J (nmintB.fl- 1 "^ 0 !" 1 '*)) 1 ^ £n-Bv i=l
«( I a-s/tu-s/tdu)(Jmin(Bs, a-^i;-^)^)' 1 " 1 „-!
0
x( J Jmin(B 2s , a-2s/kw-2slk)wdwd(pj2 -n 0
+(J min(B% a'slk u'^duj1 0
{] ] a~ Wv-W+l dvd
263
320
Wang Yuan
x( J Jmin(B2s,
a-^w'^wdwdcpj2'1
-it 0 4
a-s/k hs/k-
1 _-(n ~ 1) ^ O - W C l - 1) _~ 2'2 B 2 d - * ) ' 2
+ a ~ r i 5 < s " < I ) r i a~ 2s/ *
fc2s/*~2
a~2{'2~1]n2(s~k)(r2~1}
^ ^(s-t)(n-l)a-s/*-»+l+(1+'')'-z/'')(|—l) +
<£B
g^-k)(n-2)a-2s/k-n+2 + (l+ky-zM(-Y~2) nis k)
- a-"(a~to
zs
+
z
2zs
2z
» + a~~to~+~^)
4B«s-k)a-n-7z
The lemma follows by substitution into (37). 6. Singular integral. Let ri' =y'ico1+ ... +y'nO)n, dy' = dy'x...dy'n, t] = Br]'
C' = A Qi + • • • +
and £ = a " 1 - * ^ ' .
The Jacobians of yx, ..., yn and xu ...,xn with respect to y'_, ..., y'n and x\, ..., x'n are B" and {a~1~ky)n respectively. Define yt = at/a (1 < i ^ s). Then
and by (16), we have c9<|yf|
(38)
1 < ; < s , 1 < i < n.
Let us write f/' and C' as JJ and C again and let
Ii(0 = IE(aiyiVk0dy (1 < i < s)
and
J(0=fU(0,
where P = P(l). Then and thus j /(£, B)dx = B"( s -")a- n | /(Qdx.
(39) LEMMA 6.
J/(C)ix = I>< 1 - s >/ 2 fc-'«|Ar(y 1 ...7 s )r 1 /' [ n^n^
E
n
P
9
264
_ _ _ 321
Bounds for solutions of additive equations... II
where
in which Up denotes the domain
l
0<w,**M%
w2p = w 2 p _ 1 ±w 1 ±...±w s ,
here the sign before wt is the sign of at y\p) (cf. (22)), and where
vqi=i
in which Vq denotes the domain O^Wt^lypY
(l^i^s), VVs
-n^cpj^n
(l^j^s-l),
i
= |w}/2e ^+...+vvi/.2ie^-i|2.
The proof is similar to that of Lemma 16 in Wang Yuan [4]. 7. The proof of the theorem. We have
E N(a)4 I d2
I H
If (24) holds, then by (23), (39) and Lemmas 2, 5 and 6, we have z
Z jF(Qdx + O(HnBnis~k)a-"-4-z)
=
Vent) By
= Jo S(r, H)B*~k>a-' +
OiH-B*-*a'"-**),
where
J0 =
D^-^k^\N(yi...ys)\-llkUFPUHq P
and S=S(t,fl)=
I
Q
I G(y)E(-Xy).
Let £* denote a sum, where y runs over a reduced residue system of (ad)-[ modcT1. Then ®= I
JV(o)=l
I*G(y) I £(~Zy)+ y
xeP(H)
I
l<JV(o)«t B
= 8! + S 2 , say. We have
® x = X 1>H".
E*G(y) X E(-XY) ?
ZeP(H)
265 322
Wang Yuan
(See, e.g., [4].) If N(a) > 1, then
£ E{-Xy) = O.
/(mod a)
Therefore if the domain xeP(H) has to be split up into a union of complete residue set (mod a), plus a few other, remaining elements, say R elements, then
R^H^'Nia)1"1, and thus ®2 «
Z
^H"'1
X*H"- 1 iV(a) 1/ "
£ d3
£
W(a)1 + 1/n
Hence It follows by (38) that Jo > c16, and therefore
Z>c11HnBn(s'k)a-">
1
if a > c6(k, K, x"). The theorem is proved. Remarks. 1. The inequality (3) can be replaced by max|AT(^.)| « m a x ( l , \N(^)\, ..., \N(ocs)\f. (See [4].) 2. Consider the equation
(40)
A(X)=txi%=°>
where au .., as are given integers in K. If k is an odd number, then ai%={aiXif
(l^i^s).
If rx = 0, i.e., K is totally complex, then the singular integral Jo is always positive. Therefore in these two cases it follows by the theorem that if s > cls{k, n, s), the equation (40) has a solution in integers Xt, ..., Xs, not all zero, satisfying U\\<\A\*. 3. We can further consider the problem of the estimation of bounds •for solutions of certain diophantine inequalities in an algebraic number field (cf. [2]).
266 Bounds for solutions of additive equations... II
323
References [1] [2] [3] [4]
W. M. Schmidt, Small zeros of additive forms in many variables II, Acta Math. 143 (1979), pp. 219-232. — Diophantine inequalities for forms of odd degree, Advances in Math. 38 (1980), pp. 128151. C. L. Siegel, Sums of m-th powers of algebraic integers, Ann. of Math. 46 (1945), pp. 313— 339. Wang Yuan, Bounds for solutions of additive equations in an algebraic number field I, Acta Arith. 48 (1987), pp.
INSTITUTE OF MATHEMATICS, ACADEMIA SINICA Beijing, China
Received on 30.5.1984 and in revised form on 7.1.1986
(1429)
267
Reprinted from JOURNAL OF NUMBER THEORY
Vol. 29, No. 3, July 1988
All Rights Reserved by Academic Press, New York and London with permission from Elsevier
Diophantine Inequalities for Forms in an Algebraic Number Field WANG YUAN* Institute of Mathematics, Academia Sinica, Beijing, China Communicated by Hans Zassenhaus Received March 26, 1986; revised March 30, 1987
1. INTRODUCTION
Given a vector with complex coefficients x-(xu
..., xs) put
|x| =max \xt\ and given a form (i.e., a homogeneous polynomial) F let
1*1 be the maximum absolute value of its coefficients. With every form F of degree k there is associated a form F{xu...,xk) which is linear in each vector x, {l^i^k) Xj,..., xk and such that
and symmetric in the k vectors
F(x) = F(x,...,x). Schmidt in 1980 [5] proved THEOREM A (Schmidt). Given h^l, m^ 1, and odd numbers k^,..., kh, and given a positive number E, however large, there is a constant
c1 =
cl(ku...,kh;m,E)
as follows. If M ^ 1 is real and if Fi,..., Fh are forms with real coefficients of * Supported by National Science Foundation Grant MCS-8108814(A04).
324 Copyright © 1988 by Academic Press, Inc. AH rights of reproduction in any form reserved.
268 DIOPHANTINE INEQUALITIES
respective degrees ky,..., kh in x = (x 1; ..., xs) where s^cu linearly independent points x(l),..., x(m) in U with \x(i)\^M,
325
then there are m
l^i^m
and |#(x(/ 1 ),...,x(4 j ))|«M- £ |F,|,
Kj^h,
l^iu...,ik.^m.
If the coefficients of all forms are rational integers, he derives from Theorem A the following. THEOREM B (Schmidt). Given /z> 1, m > 1, and odd numbers kx, ...,kh, and given £ > 0 , however small, there is a constant c2 = c2(k1,..., kh;m,s) such that if Gx,..., Gh are forms of respective degrees klt..., kh with integer coefficients in x = (xu ..., xs), where s^c2, then G{,...,Gh vanish on an m-dimensional subspace which is spanned by integer points x(l),..., x(m) having
\x(i)\
1
max(\,\Gl\,...,\Gh\).
Remark 1. We will show that the conclusion of Theorem B is still true with suitable definition of G if Gx,..., Gh are forms of odd degrees with coefficients in integers in an algebraic number field K of degree n, provided only hat s ^ c3 (k,,..., kh ;n, m, e). In fact, if cou ..., wn is an integral basis of K, then G, (1 ^i^h) can be written as Gi = Giicol+ ••• +Gin(on, lsS/'
Suppose E > 0 and suppose
D{x) = d1xk1+
•••
+dsxks
is an additive form of odd degree k with dteZ (X^i^s) s^ c4(k, s). Then there is a nonzero point x in U with
£>(x) = 0
and
|x| <max(l, |Z»|£).
and
with
269 326
WANG YUAN
The next step is to prove a special case of Theorem A, namely when there is only one additive form. And the final step is to prove the theorem by an inductive argument. To consider the problem of diophatine inequalities for forms of arbitrary degrees we shall need a proposition for additive equations of arbitrary degree. If the solutions of the equations belong to a totally complex algebraic number field (i.e., a field with no isomorphic-embedding into the reals), such a result can be proved by the combination of the methods of Schmidt [4] and Siegel [6]. (See Proposition 1.) Let K be an algebraic number field of degree n over rationals. K(1\ ..., K(n) are real conjugates and K(r' + l\..., K{n + 2n) are the complex conjugates of K. Here rx + 2r2 = n. The complex conjugates are so arranged that, for rx +1 ^q^r1 + r2, the fields KM and K{
II€11 = max |£«>| and
im=max 1^11= max |Aj')| for a vector "k = (Al5..., Xs). THEOREM 1. Let K be a totally complex algebraic number field of degree 2r. Given positive integers h, m, and ^ , . . . , kh, and given a positive number E, however large, there is a constant
cs = c5(ku ..., kh; r, m, E) as follows. If M~^ 1 is real and Fx,..., Fh are forms with complex coefficients of respective degrees klt..., kh in "K=(Xl,..., Xs) where s~^cs, then there are m linearly independent points >.(1),..., X(m) in F with
HMOKJlf,
l^i^m,
and \F(X(il),...,Uikj))\<M~E\FJ\,
l
U / „ . . . , i t ^m,
270
DIOPHANTINE INEQUALITIES
327
here and below the constant implicit in <4 or 0 may depend on ku ..., kh, K, m, E, e but not on M, Flt..., Fh, andG. In particular, it follows that \Fj(Ui))\<M-E\Fj\, Suppose n o w that Gu...,Gh with coefficients in J. Let
l^j^h,
l^i^m.
are forms of respective degrees
kl,...,kh
IIGJ denote the maximum absolute value of its coefficients and their conjugates. Further let G = max(l, HCII,..., \\Gh\\). S u p p o s e t h a t s^cs(ku ..., kh; r, m, 4rkt •••khs~i) = c6(kl,..., kh; r, m, e), say, where 0 < e < l . Apply Theorem 1 with M = MoGe, where M o = M0(k{,..., kh; K, m, s) is to be chosen in a moment. We obtain m linearly independent points X(l),...,~k{m)in Js with
IIMOK M0G«,
l^i^m
and \Gj(Uil),...,Uik))\<$GM~4rk<-khe~'
(1)
On the other hand, &,!(?,(>.(/,),..., X(ik)) is an integer in K and \\kj\Gj{mx),...,
X(ikj))\\ <
GMk'<Mk0'Gl+k'.
If G'/Mi,), ...,*.(»*,)) # 0 , then \N(ki\6fi.{.ii),...,\{.ik)))\>l> and therefore
IG/M/,),..., Hik)\ > (M^Gl+k'r2r+l
>M^G~4rk'+'
which leads to a contradiction with (1) if Mo is sufficiently large. Therefore 6J(k(i1),...,k(ikj))
= 0,
and we have the following THEOREM 2. Given positive integers kl,..., kh and m, and given e, however small, there is a constant c6 = c6(k1,..., kh; r, m, e) such that i/Gj,..., Gk are
271 328
WANG YUAN
forms of respective degrees kl,..., kh in X = {Xx,..., Xs) with coefficients in J, where s^c6, then Gu ..., Gh vanish on an m-dimensional subspace which is spanned by m points X(l),..., X(m) in Js having
HMOHG8,
l^i^m.
Remark 2. Theorem 2 gives an improvement for a result due to Peck [3]. He first established by the combination of the methods of Brauer [ 1 ] and Siegel [6] the existence of X(i) ( l ^ r ^ m ) with c'6(ku..., kh; K, m) instead of c 6 , but the estimation of ||X,(j)||'s was not considered. As we have shown in Remark 1, a similar result can be obtained if the coefficients of G,,..., Gh are integers in any given algebraic number field Kx. The proof of Theorem 1 relies on the Schmidt's method [ 5 ] . In this paper, we shall prove a special case of Theorem 1, namely when there is only one additive form, and give only a sketch on the inductive argument from a single additive form to a system of forms, since it is easily treated by Schmidt's method.
2. ADDITIVE FORMS
We suppose that K is a given totally complex algebraic number field of degree 2r throughout this paper. PROPOSITION
1. Suppose e > 0 , & > 1 , and s^c7{k,r,e).
Then given an
additive form A{k) = all./Cl+ ••• + M * with coefficients in the integer of K, there is a nonzero point XeJs with A(l) = 0
and
\\X\\
\\A\n
This proposition was proved in [ 8 ] . (See Remark 2 in [8].) Now we shall prove the following proposition which is a very special case of Theorem 1. PROPOSITION 2. Given k ^ 1 and given E, however large, there is a constant cs = cs(k, r, E) with the following property. Let A{k) be an additive form of degree k with complex coefficients in at least c8 variables. Then for real M^-l there is a nonzero point keJs with
WU^M
and
\A(K)\ <M~E
\A\.
272
_
DIOPHANTINE INEQUALITIES
329
We begin with simple reductions. Suppose we can prove the conclusion for s ^ cs(k, r, E) and M^cg(k, K, E). Then in the case of 1 < M < cg, we put A, = 1
and
X 2 = - - - = X S = O,
and therefore HM = 1
\A(k)\^\A\=cEcg-E\A\
and
Hence it will suffice to prove Proposition 2 for large values of M, say for M>c9(k,K,E). Choose 5 > 0 so small that „,
E+23 .C -f
(
8
Proposition 2 is obvious if there is an a, with |a,-| <M E \A\. In fact, we may take A,= 1, /Ly = 0 (y'#/) in this case. So we may suppose that M~E\A\^\ai\^\A\,
l^i^s.
Cover the interval [ — £,0] by a finite number of intervals {/} of length 5. One of these intervals / will be such that at least \_s/([E/S2 + 1)] of the a, are of the type |a,| = M" \A\ with etel. We may suppose without loss of generality that this holds for all i. Put L = M6 \A\=MS max |a,| and choose natural numbers qy,..., qs, each as large as possible, with
Since L/|a,-| ^M5,
we have
{L^\0L,\qkt^L
(2)
if M is sufficiently large. Further, qki^LI\a.l\=M6\A\l\a.i\^M2S. Proposition 2 is true for B{]i) = a l q k 1 n k l + ••• + x s q k n k
with M0 = M ( £ + M ) / ( £ + 1 / 8 ) ,
E0 = E+\
Suppose
_
273 330
WANG YUAN
in place of M, E, i.e., there is a nonzero point \ieJs with and
||H||
\B(yi)\<M»*\B\.
Let Xi = q,ni (1 ^ K 4 Since | 5 | < L = Af* Ml, we have
| i ( i ) H M - p + 2 i | / ( £ + l f f l ( £ + 1 / V Ml « M - £ Ml and \\X\\<(maxqi)\\n\\^M2S
+ E+
^
2S)/iE+1/s))
<M
if M is large, ie., Proposition 2 is true for A(X). What is special about B is that by (2) each of its coefficients is at least %\B\ in absolute value. By what we said it is clear that if Proposition 2 is true with E +1 in place of E for form A with
iMKN<MI.
is* '<*,
then it is true with E for general forms. By homogeneity we may replace the above relation by Ml = l
and
H|oc,Kl,
1
(3)
It now will suffice to prove the following statements. (i) The conclusion of Proposition 2 is true for 0 ^ E ^ \ for forms A with (3), provided only that s^cl0(k, r, E). (ii) The conclusion of Proposition 2 is true for E for forms A with (3), provided only that s ^ clo(k, r, E) and that Proposition 2 is true for E — \ for general additive forms. 3. ANALYTIC METHOD
Let a)i,...,(o2r be a basis of / with cuJ« + r) = d>S») ( l < / ^ 2 r , 1 «S q*S,r), 5 the different and D the absolute value of the discriminant of K. We choose a basis pu ..., p2r of <5-1 such that [0,
if
i=£j,
where T(y) = ^jr=i y(0. Given a set of complex numbers c!,..., c2r with c <, + r = cg {l^q^r), there is a unique set of real numbers yu ...,y2r such that c,. = y, o>i') + • • • + ^ffli?,
1 < i < 2r.
274
DIOPHANTINE INEQUALITIES
331
Suppose that a,,..., a.s is a set of complex numbers satisfying (3). We define for each j ,
Then a, has a unique representation tf) = z1co[i) + ••• +z2ra>$ ( l < / < 2 r ) ,
z,eU ( l < / s $ 2 r ) .
Set £ = * , £ ! + ••• + x 2 f p 2 r , > 7 = > ; 1 a ) 1 + ••• +y2r^2r, dx = dxl---dx2r, and dy = dyl •••dy2r. We denote by /•"((?) the set of (yu ...,y2r) satisfying
\\i\\
5,(0=
a sum
where /. runs over all integers in K with ||A|| ^ Q. Set
Z E(ocfakl
E{a^nk)dy,
7,(0= f
5(0=115,(0
and
1
7(0=Il/,-(0,
/=i
/=i
2jt c
where £(y) = e(r(v)) with e(,v) = e " . We use the notations /sin nxM~E\2 K(x)
=
smnxM
2r
)
(
A:(X) =
and
n
K(Xjl
where A: is a real variable and x = (.v,,..., x2r) is a real vector variable. We wish to estimate the number Z of solutions of \A(\)<M'E
(4)
II^KM.
(5)
in points k e Js subject to
Our plan is to show that either Z > 1 or (i) and (ii) holds. Therefore (i) and (ii) hold in every case. Put 8 = (rk+(r+l)E+3)-l(4k _J1 m
~ \cs(k, r,E- £)
+ 5E)-\
in the case (i), in the case (ii),
(6)
(7)
n = c7(k,r,d),
(8)
h = mn,
(9)
275
332
WANG YUAN
and choose r\ > 0 so small that (10)
4lrGhri
(11)
In what follows, the constants in <^ or 0 may depend on s (in addition to k, K, E). But observe that if Proposition 2 is true for a particular value of s, then it also holds for larger values of s. We assume ¥ to be large, i.e., A/>c 9 . LEMMA
1. We have for real Q
(12) Proof. Make substitutions /? = MEa, and MEQ for g in Lemma 50 in [2]. Let En denote the n-dimensional Euclidean space. Expand Yfj=i <*jtf a s
X aj'Uj')fc = ^ ^ i " + • • • + A2rco£,
1 <«<2r.
(13)
Then by Lemma 1, we have M2rE f S(£) A:(x) rfx J
£2r 2r
2r£
=M
I
=
A e/>(A/>
'
/-co
£ ••• X n j e(xjAj)K(Xj)dxj •••
Z (l-M.IM^-.^l-M^IM^). (14)
M,,<^-£;-ie/>
Let »?7 = 7 j l c o 1 + ••• + ^ , 2 r o j 2 r ,
dYj = dyn---dyJt2r,
l^j^s,
and f
aj"»jj'">* = B, « < / » + • • • + 52rO)J?,
1 < i < 2r.
276
_ ^ _ _ _ ^ _
DIOPHANTINE INEQUALITIES
333
Then by Lemma 1, we have M2rE f I{Z)K{x) dx J
E2r
= f
•••
J
P(M)
|«/|<M- £
(l-\B1\ME)---(l-\B2r\ME)dY1--dYs.(l5)
f J
P(M)
If \Af\ <M~E, \^i^2r, then it follows by (13) that A(k)<M-£, and thus the right-hand side of (14) gives a lower bound for Z. The general idea now will be to show that the right-hand side of (15) is large and to show that the left-hand sides of (14) and (15) differ little. LEMMA
2. The right-hand side of (15) is
Proof. Let ,,]«) = uVkeilp">/k,
1 <j < s,
l^q^r.
The Jacobian of yJt (1 «£/<2r) with respect to ujq, q>jq (1 ^q^r) is
2rk-2D-1/2ft< - ' , and so the right-hand side of (15) is equal to 2rsk-2rsD-s/20,
(16)
where
cp=r fj (i - M £ 15,1) n ri « 2/ v ! ^ ^
( iv )
in which
Uy^J,
U?^r.
Then 0 = !«! • • • a,| - (2r>/ * M - < > & > / ^ ,
(18)
277 334
WANG YUAN
where
(19) in which do = Oy= i Il!| = i
A = ^ + a r § a i " (1 ^ 7 < *, 1 < ? < 2r)
and M £ B, = D,( 1 < i ^ 2r).
Then
7=1
7=1
7=1
= DlQ)[^ + ••• + Drools),
l^q^2r,
and
<»->/=l
7=1 9=1
r
where c?0 = Oj= i Yl q = i dQjq, and (W) denotes the domain O^vjc/^ \*j\ME+k, - J t < ^ < n (l
ME+k
32s
ME+k
lbs
(H;)
l^/l<^ (K/<2r) is contained in (W). The volume of (w) is ^>M<£+*:)r(-s~2). In
fact,
for any given
i) ll7 e 1 ' e "+ ••• +vs_liqe'e'-l"=Veie.
vjq, O^il^j^s—l)
in (w), we let
Then M £ + V l 6 < F < (5ME E k
^ and 0,, satisfy Kvsq-V<£l and Qsq-Q4M- - . follows. The integrand of the integral W in (w) is
+k
)/16, and
The assertion
278
DIOPHANTINE INEQUALITIES
335
Hence ip ^ ]tf(E + k)r{s - 2) + (£ + k)rs(2/k - 1) ^ ]tf2r{s - k - E) + 2rEs/k
and the lemma follows by (16)—(19). LEMMA
3. If U\\ ^ M9/1°~k, then ^
E(SXk)=\
E(^k)dy
+ O{M2r-ilw).
Proof, Expressing an integer A in terms of the o>, (1 ^ / < 2r), we obtain X = gl(Ol.+
•••
+g2rU>2r,
where the g, (I < i^2r) are rational integers. Let G(/.) denote the cube (yu-,y2r)-ri=y1(ol+
••• +y2ro}2r,
gi**yi
l^i^2r.
Then ||^*-«A*H||5|| ||f ? -;.||(|| >7 r- 1 + ||/||*- 1 )«M 9 / 1 0 -* + * - 1 ^ M - 1 / 1 0 and so E(£Xk)=\
E(Zi]k)dy + O(M-1/i0).
Since the volume of the area belonging to exactly one of \J;_EP{M) G(X) and P(M) is dominated by O(M2r~l), we have
£
£(£l*) ^ = [
and the lemma is proved. LEMMA
4.
We have
See, e.g., Siegel [ 6 ] , p. 335.
f
^ ^ ^ r f y + O^ 2 '- 1 ' 1 0 )
£(^*)rfy + O(M2r-1/10),
279 336
WANG YUAN
LEMMA
5. We have f
J
IKII«A^/10-*
S(£)K(x)dxpM2r(s-fc-2E\
(20)
Proof. By Lemma 4, we have for any q,
f
HZ)K(x)dx < M~4rE f J
(?)
k
\(M\>MV">- j=l
f] min(Ms, |£ u) | ~s'k) dx.
1
Let £ = w^e'* ' (1 < q < r). The Jacobian of *,- (1 ^ i < 2r) with respect to uq,
«M- 4 r £ ff°°
f
u-l2"Vk+1dudq>)
xff 00 )" 1 min(M2s, w^(2j)/*) wrfwrfe) •^ M ~ 4 r £ +
<9/1
° ~ k)( ~(2s)lk
^ Jrf2rU -k-2E)-
+ 2)+ 2 ( r
" ' ) U ~ k)
(9s)H5k) + 9/5 ^ J^X* - * - 2£) - 1
if s is sufficiently large, i.e., s > 2k. Therefore it follows by Lemma 2 that f
I{0K(x)dxpMMs~k-2E).
(21)
It remains to compare the integral (21) with the one in (20). By Lemma 3 we have for U\\^M9/1°-k,
|S(O-/(S)I= ft (/,(£) +O(M 2 - 1 / 1 0 ))-/(O
and there fore the left-hand sides of (20), (21) have a difference J + M~4rE\
mm(M2is-l\u'ms-im)ududcp)
M«2r-l/l0)dx
a D1OPHANTINE INEQUALITIES
U ~ (^ " ' W* + ' du Y
' M2° ~ 2U du + T
M
<M-4rE+2r-1/wfr
+ M2r(s~k~2E)~s/w
337
+ (9r)/5
<^M2r(s~k~2)~ino if s is large, i.e., 5 > 20r. The lemma follows. 6. We have for any q
LEMMA
S(Z)K(x)dx<M2ns-k-2E)-i.
f
P r o o / I t f o l l o w s f r o m \^q)\> Mrk + lr+1)E+1 that a t least o n e of t h e x , i n t h e e x p r e s s i o n £ = xlp]+ ••• +x2rp2r satisfies \x,\>Mrk
+ ir +
l)E+1
.
Since |^(x)l^min(M- £ ,7i- 1 |x|- 1 ) 2 , we have
f
S(Z)K(x)dx 2rs
dx\2f
/ rx
^
<M {\
r-s-
\2r-2
min(M-2E,x-2)dx)
\ J M r k ~ t , + l)E+l X J \ J Q <^ ] ^ p - r s - 2 r k - l { r ^ l ) £ - 2 - 2 ( r - l ) E
J
The lemma is proved. For 1 ^q^r, let (Uq) denote the domain U\\^Mrk
+ lr+1)E+
\
\^\>M9/i0-k.
Suppose for a moment that the r integrals satisfy f
\S(i)\dx^M2rls-k)-\
l^q^r.
(22)
281 338
WANG YUAN
Then it follows from Lemmas 5 and 6 that f S(Z)K(x)dx=\ J
S(^)K(x)dx + J
E2r
O(M2r{^k-2E)-1)
\\i\\
and in view of (14) it follows that Z^M2r(s^£)>l.
4. THE PROOF OF PROPOSITION 2
We may thus suppose (22) is false, i.e., there is a (Uq) such that f
J
(t/,)
\S{O\dx>M2ris~k)-i,
(23)
l^q^r.
Let <x(i) be a transformation such that
where \ = (iu ..., ir) is a permutation of (1,..., r). Since S(£) is invariant under the group {ff(i)}, (23) is true for any q. In particular, it holds for g= 1, i.e.,
f
\S(Z)\dx>M2rU-k)-\
(24)
There is a £(= £(1>) satisfying |{|a
r t +|r+1|£+1
|£|>M9/10^
,
(25)
and \S{£)\ ^M2r{s-{r
+ 1){E+k) 1) 2
(26)
- '.
Otherwise, we have
[
\S(i)\dx^M2rU'ir+1){E+k)-l)-2
J((y,)
f J | | 4 | | < A / r t + (r+l)£+l
<M2Hs-k)-2
which leads to a contradiction with (24) if M is large.
dx
282 339
DIOPHANTINE INEQUALITIES
We may suppose without loss of generality that
1^(01^ ••• >\SAl)\. The left-hand side of (26) is
<\Sh(Z)\'-h
+i
lrf«h-l\
and thus we have by (11) \Si(£)\^M2r-(2Hr+1HE+k)
+ 2r + 2Ws h
-
+ i)
^M2r-'1,
l^i^h.
(27)
7. Suppose that n>0, M> cn{k, K, n), and C^ M2r-l/G + \ where G = 2k~1. If |£;. e /> (M) E{t^kk)\ ^ C, then there are two integers a, fi of K such that LEMMA
\w-n<\^r\ M"-k and 2r G
(M \ 0<||a||«(— J M\ See, e.g., [ 7 ] . In view of (10) and (27) we can apply Lemma 7 with C = M2r~n and C = £oe, (l^i^h). We obtain integers au...,<jh and f}u...,ph of K with 0<||(7/HM2G" Setting
and
||a,ff,$-/J,HA/-*
and (O-J5,)/O-, =
T,
+ 2c
l^i^h.
(1 < / < / I ) we have
| | a , c r ^ - T , . H M ' ' t + 2G/"),
and
',
l^i^h.
(28)
Set
Then 11
|a(2)---a(>)|
and by (10), (25), (28) we have ly/Kla^"1
M-k
+ 2Gh
*4Mk-9/w
+ 4rGh
"~k<M-4/s
(29)
283 340
WANG YUAN
if M is large. So with X = (/Lu ..., Xs) = {nu ..., jxh, 0,..., 0) = (fi, 0) we have A(X)=\a^rlB()i) + C(ii), where £(H) = T 1 /4+
•••+Xhflkh
is a form with coefficients in / and C(\i)
= y l f i k 1 + •••
+yhnl
From (25), (28) we obtain ||T,||
<Mrk
+ (r+l)E+1
+2Gh
",
l^i^h,
and therefore ||5||<Mr/t + (r+1)£ +2
(30)
by (10), if M i s large. (i)
First, suppose that 0 < E^%. In view of Proposition 1, and since h = mn = n = c7(k, r, 9)
by (V), (8), (9), there is a nonzero vector fieJh with fi(fi) = 0
and
| | n | | « m a x ( l , \\B\\e)<Mirk
+ ir+l)E
+
2)e
<Ml/i*k)
by (6) a n d (30). With X = (\i, 0) we have for large M, \\X\\ «£ M a n d
\A(l)\ = \C(\i)\^h\C\ ||nf <M~4/5M1/4^M-E by (29). (ii) Next, suppose that Proposition 2 holds for E — \ in place of E. We have h = mn by (9). Write H=(|ii,...,|i m ),
S(>i) = fi1(n1)+ •••
+BJ11J,
C(n) = C 1 ( j i 1 ) + - - - + C m ( j i m ) , where each n, has n components. Since n = c-,{k,r,6) by (8), there are nonzero vectors ]iu ..., ]im in J" with 5,(11,) = 0
and
|||i,||«max(l, \\B\\B)<Mirk «min(M 1/(4/c) , Mm5E)),
+ {r +
l)E+2)e
l^i^m.
(31)
284 DIOPHANTINE INEQUALITIES S e t t i n g X = vl\il ^,11,+
+ ••• + vm\im ••• +vmliJ
we have = C1(fi1)V1+
say. Since m = cs{k, r, E—\)
341
••• +Cm(]iJvkm
= D(x),
and
|£>| <max !C,.(n,)|
and \D(v)\ « M ~ ( 1 ~ » / f 4 £ ) ) ' £ - W \ D \ ^ M " £ + 1 / 2 - i / ( i 6 « - 1 / 2 < M - E if M is l a r g e . C o n s e q u e n t l y , i n view of X = vl]il
+ ••• + v m ( i m w e h a v e
\A(X)\ = \D(v)\<M~E and by (31) ||X|| < ||V|| llfiH « M 1 - l/<4£)+l/(5£) <
M
if M is large, i.e., Proposition 2 holds for E for forms with (3), and thus the proposition is proved. 5. THE PROOF OF THEOREM 1
Theorem 1 will be proved by induction on the values of k = max{kl,..., kh). The case k = 1 is proved by the box principle. We may suppose that |fj|= ... =\F,,\ = l. Given A/>2, consider the points leJs with li€P{M/2) (l^i^s). Then there are cl2(K) M2rs such points X. (See, e.g., [7, Lemma 11].) Given such a X, the point p(X) = (F{(X),...,Fh(X)) lies in the domain D: (pu ...,ph), where p; -Xj + yjt x,, y}£ U, \xj\ < (sM)/2 and \y}\ ^ {sM)/2 (1 ^j^h). Let v be the rational integer in ( C l 2 M 2 ") 1 / ( 2 A ) -lO<(c 1 2 M 2 ") I / ( 2 A )
285 342
WANG YUAN
and divide each region of pj in D into v2 subsquares of side (sM)/v. Since v2h < c12M2rs, there are two points p(X) and p(X/) such that for j= 1,..., h, Fj(X) and Fj{X') will lie in the same subsquare. Then \ = X — X' has
0<||AK||M| + ||M<M and for j= 1,..., h,
lFJ(A)\ = \FJ(X)-FJ(r)\^yJ(-vj
1/sMV
TsMV 2sM __ E£ +{—) < — « M
if s^zcn(h,
r, £ ) = c 5 (l,..., l;r, \,E) = u,
say. So Theorem 1 is true for k = m = 1. Let c 5 (l,..., I; r, m, E) = mu. Replace the indices 1 < / < mu by double indices 1 < / ^ m, 1
1 s£ i ^ m.
They are linearly independent and satisfy \Fj(X(i)\4M-E,
||3L(I)KM,
1<7
So Theorem 1 is true for k= 1. If F=F(l) = F(A,,..., A,) and G = G(p) = G(n1,...,/i,)
l<»<m. are forms, write
if there are / linearly independent points X-j,..., X, in /•" with GOi,,..., /x,) = F(^, > . ! + • • • + |x,Xr).
(32)
Put >MF,G) = minmax(||M,..., ll>-f||), where the minimum is over 'kl,..., X, with (32). Put \)/{F, G) = oo if F-h G. We use s{F) to denote the number of variables of F. Suppose that k > 1 and that Theorem 1 has already been proved for forms of degrees less than k. First we treat the case of a single form F of degree k.
286 DIOPHANTINE INEQUALITIES
343
LEMMA 8. Suppose k^l, / > 1, £ > 0 . If F is a form of degree k with s(F) > cl4(k, r, I, E), and if M^l, then there is a form G with F-*G and i]/(F, G)^M, and where G is a form in / + 1 variables, of the type
G =
+ H(v,vl,...,vl)
(33)
with \H\<$M~E\F\. Proof. Let e(l),..., e(/+ 1) be the first / + 1 unit vectors in Es. Consider the forms KPh...,Pk-u{y.) = HK .... K e(/»,),.-, e(pk-u)), <
M
(34)
•
where u takes all the values from 1 to &— 1, and plt -,pk^u all the values from 1 to / + 1. The total number of the forms (34) is less than k(l+ l)k, and each is of degree ^k— 1 in X. Each form (34) has \Kpi_ptJX)\^\F\. So by the part of Theorem 1 which we already know, we see that there exists c14(fe, r, /, E) such that if s ^ c14 there is a point Xo ^ 0 in Js with
IIXoKM and
\Kpt
ptJl0)\<M-
E
(35)
\F\
for all the forms (34). We may suppose without loss of generality that k0, e(l),..., e(/) are linearly independent. Writing G{n, v,,..., v/) = F(/iX0 + v I e ( l ) + ••• +v,e(/)), we have F^G and t(F, G) ^max(||J.o||,
||e(l)||,..., ||e(/)||)^M.
G is of the type (33) with a = F(k0),
F 1 (v 1) ...,v / ) = F ( v 1 e ( l ) + - - - + v / e ( / ) )
and H{n,vi,...,v,) =
Y (k)F(nk0,...,fiX0,v1e^)+
H fi
V"/
*—"—>
*
••• + v / e ( / ) , . . . , v , e ( l ) + ••• + v , e ( / » .
*-"
y
287 344
WANG YUAN
It follows from (35) that \H\<M~E\F\, and the lemma is proved. We omit the remaining part of the proof, since it is parallel to the corresponding part of the proof of Theorem A. ACKNOWLEDGMENT I am grateful to Professor W. M. Schmidt for his suggestion to consider the present problem and for all of his help.
REFERENCES
1. R. BRAUER, A note on systems of homogeneous algebraic equations, Bull. Amer. Math. Soc. 51 (1945), 749-755. 2. H. DAVENPORT, "Analytic Methods for Diophantine Equations and Diophantine Inequalities," Lecture Notes, University of Michigan, 1962. 3. L. G. PECK, Diophantine equations in algebraic number fields, Amer. J. Math. 71 (1949), 387^M)2. 4. W. M. SCHMIDT, Small zeros of additive forms in many variables II, Ada Math. 143 (1979), 219-232. 5. W. M. SCHMIDT, Diophantine inequalities for forms of odd degree, Adv. in Math. 38 (1980), 128-151. 6. C. L. SIEGEL, Sums of m-th powers of algebraic integers, Ann. of Math. 46 (1945), 313-339. 7. WANG YUAN, Bounds for solutions of additive equations in an algebraic number field I, Ada Arith. 48 (1987), 21^8. 8. WANG YUAN, Bounds for solutions of additive equations in an algebraic number field II, Ada Arith. 48 (1987), to appear.
a Vol. 32 No. 5
SCIENCE
IN
CHINA (Series A)
May 1989
ON HOMOGENEOUS ADDITIVE CONGRUENCES WANG (Institute
YUAN
( £ xO
of Mathematics, Academia Sinica,
Beijing)
Received August 26, 198S. ABSTRACT In this paper, the additive equations of the type a,A,* -f- ••• + cttX$ = 0 are studied, ajs being integers of an algebraic number field K of degree n. The main result is as follows: If J ^ ( 2 ^ ) " + I (or s^cl^nlogk. for 2 + ^ ) , the equation is solved nontrivially in any ^3-adic field, where ^5 is a prime ideal of K. Key words: additive equation, prime ideal, congruence, singular series.
I.
INTRODUCTION
Let K be an algebraic number field of degree «. For any X € K, we use /l ( ; ) (l ^= / ^ r ) to denote the real conjugates and Xim>{r + 1 ^ m =?S n) the complex conjugates of X. Let \ be a rational integer Js 1 and suppose « i , • • • , « , is a set of integers in K. Consider the additive equation 0 « , i f + ••• + nr^* = 0 .
(1)
One can apply SiegeFs generalized circle method15'65 on Waring's problem in algebraic number fields to prove that if s ^ cx(\, n) (hereafter we use
\X'im>\ < P . l < / < / .
(2)
The main term in the asymptotic formula is essentially a power of P multiplied by the so-called singular series connected to « , , • • • , « , . Ler ^3 denote a prime ideal of K and r be a rational integer the congruence
2& 1.
ff^f + • • • + axX) ss 0(mod?)3r). A set of numbers /l15 • • • , X, of K satisfying ( 3 ) is called a nontrivial ( 3 ) if Xt, •••, X, are integers, not all divisible by ^ .
Consider (3) solution of
1) We suppose t h a t the coefficients of additive equations and iofms a r e nonzero i n t e g e r s in K. t h r o u g h o u t this p a p e r .
(289) a No. 5
HOMOGENEOUS ADDITIVE CONGRUENCES
525
In order to derive from the asymptotic formula for the number of solutions to Eq. (1) in (2) that (1) either has infinitely many solutions or has a small solution in (2) with P depending on \, K, al} • • • , as (cf. [ 8 , 9 ] ) , we need to know that the value of singular series is positive. This is equivalent to the congruence condition: For any prime ideal power $% the congruence (3) has a nontrivial solution. Let r * ( ^ , K, S)3) be the least number such that if s > T*(^, X,^3r) and «!,•••, as are any given integers in K, the congruence (3) has a nontrivial solution. Write
r*a, K) = maxr*U, K,V). Peckt4] first established that T*(k, * O < *k2n+i + L However for the case of rational field Q, Davenport and Lewis[2] proved that T*(^, 0 ) ^ /^2 + 1, with equality whenever ^ + 1 is a prime. Chowla and Shimulacu showed that F*(^, Q) <1 c^log^ if \ is odd, which is best possible apart from some possible improvements on the constant c. Their methods can be generalized to treat the problem for the estimation of r * ( ^ , K ) , and we shall prove the following Theorem 1. We have
r*(k r>
if
* " odd'
~" 1(2^)" +1 , if -^ is an integer > 1. II.
PRELIMINARY LEMMAS n
For r 6 K, we denote 5 ( r ) = _^ rU) and E(r) = e ( 5 ( r ) ) , where e(x~) = e2*ix. i=l
Let d be the different of K. Suppose that a € K and aS = ^ / ^ 5 , where
s(«a*. v) = _Z £(«!*)» I (modfs)
-where X(mod'ip) denotes X running over a complete residue system. Lemma 1. Let d = U , N(^5) — 1).
Then
|S(«i*, ? ) | < (^ - lViVC^)". Proof. Choose JJ to be a primitive root mod^5. Then ^ = 7)ind'"(mod^5) for any ^t with ^P-f"^. The necessary and sufficient condition for the solubility of the congruence i* == ^(mod^3) is ^ind^, = ind^(modiV (^5) — 1) or d\indfi. If the congruence is soluble, it has d incongruent solutions mod^3. Suppose d = 1.
Then M(mod¥)
(see H e c k e [ 3 ] , p . 1 9 7 . )
N o w suppose t h a t d > \ .
Then
5(«A*. V) = 1 + E * E(«A«) 2 - ( ™ ^ Y M(mod1!)
m=0
\
d
'
290 526
SCIENCE IN CHINA (Series A)
Vol. ?2
where 2 * denotes a sum in which ft runs over a reduced residue system mod^B. Since
?, E(ctfi) = — 1 , we have M(mod$)
and by the Schwarz inequality we admit
|s(«a*,*)|'
|5(«l«,?)|J<(rf-l)S S* S* ,(^)£(«Cr-l)^)
- o* - 0 s (wcw -1 + 5T* * ( ^ - ) OT=1 ^
i*x (mod P)
(Tmodp)
\
d
'
'
= w -1) S (w(W -1 - £**' f^-))«i=l ^
T(modp)
X
d
/J
where 2 * * denotes the sum 2 * with T = 1 deleted. Since **ry** / windr \ __ > , -"c-** /mindr \ T(modp)
\
d
r(modp)
J
\
d J
-l+^)-'tt(=l)-l, we have |S(ol*,qs)| < ( r f - D \/iVC?). The lemma is proved. Let * be an integer of K such that sp||jir. If 5 is a set of complete residue sys tem modsp, then any integer a of K has a unique ^3-adic representation CO
<=o In fact, ,K0 is uniquely determined by ^0 = a(mod^5), /*, is then determined by a — ^O(mod^52) and so on. Consider two additive forms:
JT^I—
^
291 No. 5
HOMOGENEOUS ADDITIVE CONGRUENCES
527
A = 2 «/*? and S = 2 /W* <= 1
i= 1
with integer coefficients. We shall say that A is ^p-adically equivalent to B if A can be changed into A' by a transformation ^,- = r,-^,(l ^ * ^ s) such that A' = y 5 (modspr) for any r > 1, where r,-(l < i < s) are nonzero integers of K. It is easily seen that the relationship is reflexive and transitive. Now we proceed to prove that the relationship is symmetrical. In fact, suppose that
2 <%(*/*)* = r SftA*?(mod?')
(4)
holds for any r ^ 1. Then set ^,- = rt,- • •r,_ir/+1' • •tsli{\ ^ < < s). f
We have
f
2 ftr*(ii. • -r^r,-^- • -tsyx} = r^ 1 2 " ^ ( ' r - •r,_1r,+1- • -r,)*i? *= !
«= t
If the congruence (3) has a nontrivial solution for every r, then it is said that the equation A = 0 is solved nontrivially in sp-adic field or that the form A represents zero ^J3-adically. We may prove that if A and B are ^5-adically equivalent, then they can or cannot represent zero simultaneously in the "Sp-adic field. In fact, from (4) the solubility of A = 0(mod^3r) is found to follow from the solubility of B = 0 (mod^J5r') for some sufficiently large r'. Lemma 2. Any additive form A is equivalent ty-adically to a form of the type F = pw + xpM +
1- w «-iF ( <-»,
(5)
where Z7''' is an additive form in Vj variables {the variables in distinct forms Fiiy being distinct) with all coefficients ^ 0 (mod^3), and where va^$l\. Proof. Represent every a,- by a series of it. By a substitution A,- = ^'fi,- with suitable r , ( l =£~/ s/\. Put A,- «= Tt/z,- for the variables in Fm. Then F is equivalent sp-adically to the form Hence we can choose a cyclic permutation of F ( o ) , • • • , F(*~i:> to ensure that the number of variables in F (o) is ~^ s\\. The lemma is proved. III.
THE PROOF OF THEOREM 1 (ODD DEGREE
K)
We use the notations: II = J J a,-, e means any pre-assigned positive number <= i
and \ means an odd integer ^ c ( s ) .
Lemma 3. // *$\Tt, then
292 52S
SCIENCE IN CHINA (Series A)
Vol. 32
r*tt,K,qO<(-2-+s)log*. V log 2
/
Proof. 1) Suppose iV(sp) < k * . Consider the 2s — 1 sums 2+ n
«,-(1 < i < f ) , a,- + a,( 1 < i < ; < * ) , • • • , « ! + • • • + a^. If 2'— 1 >2V(?P), then at least two of the sums are congruent mod^3. Since —1 is a ^th power residue mod^P, the congruence A = 0(mod5p) has a nontrivial solution if 2' > * 2+£/2 (l + kT1-*'1) or Hog2 > (2 + —^log^ + log(l 4The lemma follows. 2) Suppose N(ty) > ^2+E/2.
k'2'en).
We shall prove the following stronger inequality
r*u,
K, ?) <
[!•] + 4.
Let M(^5) denote the number of solutions of the congruence ail\
+ ••• + «,_,!*_! = — a,(mod?P), ^,(mod^), 1 < /' < s — 1.
Then
M » ) = ^ S "V-Py
c
2 ••• S JjCmodp)
£(*(«.*?+---+«,-itf-i+ «,)),
l._i(modt)
where jt runs over a complete residue system of (^P^)"1, mod 8~l. Therefore
W(?) = W(?)''! + W(?)-'S' £(fw,) X *(/«/^» ¥). By Lemma 1, we have
Consequently, if
(^ - i y - w ( ? ) ~ r < zv(qsy-2,
^
y
then
MOP) > 0. If ^ satisfies
N($) > k^ •= V+T^I, t h e n ( 6 ) h o l d s . ( 7 ) follows from A T ( ? ) > * + 4 , w e can get a nontrivial
2+e/2
if s — 3 > 8 / e . H e n c e if
(7) s^[S/e]
solution
of t h e congruence A = O(mod^P).
T h e l e m m a is p r o v e d .
Lemma 4. Suppose that ^Tl^'^p
and pb\\ki where p is a rational
prime,
293
No. 5
HOMOGENEOUS ADDITIVE CONGRUENCES
^ ^ 1 and £ > 1.
529
Then if w > (b + 2)e, we have
\ log 2 Proof. Suppose u> = {Jb + 2)e. 2
/
Let j be an integer such that 2
2' — 1 > ATU) > NO) * > 2V(?)2*e > iV(^3)(*+I)'.
(8)
Then at least two of the sums «,-(1 ^ / ^ s), a,- + «,-(1 =^ /
J)
, • • •, a t + • • • + as
( +I)
^re congruent moctf)3 * % i.e. the congruence ^ = 0(modq3(*+1)e) l a s a nontrivial solution. From (8) it follows that the condition of .y is H o g 2 > l o g ( ^ » + 1) or
The lemma holds for the case w = (3 + 2)e. the congruence
Suppose that w ^ {b + 2)e and that
y* = 0(mod (?*"') has a nontrivial solution 215 •••, ^ . Without loss of generality, we may assume that ^Pl^i. We now proceed to prove that A = 0(mod SP"O also has a nontrivial solution. Set Pi = lt + v.it"-1"-*, K
/ < s.
We start to show that
2 am1} = 2 «A* + ^"'~*5""': 2 «/»'/^"1(mod?"'). <= 1
i= 1
i= 1
In fact, consider the (h + l)-th term ( ) ifi.Viit"'1"'')'' \h / of
(9)
in the binomial expansion
^ - (1/ + v,**-*'-')*, where A > 2.
Suppose /> a p.
Since (
\h i
) = -f ( ,
)> the (A + l)-th term is
h \h — 1 /
divisible by SJ3*, where g = h(w — be —
To prove (9) it suffices to show that g ^ w or (A — l ) ^ — £ 0 ^ A e + oe. Since w ^ (b + 2)e, it suffices to show that
294 530
SCIENCE IN CHINA (Series A)
Vol. 32
2(A— l ) e Ss (/»•+• a)e, i. e.,
(10)
h^a+2.
Since 2 ^ and p\\, we have 3 " < f ' ' < A , and therefore (10) follows by 3° = (1 + 2)" ^ l + 2 a ^ a + 2 for the case a ^ 1. If a — 0, (10) follows immediately from h > 2. Write
/t = **> and 2 «>•*.*
=
"^"'^
<= i
in their ^3-adic representations, where ^ q o .
Then from ( 9 ) , we have
Since (qoa^i" 1 , ^5) = 1, the congruence (/> + cpa^-'Vi = 0(mod^O has a unique solution v1(mod^3'0. A*IJ
••••>!**
P u t "/ = 0(2 ^ « ^ i ) .
Then we have a solution
of
2 «••*•? = OCmod?"), i-l
where ^Pi.
The lemma follows by induction.
Proof of Theorem 1 (odd degree ^ ) . the forms F of the type ( 5 ) .
By Lemma 2, it suffices to consider only
1) Suppose ?P|^. Then it follows by Lemma 4 that the equation FC0) = 0 can be solved nontrivially in ^3-adic field if [ 2 \ v<,> [+ s nlog^, \ log 2 / and therefore F = 0 can be solved if V log 2
/
The theorem follows. 2) Suppose ^ \ \ and ¥>\TL. If * > (
\ log 2
+ ellog^ and w > 2, then we may /
assume that F = 0(mod<)3"/-1) has a nontrivial solution with ^Plili. Now we proceed to show that also has a nontrivial solution. Set
fii = Xi + Vi*f»-\ K
/ < s.
F = 0(mod^ K ')
295 No. 5
HOMOGENEOUS ADDITIVE CONGRUENCES
531
Then
s
Since 2 «/*? i s divided by (^"^f"" 1 , ?"0 = ^w~l, the congruence x
2
«,•!? + V " 1 ^ " 1 ^ = 0(mod'?5"')
i= l
has a unique solution i>l(mod'$5). Put Vj •= 0(2 ^ z ' ^ . f ) . We have a solution of F = 0 (mod^P"') with ^ ^ O(mod^P). The theorem follows by induction. 3) Suprose SP-I1^ and ^3|II. It follows by 2) that Fm = 0 is solved nontrivially \ / 2 in ^(3-adic field if v0 ^ ( + e) log^, and therefore F = 0 is solved if \ log 2 / \log2
'
The theorem is proved. IV.
THE PROOF OF THEOREM 1 (ARBITRARY DEGREE ^)
Let p denote a rational prime and w{a) the exponent with which ^3 enters into the canonical factorization of a, i.e. ^3w<;'')||o! and
l
(11)
>~[jz-i\ + be + l>
where pb\\\ and ^||/>. Lemma 5. The series
±4-
•=it\
are convergent provided to(a) >
and ±{-iy°-
P— 1
<=i
, and they are denoted
log(l + a) respectively. See Ref. [ 7 ] . Lemma 6. Suppose ^3^/?.
i
// the congruence ftk = a(mod$'°)
Aoi a nontrivial solution, then so has the congruence / V = a(mod^5') for any I > /„. Proof. Since *P-f(?, there exists ^ such that /?^= I(modq3').
by exp a and
296 532
SCIENCE IN CHINA (Series A)
Set r = ap and 7 — f* = Sp'°.g>, where ^ l
follows from £* (— — \\=^> o^ Vj* /
Vol. 32
is an integral ideal.
that
Since (f, ^3) = 1, it
Therefore by Lemma 5, there
exists
lDB 1 +
(
^-1))"l0gF-
Since
" (| l 0 g ^) -
U
( l0g ^) ~ »W >U-be->
-f_,
there exists also exp( — l o g ^ ) = A, say, by Lemma 5. Then
r = U£)* and therefore /4f is an integer.
Choose an integer IJ such that
»Uf
-n)>l.
Then M(IJ*-?)-«(IJ*-UI)*)>/,
i. e. JJ* = y(mod^'). The lemma follows. Lemma 7.
Le» (/ = (^,N(sp) — 0 . Suppose that <*i---(*d+i^ 0(mod^3).
Then «^{ + • • • + ad+ll%+1 = 0(mod^3) has a solution Proof.
with Xt ^ 0 (mod^5).
It is well known that the congruences y* = m (mod^5) and yd = m (mod'p)
are soluble or not soluble simultaneously. Hence if the conclusion of the lemma does not hold, then for any X2, •••, %d+i> we have r- ad+l lkd+L H 0 ( m o d $ ) ,
«i + CHX\ + i.e.,
( a , + cnli + ••• + ad+1Xi^Ym~l
= 1 (modq3),
which should be an identity. But the expansion of the left-hand side contains a term ffv.^2
^d+i)
d
—ayt>2
Ad+1)
297 No. 5
HOMOGENEOUS ADDITIVE CONGRUENCES
533
where ^>\a, and every other term contains at least one of the variables X2, •••,Xd+l with a power less than AT(^3) — 1, which leads to a contradiction, and the lemma follows. To prove the theorem it is sufficient to consider only the forms F of the type ( 5 ) . First we define an operation of contraction as follows. Consider a sum of d + 1 terms in F (0) .
Since at- • -ad+l^
ayl\ + • • - + ai+lXi+i 0 (mod^p), the congruence
(12)
«ii«* + • • • + ad+lfid+1 = 0 (mod^P) has a solution with (ii ^ O(mod^p) by Lemma 7. We can assume that the left-hand side of the congruence is equal to a nonzero number a, or otherwise it follows directly that F = 0 is solved nontrivially in the ^3-adic field if s ^ d + 1. Set A,- = nil (1 < i < d + 1). Then there holds M ? + • • • + aJ+lXd+l = (oti^f + • • • + ad+1fj.d+l)\*- = a^*> where sp''||as, / ^ 1.
(13)
Thus a has a $p-adic representation a = VV,+ • • • , Wvi.
(14)
Therefore the operation of contraction consists in replacing the sum (12) by a single term (13). Notice that X^O (modsp) implies 1 , ^ 0 (mod^5), where 1, is called the distinguished variable in (12). Operations of contraction are applied to groups of d + 1 terms in Fw and here any one of the variables can be chosen as the distinguished variable. The number of remaining variables in F (o) is t h e n ^ d , and we put them to be equal to zero. Hence there results a form of the type
*Gia + ^G(» + • • •, where G<;) contains the original Vj terms of Fw together possibly with some additional terms, each arising in the form (13) from one of the contractions. The variables in these additional terms are called the derived variables. If i ^ \ , we define #,- = 0. The next step is to apply contractions to groups of d + 1 terms in G(1), subject to the condition that each group contains at least one term with a derived variable; such a variable is chosen the distinguished variable in the contraction. The process is continued, and at each stage we take care that at least one of the variables in a group is derived, either directly or indirectly, from a variable in Fm. Suppose that, after a series of permissible contraction, we get a form H such that H = 0 (modsp'o) is soluble with at least one of the derived variables in H, not divisible by ty. This implies a solution of F = 0(modsp'°), and on tracing back the derived variable to its ancester in F(0>, we see that the solution is nontrivial. Finally, if we get a form H = 7t^HM +
W «+IH<«+»
+
•••
(15)
298 534
SCIENCE IN CHINA (Series A)
Vol. 32
in which any one of the forms HW|)), H u ° + 1 ) , • • • contains a derived variable, then we take the derived variable to be 1 and all the other variables equal to zero.
Thus a
nontrivial solution of F = 0(mod^)3'») is derived. Lemma 8. We can obtain a form H of the type (15) from a form of the type ( 5 ) by repeated contractions, where H U ) j Hiu+i>, • • • are additive forms in distinct variables with all coefficients ^ 0 (mod'ip). For i^k.— 1» the form H (j) includes the terms of F{'\ and possibly some additional terms contain derived variables; for j ^ \ the form H^ can contain derived variables only. Further, if Su denote the total number of derived variables in H, then
We divide the v0 terms of Fm into m sets
. Proof. Suppose first that « = 1. of d + 1 terms, where m
= I" v" 1 ;=> _ i ^ ld+
1i
d
d+ 1
> _^>
d+ 1
1
d+ 1
The lemma holds. Suppose that u ^ 1 and the lemma is true for «, We proceed to show that the lemma holds for « + 1. Let w denote the number of derived variables in H
(w,
\\*z±JL\t
dw,
MvH
(L d + l J The remaining variables in Hiu) are equal to zero. contracted into a single term xsalk,
£>»4-l,
Each set of d + 1 terms can be
a^5=0 ( m o d ^ ) .
Adding such a term to the corresponding form itgH{g}, we obtain a form of the type ^.a+ipCB+O +
x«+2pl.u+2)
+
. . .,
where each Pw> contains the variables in HC'J and derived new variables of the form a?al', ^\a. The total number of derived variables in P(a+I>, • • • is expressed as
1
u>,
if vu ^
dw,
[^t«L], if Vu
In the first case, we find c A
-+
i
c --> *"^(d+
In the second case, we find
v
"
i)«
— 1 •>
v
° — 1 (d+iy+i
299 No. 5
HOMOGENEOUS ADDITIVE CONGRUENCES
535
1-4 + 1 J ^ 5B — #/ + — 4+1 Since w^Su,
-— 4+ 1
it follows that
58+I >
5
" + dw ~w <£+l ^+1 i
Su
f«
4+1 •->
+ v"+w 4+ 1 4
4 + 1
^
! _ 4+1 4
Su
4 + 1" 4 + 1
fo
1
4
(4+l)«+1
4+1
4+1
=
4 + 1 ^0
j
(4+l)»+1
and the lemma follows by induction. Lemma 9. If s ^ (2^)" +1 , the congruence F = 0 (mod
0
.
If 2V(
d = (^, iV(?) - 1) = (*, /»' - 1) = (it,, />' - 1), d + 1 < min(2^0, pO and (4 + 1)'« < (4 + l)-^.+*»+i
< (.P^W • (2/t.) < 2» + l (V*) / e < 2»+^». Therefore, Lemmas 2 and 8 lead to '•
(4+l)'»
/^(4+l)'o
^(2«"+1
The lemma is proved. Proo/ o/ Theorem 1 (arbitrary degree /^). Suppose that lt, •• • ,XS is a solution of a^f + Avith sp|i,.
r- eg,* = 0 (mod'ip'o)
Let / > /0 and (1.1 = l i + 7tl"v,
2 < i < j ,
where v runs over a complete residue system mod ^5;"'«. ence a^
By Lemma 6, the congru-
= — a2fi2 — • • • — asfi^ (mod ^5;)
-has a nontrivial solution X for any given fi,,--',
fis, i.e., the congruence
a 536
SCIENCE IN CHINA (Series A)
+ • • • + a,X) = 0 (modsp')
aiX* u 7|)><
has no less then JV(sp) ~
Vol. 32
'~I) nontrivial solutions.
The theorem is proved.
EEFEEENCES [ 1] [2] [3] [4] [5] [6] [7] [8] [9]
Chowla, S. & Shimula, G., On the representation of zero by a linear combination of £-th powers, Norsfa Vid. Sets. Fork., 36(1963), 169—176. Davenport, H. & Lewis, D. J., Homogeneous additive equations, Proc. Roy. Soc, A 274 (1963) 443—460. Hecke, E., Lectures on Theory of Algebraic Numbers, Springer-Verlag, 1980. Peck, L. G., Diophantine equations in algebraic number fields, Amer. ]. Math., 71 (1949), 387— 402. Siegel, C. L., Generalization of Wiring's problem to algebraic number fields, Amer. ]. Math., 66 (1944), 122—136. , Sums of m-th powers of algebraic integers, Ann. of Math., 46 (1945), 313—339. Tatuzawa, T., On Waring's problem in algebraic number fields, Ada Arith., 24(1973), 37—60, Wang Yuan, Bounds for solutions of additive equations in an algebraic number field I, Acta Arith., 48(1987), 117—144. • , Bonuds for soluitons of additive equations in an algebraic number field II, ibid., 48(1987), 307—323.
301 Reprinted from JOURNAL OF NUMBER THEORY
Vol. 45, No. 3, November 1993
All Rights Reserved by Academic Press, New York and London with permission from Elsevier
Small Solutions of Congruences WANG YUAN Institute of Mathematics, Academia Sinica, Beijing 100 080, China Communicated by Alan C. Woods Received June 10, 1990; revised February 28, 1992
be a system of quadratic forms with Let Qj(k) = Ql(ll,..., Xs) (l^j^h) coefficients in integers of an algebraic number field K of degree n, and let a be an integral ideal of K. The purpose of the paper is to prove that the system of congruences Qj(k) = 0 (mod a) (l^j^h) has a nonzero solution X satisfying max,j. |A]''| <£N(a)1/2 + i provided that s^c(h,n,s). This improves a result of T. Cochrane (1987, Illinois J. Math. 31, 618-625) and also gives a generalisation of a result of R. C. Baker (1980, Mathematika 27, 30-45). The small solutions of additive congruences are also considered. © 1993 Academic Press, Inc.
1. INTRODUCTION
Let K be an algebraic number field of degree n over Q with a ring of integers J. Let a ^ H E . ^ K . ^ O l U K,(/> = °%0), 1 ^j^h) be a system of h quadratic forms in k = (A,,..., 2.s) with coefficients in J. Using a generalised geometrical method of Schinzel, Schlickewei, and Schmidt [5] on the small solutions of quadratic congruences, Cochrane [3] proved that if a is a nonzero integral ideal of K and h < s/2, then the system of congruences Qj{-k) = Q
(modo),
l^j^h
(1)
has a solution X in J with 0 < max \N(X,)\ ^ c, (K) N(a)s,
(2)
where a + 3/2(s-l), \h/(h+l) + h/(h+l)(s-r),
if h = 1,2, if h>2.
c(f, -., g) denotes hereafter a positive constant depending on /,..., g, N(X;) 261 Copyright © 1993 by Academic Press, Inc. All rights of reproduction in any form reserved.
_ ^ _ _ _
302
262
WANG YUAN
and N{a) are norms of A, and a, and r is the remainder on dividing s-h by h + l. However, in the rational case, Baker [1] has given a result in which d -> \ rather than h/{h + 1), as s -> oo. Baker [2] proved also that for any given £ > 0 and integers k, q^2, if s^ c2(k, h, e), then the system of h congruences s
X a,y** = 0
(mod?), U / ^ A
./= i
with coefficients in Z has a solution in nonnegative integers satisfying 0 < m a x x , ^ 9 1 / A r + £.
xl,...,xs (3)
We may use the form xk + ••• +xks (k^2) to show that the limiting exponents \ and \/k in (2) and (3) are best possible, even when K= Q and h=\. The proof of Baker's results is based on a discrete version of Schmidt's method on small solutions of additive equations in many variables [6]. In Baker's argument, the relation «
>
„_,
exp
/
,au\
fl,
if
q\a,
q)
10,
if
q\a,
2TT/— = <
V
is used instead of the integral over [0, 1) in the circle method, and the case h— 1 of Baker's estimation (3) is derived immediately from the following Schmidt theorem: Let au ..., as, bu ..., bs be natural numbers. Then there is a constant c3(k, e) such that if s^c3(k, E), the equation a , x ? + •••+asxks=b1y\+
•••+bsyk
has a solution in nonnegative integers x { ,..., x s , y l t . . . , y
0 < max (xh yj) < max (a,, bj)[/k i.j
s
with
+e
.
i,j
(See Schmidt [6].) Since Schmidt's theorem has been generalised to algebraic numer fields by the combination of his method and Siegel's version of the circle method [7, 8], we could generalise the above Baker theorems to algebraic number fields by a similar discrete method. Let K{l) ( K / ^ r J be the real conjugate of K, and K{m) and K(m + r2) (rt + 1 ^ m < rx + r2) the complex conjugates of K, where rt + 2r2 = n is the degree of K. Throughout this paper, the indices / and m are over the sets of integers cited above. For yeK we denote by y(l> (l^i^n) the
3ra SMALL SOLUTIONS OF CONGRUENCES
263
conjugates of y. A number y of K is called totally nonnegative if y ( / | ^ 0 , and we denote by P the set of all totally nonnegative integers of J. We use a, q,... to denote the nonzero integral ideals, p, px,... the prime ideals, <5 the different (ground ideal) of K, and e any pre-assigned positive number. The constants implicit in the symbols <^ and 0 may depend on k, h, K, e,... but not on a, q,.... THEOREM 1.
Let
F ( W=iv;,
Ui<*
(4)
7=1
be f o r m s with that
coefficients
i f s ^ c 4 ( k , h , n , e ) , there
in J . Then
(mod a),
Fi(k) = 0
is a X e P s
1
such (5)
and 0<maxN(Xj)<$N(a)l/k j
+e
.
THEOREM 2. Suppose s~^cs{h,n,z). Then the system congruences (1) has a solution X in Js with
0 < m a x \N(A.j)\
+e
of quadratic
.
We prove Theorem 1 first and then derive Theorem 2 from Theorem 1 by the method of algebraic diagonalization with the aid of the box principle. Since the conclusions of Theorems 1 and 2 can be improved to N(a)s if K is a totally complex algebraic number field, we assume in the following that r, is positive. See Wang [8, Chap. 11].
2. LEMMAS
Let co1;..., con be a basis of / and D the absolute value of the discriminant of K; i.e., D = |det(wj' ) )| 2 (1 < / , . / < « ) . Let y, (1 ^j^n) be numbers of K and Xj (l^j^n) be real numbers. We set £ = Z"=i xj"?j a n d define w x Z, = Tj-\ jy? (1 < ' ^ « ) - W e u s e t h e notations
ll^maxl^'l,
S(£)=t
Z W>
E(Z) = exp(2niS(Z)),
and P{T) the set of integers in P satisfying ||A||< T.
304
264 LEMMA
WANG YUAN
1. Suppose a eJ. Then
Z£K«)={^(0)' ^
(.0,
V"; otherwise,
where t; runs over a complete residue system of (ad)~\
m o d <5~'.
See, e.g., Wang Yuan [8, Chap. 2]. LEMMA 2. Le? T, (1 < / < « ) be positive numbers such that Tm + n= Tm. Let fieJ and N(a, T) denote the number v of P satisfying
vsju(moda),
|v (m) |^r m .
0^v{l)^T,,
Then
where To = max(Af(a) 1/n , (Ty • • • Tn)l/"). See, e.g., Wang Yuan [ 8 , Chap. 3 ] . Remark. Take Ti = c6(K) N(a)l/" (1 < / < « ) , where c6(K) is sufficiently large. Then it follows from Lemma 2 that N(a, T ) $> 1; i.e., there is a veP satisfying v = ju (mod a) a n d 0 < ||v|| ^ ^ ( a ) 1 7 " . Therefore we may choose a complete residue system R(a), modulo a, such that yeP and
0<||y||
3. Let /(A) = akXk + • • • + a, A be a polynomial with coefficients in J and a^/O and let S(f,T) = YjieP(T)E(fW€)> where A runs over all integers in P(T). Suppose that T^c7(k,K,e), C^T"~l/r~l + \ and \S(f, T)\ ** C. Then there exist aeP and fieJ such that LEMMA
II««*£-0M(^-J
T'k^
and
O
T\
See, e.g., Wang Yuan [8, Chap. 3]. LEMMA
4. Let F(X) = £ J = i a, Af be an additive form with nonzero integer
a
SMALL SOLUTIONS OF CONGRUENCES
265
coefficients in J. Suppose that s^cg(k,n) and that a-" ( 1 ^ / ^ s ) have different signs for each I. Then the equation F{X) = 0 has a solution "kePs with
0<max m.-Mmax ||a,-|]C9(*'") See, e.g., Wang Yuan [8, Chap. 8]. LEMMA 5. Let c = c%{k,n) and r be a natural number. Then there is a matrix Gr=(gij) (l^i^r, l^j^c1") whose entries are + 1 having the following property. Given integers /?,-, with ||jS,y||^5, there is a k in Pc' satisfying
(6) .7=1
and
0<max m,-||«5fl0(*'r''1).
(7)
Proof. We take c > 2r t , and for each j , 1 ^ j < r ls we choose ^I J2 _/- i and gU2J such that gt.2,-1, P[%-i and gi,yPluv h a v e different signs. Note that if there is a Ptj = 0, then the assertion of the lemma is obvious for r=\. So the case r = 1 of the lemma follows from Lemma 4 by taking clo(fc, 1, n) = cg(k, n). Now suppose that r^-2 and that the lemma is true for r— 1. Let Gr be formed by Gr , as follows:
0,The vertical lines here divide Gr into c blocks. The variables / ; ( U j < c ' ) are also divided into c blocks with indices Bl = {1,..., cr~'},..., Bc = {(c— 1) c r x + 1,..., cr}. By induction we can choose gtJ ( l < / < r — 1 , jeBv) such that there are ^7 (je Bv) in /" satisfying
£ g,jPvrf = O,
i^i^r-l
(8)
ye i^-
and 0<max||^.||«5 cl0( * r - r - 1 -" ) .
(9)
306 266
WANG YUAN
For 1 < v < c, let aD be the integers given by « , = I PrjHkJ. JeBr
T h e n w e m a y c h o o s e grj=gr(Bv)
s u c h t h a t g,(BL)
(jeBv)
a[n
(\^v^c)
have different signs for each /. Consider the equation gr(Bl)xlvkl+
acvkc=0.
•••+gr(Br)
(10)
It follows from Lemma 4 that (10) has a solution v in Pc satisfying 0 < m a x ||v,-Hmax \\a.\\'^k-"^
B(]+k'mik-r-l-"))c''(k-").
(11)
Define A,- = vvn, (ieBv). It follows from (8) that (6) holds for / = 1,.., r- 1. From (10), we deduce that (6) also holds fory'=r. Equation (7) follows from (9) and (11) and by taking
cl0(k, n, r) = (1 + (k +l)cw(k,r-l,
n)) cg(k, n).
The lemma is proved. Remark.
Lemma 5 was established by Baker [2] for K= Q and j8,y>0.
LEMMA 6. equation
For any given 2s nonzero integers a l 5 ..., as, jS l5 ..., fis in P, the
has a solution in integers ll,..., Xs, nl, ..., ns of P, not all zero, such that
max (JV(A,), AT(^))«max (JV(a,), 7V(^))1//r + £
provided that
s^cn(k,n,e).
See Wang Yuan [7, 8]. LEMMA 7 (The case /J = 1 of Theorem 1). Suppose that s^cA(k, am/ a , e / (1 ^ / < ^ ) . 77;en //;e congruence
£ a,A* = 0 i=
(mod a)
1, n, e)
(12)
I
Ziai1 a solution l.ePs such that 0<msixN{li)
+e
.
(13)
307
SMALL SOLUTIONS OF CONGRUENCES
267
Proof. B y t h e r e m a r k of L e m m a 2, w e m a y s u p p o s e t h a t u,eP a n d N(oLj)<$N(a) ( l ^ K s ) . Take a e a such that aeP and N(a)^N(a.)
(14)
Take c4(k,l,n,e) = cn(k,n,e). Then it follows from Lemma 6 that Eq. (14) has a solution in integers A,, ..., Xs, ^ l 5 ..., (is in P, not all zero, such that max (N(A,), N{nj)) < max(max N(«,), N(a))l/k
+s
4 N(a)1/k
+
\
It is clear that >. # 0 , since r, > 0 . Hence from (14) we have a solution X of (12) satisfying (13). The lemma is proved.
3. REDUCTION PROPOSITION 1. Let l/k^x^l and s^ci2{k, h, n, x, e). Let FjCk) (l^j^h) be forms as in (4). Then for each ideal a, there are integers Aj,..., Xs in P satisfying (5) and
0<maxN(li)
+e
.
The case x = l/k is Theorem 1. It is obvious that Proposition 1 is true for x = 1, because we can choose by Lemma 2 an integer a satisfying aeP, a|a, and 0
N(a)^cii(k,
h, K, x, e). In fact, if N(a)
then (5) has a
solution A, = ••• = A J = a with a stated above such that
N{a)
+
'.
Let X be the set of x such that Proposition 1 holds. X contains 1, and so X is not empty. It is clear that X is a closed set. Hence the proof of Proposition 1 is reduced to proving that if 1 > x > l/k and x e X, then there exists an x' e X with x' < x. We assume that Theorem 1 holds for h— 1 forms (h^2) by the induction, since Theorem 1 is true for h = 1 by Lemma 7, and that Proposition 1 is true for a number x of X. We proceed to show that Proposition 1 is true for a number x' with x' < x. Suppose that j is a rational integer ^ 2 and that there is no prime ideal p such that pj\a. Then a is called ay-free ideal. Now we first show that
308
. 268
WANG YUAN
if Proposition 1 is true for square-free moduli in at least ct4(k, h, n, x', e) variables, then it is also true fory-free moduli, provided that s^c'M '. The rational case of this assertion was established by Baker [ 2 ] . Now we use the induction and assume that the assertion is true for (y — l)-free moduli, where y > 3, and we proceed to prove the assertion fory-free moduli. Write a ./-free ideal a as a = n ?= i P f, where 1 ^ a, ^ j - 1 (1 ^ / =$ q) and where pt, pr (i¥=r) are distinct prime ideals. Let * ; = a,if l^ai<j-\ and and 6, = a , - l otherwise. Let al = YlUiP^ a2 = Y\Ui PT~h'- T h e n 0 = 0 ^ 2 , where ay is (j— l)-free and a 2 is square-free. We may choose an integer nu) such that (nli), a) = fp, (i=\,...,q). See, e.g., Hecke [4, Theorem 74]. Set n^ = Y\Ui ^i)b' and n2 = Y\t=\^U)"'~hl- Then a, j TI,, a217r2 and n2\iti. Any a in K which is a p,-adic integer for all / has a unique representation « = Z Pit'i' ;=0
M,e^(
where Ria^ is a given set of complete residue systems mod a,. In fact, there is a unique fi0 such that a = /i 0 (mod a j by the Chinese Remainder Theorem, and nx is then determined by 7 1 , ^ =a — pL0 (mod a\), and so on. Set M = C 1 4 2 , i> = c 1 4 , and s = uv. Rewrite the forms (4) by F , ( A n , ...,*„,)= t i=
( a i l ( ; ) A * 1 + ••• +aiu(j)Xl),
l^j^h.
I
By the assumption of induction, we may choose integers X'n,..., X'iu in P such that «« U) A/i* + • • • + a,B (;) A;* = 0
(mod a), 1 < j ^ h
and 0<maxJSr(A;,)«JV(o,)JC' + *. Write a, (;) = «/i (y) A;,* + • • • + «,„ (y) A;B*.
Expand a,(y) in a power series of nlt
where 1=0
I
< / < v, 1 < j < A.
309 SMALL SOLUTIONS OF CONGRUENCES
269
Since it2\n1, /?,(./) is also a power series of n2, that is,
j8/(7)=f v}JV2J
vfeR(a2).
1=0
Since a2 is square-free, we derive by the inductive hypothesis that the congruences
t PiU)rf= t v ^ N «
1=1
(moda2), 1
i=l
have a solution in P" satisfying
0<m!nN([ii)
l^t^u).
Fj(kn,...,kvu)
+s
.
Then we have =0
(modo),
l^j^h
and 0<maxN(Xil)
+e
.
This completes the proof of the induction step. In particular, Proposition 1 holds for x' if s ^ c\^y and a is fc-free. Now we may write any ideal a as a = 93*(£, where £ is A>free. So if we set s = ckl4i, we have integers k\,..., A^ in P such that F , ( r ) = Fy(Ai,...,A;) = 0
(modK), 1 < ; < A
and 0<maxiV(A,')<|A r (e) v ' + e. We take an integer ^ by Lemma 2 such that PeP, Af(/9) <^iV(»). Then X = 0k' = {pX\,..., fik's) satisfies
Fj(X) = pkFj(X') = 0
(mod a),
5B/j8, and ./V(2$)«S
l^j^h
and 0 <max N(Xf) = 7V(/S) max N(X-) < (N{®)k N(<S,))X' + B < N{a)x' i
i
Therefore we may assume that a is square-free in Proposition 1.
+
*.
310 270
WANG YUAN 4. DIVISION INTO TWO CASES
Set y = (x-^\cio(k,h,n)-l(2(h z = (x-l-\y(2h
+ 6)n)-\
+ 6)-[
(15) (16) (17)
x' = x-z. We have - + cl0{k,h,n)2y(h
+ 5)n
= ^+(x-^\2(h
+ 5)n(2(h
+
6)n)-1
=-j;+ (*-£) (/i+ 5)(/! + 6r 1 = x-(x-^\(h
+ 6)-1<x-z = x'.
(18)
Write a , = (<xu,..., aht) (l^i^s) for the coefficients of A* in (4). For any two vectors a = (au ..., ah) and P = (/?i,...,fih),we denote by aP = X''= i a,-)?,• the inner product of a and p. A vector y = (y,,..., yh)eJh is said to be prime to an ideal q, if the greatest common divisor (y 1; ..., yh, q ) = 1, where y, is regarded as the principal ideal generated by y,. Condition D(k, h, t, a, y). For any t coefficient vectors, say a,,..., a,, any divisor q of a, and any y in Jh prime to q such that ya, = 0
(modq), 1 «£/=%/,
we have N(q)^N(ay. PROPOSITION 2. Let t be an integer ^ 1. Let the forms Fj(k) (l^j^h) have the property D(k, h, t, a, y). Then if s^cl5{k, h, n, x, t), the system of congruences
FjCk) = Hj
(mod a),
l^j^h,
m SMALL SOLUTIONS OF CONGRUENCES
has a s o l u t i o n k l t . . . , ks, [ i x , . . . , \i h in P s + h 0 < m a x | | A ( M ^ ( a ) ( 1 / * + -v)(J/'" i
271
satisfying max \\^\\
and
+ 4)
.
i
Let d=c4{k, h — l,n, z), w = cl2(k, h,n, x, z), and t-dw. We proceed to show that Proposition 1 is true if F, (1 ^j^h) satisfy the condition D(k, h, t, a, y). Write c — cg, u = cl5,
and s = chu. After
a change of
notation, (4) becomes < • *
I
(«rlU)X-krl+-~+"n.U)*kJ = O
(mod 0 ), 1 < j ^ h.
(19)
r= 1
For each r, 1 ^ r < ch, consider the system of h forms gjr(*riU)*k1+
••• +«™a)A*J
(l^j^h)
in u variables. These forms satisfy the condition D(k, h, t, a, y). Note that the introduction of gJr ( = ± 1) does not affect the condition (7,,..., yh, q) = 1. We choose gjr (1 ^j^h, i^r^ch) for the case such that f}jr are all totally nonnegative in Lemma 5. It follows from Proposition 2 that the system of congruences «>(«MU) K1+ •••+ OLruU) Ku) = vrj
(mod a), 1 < y < A,
(20)
has a solution X'rl,..., X'ru, v rl ,..., vrh in pu + h satisfying 0<max||A;,.HAf(a) <1/ * + -v)(1/"), i
max \\vrJ\\
+4
\
(21)
j
By Lemma 5, the system of equations X gJrvrjlikr=0,
(22)
l^j^h
r= 1
has a solution /xr (1 ^ r ^ c * ) satisfying 0<max\\fir\\^N(a)E,
(23)
r
where £=ciO()fc, h, n) 2y{h + 4). By (18), (20)-(23), we see that kri = nrk'ri (l^r^ck, solution of (19) satisfying krieP and 0<maxN(krl)
I^r^c\
l^i^u)
is a
l
We are left with the case where the condition D(ky h, t, a, y) fails; i.e.,
312 272
WANG YUAN
there is a divisor q of a, / coefficient vectors a,..., a, (say), and an integral vector y = (y,,..., yh) such that N(q)>N(a)v,
(24)
(y,,..., y , , q ) = l
(25)
and ya, = 0
(modq), 1 < / ^ r .
(26)
We set all the variables, except the first t = dw, equal to zero, and so Fj(V=
I
+ ••• +<xmr(j)XkmX
(*uAJ)til
K
i
a
(27)
u= 1
A consequence of (26) is that the congruence y.fiW+-+y»f»W =0 (modq) (28) holds for every integer vector k. Now we show that there is a divisor q' of q such that N(q)>N(q)l/\ and some j , 1 ^j^h,
(29)
for which (y,,q') = i-
(30)
In fact, since q is square-free, we write q = pl • • • pa, where pt are distinct prime ideals. Because of (25), no pj divides every element of yu...,yh, hence
Thus N((yj,q))^N{q)l-l/h for s o m e ; in l^jKh, and (29) and (30) follow by taking q' = q/(y/, q). We may suppose without loss of generality thaty in (30) is j= 1. From (28), we deduce that if F2(X)=---=Fh(l) = 0
(modq'),
(31)
then Fl(X) = 0
(modq').
(32)
313
SMALL SOLUTIONS OF CONGRUENCES
273
For each u ^d, it follows by the definition of w = cl2(k, h, n, x, z) that the system of congruences ««i(y)^+"-+a W B .(7)A' H *. = 0
(m°d^), U
M
(33)
has a solution l'ul,..., X'un, in Pw such that
0<max7V(AL,)«^fA)
+
( 34 )
•
We denote by au(j) the expression on the left-hand side of (33). Since d=cA(k,h — l,n,z) and Theorem 1 is assumed to be true for h — 1 forms by induction, i.e., the system of congruences a,(./)Ai?+-"+arf(;)/i*s0
(mod <,'), 2 < j ^ / * ,
(35)
has a solution /<j,..., \id in P d satisfying 0<max^(/x,)«Af(q') 1/Ar + -".
(36)
I
Write kui = ixuX'ui (l^u^d, l^i^w). (31) holds. So by (32), we have
From (27) and (35), it follows that
Fl^) = F2{X)=--=Fh^) = 0
(modq')
(37)
(mod^j.
(38)
and by (33), we have
Fl(X) = F2(X)=--=Fh(k) = 0
By the combination of (37) and (38), we deduce that Ft^^FAV^
•••^Fh(X)
=0
(mod a),
since a is square-free. And by (34) and (36), we have / Ma ) \
x + :
0<maxtf(A,y)«JV(q')>/* + -- ( j ^ j «JV(a)-v + z Ar(q')- < *- 1/ * ) .
(39)
Now Af(q')^Af(q)1/A and iV(q)> AT(o)-v' from (24) and (29). Therefore the right-hand side of (39) is < N(a)x + z-lx~1/k)y/h
« N(a)x + Z~2z < N{a)x'
by (16), and Proposition 1 is proved for x'.
314 274
WANG YUAN 5. PROOF OF PROPOSITION 2
Let
T=N{a)Wk
+ y)Wn
\
Z = N(a)2yih
+4
s>t\\ +z-12*(/* + 2)). (40)
\
Let W denote the number of solutions of the congruences (mod a),
Fj(k) = nj
l ^ A ,
and HjeP(Z), l^j^h, where Fj{X) satisfying ^.,-ePiT), l^i^s, (1 < 7 < A ) are forms stated in (4). By Lemma 1, it follows that
»,.,-i«w-,,»«-{i;
l^"'^
where /? runs over a complete residue system of (ad)~ ', mod <5~'. Hence
W=N(a)-h
I
I
I
E(Z(F,{k)-nr)fi\ ft, 1 *
Ai£P(T) ft/sPiZ) 1 < I sg j 1 aS / * /i
\r=l
/
where each ^ runs over a complete residue system of (ad)~\ l^q^h. We may write W as
fr=^V(a)-" X
1
; . , £ / > ( r ) /ijeP(Z) 1 « i sS s 1 sS.; «
1
mod<5~',
VE
q|n y, fc He/*''
x(^X ( F r ( » . ) - / i r ) y r ^ where X!* denotes a sum such that each yq runs over a complete residue and satisfies ((q<5) yu ..., {qS) yh, q) = 1. system of (q<5)"', mod 5~\ l^q^h, We write #(a) A
W = X S(q), q|a
where
S(q)= Z
I
>.ieP{T) njeP(Z) K/SSJ
ISSJSS/I
I* £ y, l^?«/i
x ( i (Fr(>.)-Air)yA
(41)
315 SMALL SOLUTIONS OF CONGRUENCES
275
By Lemma 2, it follows that £
l ={-^-T"+ O(T'-1),
and so S{l)>TmZh".
(42)
Now suppose that N(q)> 1. By Lemma 1 we have X
£(-|iy) = 0,
fi (mod q)
where fi runs over a complete residue system, mod q, and where ye(qi)" 1 , 7^(5"'. By Lemma 2 we see that the number of integers fi satisfying H G P(Z) and )i = v (mod q) is equal to
y/DNM
Km*)1-1'")'
Hence if the domain \i e P(Z) is split up into a union of a complete residue system, mod q, plus a few other, remaining elements, say R elements, then
Therefore S{(\)
£ q|a i<«(q)a(«)!'
X
2
^(q)* +1/B
l
+ i)
4Ts"ZhnN(a)--\
(43)
We claim that \S(q)\4r"Zh"N(a)-1.
Y 2 1
/V(q)>/V(a) >
(44)
316 276
WANG YUAN
Otherwise there is some q satisfying q | a and N(q)>N(a)2y
such that
\S(q)\>TmZ'mN(a)-2. Then we have I
^ ( Z y / y W ) >TmN(a)-2N(q)-h
>.ieP(T) \=l 1 « i: ss i
)
f o r s o m e y w i t h ( ( q ^ ) y 1 , . . . , (qS)yh,
q ) = 1, t h a t i s ,
fl |5,|>r"W(a)-27V(q)-",
(45)
/= i
where
=
X
£(YO,A*).
;.EP(D
We may suppose without loss of generality that \SA>
•••>\ss\.
Then the left-hand side of (45) is •4Tu-l)n\S,\{s+l~').
It follows that for
i=\,...,t, \Si\^\Sl\>T"{N(a)2N{q)h)-llu
+ l
-').
Take t sufficiently large. We have by (40), (N{a)2 N(q)h)[/is+l
"]< N(aY>' + ms+l""
< N(a):"'-2k
Set C = 7"W(a)-z/'22*. We have C ^ T" 1/2*-' + --/'-2*_ Hence by Lemma 3, we have ff,6? and T,-E/ ( l < / < / ) such that ||(ya,.)(J,-T,.|| «Af(a) z/2 ' ; j - * + --/2'--
UK/.
317 SMALL SOLUTIONS OF CONGRUENCES
277
Therefore ^((ya,)^—T,)«iV(a)- I -*' + " /|2 <(7)Ar(a))- i ^(DN(q))'1,
because N{a)^cn{k,
l^i^t,
h, K, x', e). This means that (ya,)cr,= T,,
ls=/
Since ((qS) y 1; ..., (qS)yh, q)= 1, we may choose fieqS such that (/Jy 1 ,..,/?y*,q)=l. Since qK/Jya.-Jff,,
K i « ,
we have q'|/Jyo,,
1
where q' =
5 ( q , °i
. •••<*,)
Therefore we have N(q')>-~^~> N{al--al)
N(q) N(a)'"z"> N(a)2"-nz"> N(aV.
This gives a contradiction, and we have proved (44). By (41)-(44), we have W>N(ayhTsnZhn>\. Thus Proposition 2 and also Theorem 1 are proved.
6. PROOF OF THEOREM 2
With a quadratic form 2(M= we associate the bilinear form
£
ay-M./
(<Xy = a,v)
318 278
WANG YUAN
Then £>(>., H) = 0
and
Q(X,X) = Q(k).
By the remark of Lemma 2, we may take R{a) as a complete residue system, mod a, such that for any aeR{a), we have 0<||a||=$ cl6{K) N(a)l/". We assume that the coefficients of Q,-(X) (1 < / ^ / i ) belong to R{a). Set u = c 4 (2, h, n, e/2), v = cl7(h, n, e), and 5 = uv. Let
(l-I)D
(M-l)t'
Now we define ^.l,..., fiu by induction such that (1) (2) (3)
Hy (l^i^u, 1 ^ 7 < u ) are integers in / , ll^-ll <^V(a)e/2" (1 < i s s « , 1 < y < p ) , <5y(|lr,|l,) = 0 ( 1 <_/<*, l < r < ^ < « ) .
We define H! by /i,, = ••• = ^ i u = 1, which satisfies conditions (1), (2), and (3). Suppose now that (i t ,..., | i , _ i have been defined. Consider the h(q- 1)(/-] +r2) linear forms
Consider the points | i e ? " with components [i,eP(N(aY/2"/2), l^i^v. Then by Lemma 2 it follows that there are clg(K) Nvc/2 such points. Given such a ji, the point
x(fi)=(e(1i)(m,n),-,eiri)(^-.^),e
. . . , / A ( , _ i ) r i , g i , - , g/,(<,-i)r2). where
-c 1 6 A^(a) 1 / " + e / 2 " t ; < / > < c 1 6 A r ( a ) 1 / " + £/2"f, gj = Xj + iyj, l/
\Xj\^cl6N(a) "
1 <j
+ e/2
"v,
\yJ\^c16N{a)1"' + '/2"v,
l^j^h(q-l)
r2.
Let t be the rational integer in
(clsN(ar/2)m"-l)"-l^t<(clsN(a)^2)l/"("-l)" and divide each interval of /} in D into / subintervals of length
319
SMALL SOLUTIONS OF CONGRUENCES
2cl6N(a)l/" 2cl6N(a)1/n
279
+ F /2n
-
+ e/2
vt~l and each square gj in D into t2 subsquares of side "vt-\ Since th("-l)n
it follows by the Box principle that there are two distinct points x(n') and X(n") such that Qjl)(]ir, \i') and Qj'\pr, V") will lie in the same subinterval for each/, r, I and Qjm\\ir, |i') and 2j m) (fi r , Ji") in the same subsquare for each j , r, m. Set
m, =»»' - n" Then O
Kj^A, U r < ? ,
if we take u = c17 sufficiently large. Hence \N(Qj(nr, p,))| < 1,
l
Since g7-(|ir, n , ) e / , we have ^(H r ,|i,) = 0,
U;<*,Ur
So \iq is well defined, and thus we obtain |ij,..., ]iu by induction. Set k = Vl]ll+
••• + V u J l M .
Then
= I QA^,^)^,
i
By Theorem 1 it follows that the diagonal congruences
£ &(l»,.|i,K = 0
(mod a),
1^7
320
280
WANG YUAN
has a solution v in P" satisfying 0 < max N(v,) < N(a)l/2
+ E/2
,
and therefore the system of quadratic congruences
Qj(X) = 0 has a solution 'k = vlfil+ 0 < m a x \N{Xi)\=max
(mod a), 1 ^j^h,
••• + vu\iu satisfying |Ar(v,Ai,,)| «7V(a) 1 / 2 + £/2 + £ / 2 « N ( a ) ] / 2
+ i:
.
Theorem 2 is proved
ACKNOWLEDGMENTS
I am grateful to the referee for his useful suggestions and help, and to the NSF of the People's Republic of China and a Glorious Sun Fellowship through the Committee for Educational Exchange with China for financial support.
REFERENCES 1. R. C. BAKER, Small solutions of quadratic and quartic congruences, Mathemalika 27 (1980), 30-45. 2. R. C. BAKER, "Diophantine Inequalities," Oxford Sci. Pub., Oxford, 1986. 3. T. COCHRANE, Small solutions of congruences over algebraic number fields, Illinois J. Math. 31 (1987), 618-625. 4. E. HECKE, "Lectures on Theory of Algebraic Numbers," Springer-Verlag, New York/Berlin, 1980. 5. A. SCHINZEL, H.-P. SCHLICKEWEI, AND W. M. SCHMIDT, Small solutions of quadratic
congruences and small fractional parts of quadratic forms, Ada Arith. 37 (1980), 241-248. 6. W. M. SCHMIDT, Small zeros of additive forms in many variables, Trans. Amur. Math. Soc. 248 (1979), 121-133. 7. WANG YUAN, Bounds for solutions of additive equations in an algebraic number field, I, Ada Arith. 48 (1987), 21^18. 8. WANG YUAN, "Diophantine Equations and Inequalities in Algebraic Number Fields," Springer-Verlag, New York/Berlin, 1992.
321
wB^\
Acta Mathematics Sinica, New Series 1993, Vol.9, No.4, pp.382-389
>*^
^ ^ ^ ^ ^ ^
ii
On Small Zeros of Quadratic Forms over Finite Fields (II) Wang Yuan C£ 7C) Institute of Mathematics, Academia Sinica, Beijing, China Received October 24, 1991
Abstract. Let Q(x) — Q(xi,'-- ,xn) be a quadratic form with integer coefficients and let p denote a prime. CochraneM proved that if n > 4 then Q(x) = O(modp) has a solution x ^ 0 satisfying |x| <| ^/p, where |x| = max |x,|. The aim of the present paper is to generalize the above result to finite fields.
§1. Introduction Let 1F(= Fq) be a finite field of order q = pm, where p is an odd prime. Let 7Z9 = Z/pZZ = JO, ±1, • • • ,±^—j— j- be a complete residue system modulo p. Let a»i, • • • ,wm be a basis of P . The elements of 2F can be represented by z = a1w1 + --- + a m w m ,
ajG^p,
1 < i < m.
Set |x| = max|o,-|.
(1)
If x = (xi, • • •, xn) is a vector with components in W , then we define \g\ = ma.x\xi\.
(2)
Note that |x| and |x| depend on the basis of IF . A quadratic form over JF is a polynomial in W[x\ of the type,
where qij = qji,g— (xi, • • •, xn) and Q = (?ij)(l < i,j < n) is the matrix associated with Q(l).
322 Wang Yuan
On Small Zeros of Quadratic Forms over Finite Fields (II)
383
For the Case F = 2Zp, Heath-Brown!2' first established by his celebrated method based on Fourier analysis that if n is an even integer > 4, then the congruence Q(x) = O(modp) has a solution satisfying 0 < \g\ < Vplogp. (3) The right hand side of (3) is of best possible apart from a possible improvement on the second order log p. CochraneM proved that
III < VP
(4)
under the same condition by the use of finite Fourier analysis. Wang Yuan'3' has generalized Heath-Brown's result to all finite fields with the norms defined by (1) and (2). In this note we shall show that (4) is also true for all finite fields. Theorem 1. Ifnis an even integer > 4, then Q{g) with coefficients in IF has a zero x in IF" satisfying
0 < \g\ < VP, where the constant implicit in
det
,ep(x) = e^1!", V the set of zeros of Q(g) in Fn, E £ a sum
®)
where x runs over Fn , and Q*{x) the quadratic form associated with matrix Q" 1 , Let I be the trace from IF to ZZp and
'(T)=£^C»(5)A region of points S in JRm" is said to be star-shaped about the origin O if for any point P in S the line segment joining P and O is also contained in S. Lemma 1 (Cochrane [1]). Let S be a star-shaped region about O with y < p/2 for all g£ S. For 0 < r < 1, let rS = {rg\g G 5}. Let V be the set of zeros modulo p of any given form in x . Then
\rsnv\
=
y
I — q?
X
A,
otherwise
(5)
and
'£l = \V\ = qn-1+qS-l(q-l)A. V
(6)
323 Acta Mathematica Sirica, New Series
384
Vol. 9 No. 4
Proof. Suppose that y ^ g. Then
^ e p ( I ( i p = g-1A, v
where
A = E e p( I (i£'))E e p(^(i))Let Q = TDT1, where D = [rfi, • • •, dn] is a diagonal matrix. Let £T = g and yT1"1' = MSince
we have
•EF'fel \zief
/
-5n*«-i))E^*o>('+(=))
•tjH«(3))£Mi) = g»/2A X ) ep(I(-(4a)-1Q*(E)))
and thus (5) follows. The proof of (6) is similar. Let a(^) be a complex function of F " with Fourier expansion
Then
X) «(i)eP(-Z(£ £')) = E E «g)epV(£(i ~ E'))) = 9"c(E). £
£
£
Lemma 3 (Basic identity). If A / 0, then
E a (i) = ?" 1 E a d)- A 9 f " la (S)+ A 9 f E c(^)V
£
2
Q'(»)=o
Proof. By Lemma 2 and (7) we have
E a d ) = EEc(M)epWig'))
(7)
324 Wang Yuan
On Small Zeros of Quadratic Forms over Finite Fields (II)
V
=
3SS
V
y^O 1
c(g)(q"- +~g*-»(g - 1)A)+ ,!-»(«, - 1) A
£
c(g)
Q*(£)=0
£
£
*
The lemma is proved Let £ be a box of points in Wn of the type B= lg\geFn,
Xi = ^
>
I
XijUj, a y < x{j < aij + my, 1 < » < n, 1 < j < m I , (8)
=1
J
where aij,niij G .ZJ and 0 < ay + my < p , characteristic function of B and
1 < i < n,
1 < J < m. Let XB(S) be the
i m
Set j/j = ^^y>juj> i=i
where {u{, • • • ,u>'m} is the dual basis of {wi, • • • ,u>m} i.e;
I
^i)=lo>
otherwise.
Then / n
i(s i')
\
n
m m
= i ( £ *'» 1 = E E E *«W*2:(WJW») \i=i
z
/
= EE ^ n
m
i=i
;=i
i=i
<=ij=i
and by (7), we have
cafe) =r"Ewd)«pK(ip £ n
m
1\
\
S m
—L
--»nn^(-(..v+^-0^)^fe/
/
Let a(x) be the convolution of XB(*) w i ' n itself, i.e; a(i) = XB*XB = ^2 XB(g)XB(g - g)-
(9)
325
386
Acta Mathematics Sinica, New Series
Vol. 9 No. 4
Then the Fourier coefficient c(j/) of a(g) is
£
£
£
2i £
= 9- £EEE^(s) e p( I (Ms'))^(M) e p( 2: (i(i-M)'))ep(-i^M')) li
(10)
= EEcB(£)Vk(£-g)') = ^te) 2 Lemma 4. Suppose that A = — 1. Let f? be a box of tie type B = {gig £ F " , |*y | < Bij, 1 < * < n, 1 < ; < m}
(11)
for some honnegative integers Bij. Let t be a positive integer satisfying (tm + 2)log2 < ( § - l ) log 9 . If Bo- < 2-(n+«+3)mp for all i,j or \B\ > 2-(n+t+2)nm'q*, then \B n V\ < 2(n+i+3>m3+1q-1\B\
+
2-tmq$-1.
Proof. Take a^- = — Bij and mjj = 2By + 1. It follows from (9) and (10) that cs(y) is real and c(y) is positive. Since
£.
H.
£
and «(S) = £XB(M)XB(-M) = |B|, we derive from Lemma 3 that
£ > ( i ) < , r W + ?*-1|siOn the other hand we have
£<*(£)> S 2-nm|B| = 2-"m|5||BnF| £€V
£eBnV
and |J5nV|<2nm(g-1|B| + 9'-1). Set r = 2-("+<+2)"\ Suppose first that |B| > r n m 9 * . Then 9f-i<2(»+'+
and so \BHV\
2 nm3
)
,-1|B|,
< 2"^q-1lB\ + 2nm+(n+'+2')"m:'q-1[B\ < 2(n+t+3)oma + l g - l | f l |
(12)
326 Wang Yuan
On Small Zeros of Quadratic Forms over Finite Fields (II)
387
The lemma follows. Suppose now that \B\ < rnmq% and 2?y < yrp for all i,j. Set
r~lB = {g\g £ F " , \Xij| < r-lBij, 1 < « < n, 1 < j < m}. Then for any g € r - 1 5 we have \g\ < p/2 and thus by (12), | r - x 5 n V| < 2 n m ( r - " m g - 1 | B | + «*-*) < 2 " m + 1 j * - 1 We obtain by applying Lemma 1 to the region r~lB that iy—tm—1
\B n V| < 1 + T-— 2 n m + 1 ? ? - 1 < 1 +
q*-1 <
2-tmq'-1
since (<m + 2) log 2 < f ^ — 1J log q. The lemma is proved. §3. Proof of Theorem 1 If A = 0 or A = 1, then the theorem is proved in Wang'3'. So we assume that A = — 1. Theorem 1 is a consequence of the following Lemma 5. Suppose that p > 12 2nm • 2 1 O n m + 1 2 m and that B is a box of the type (8) satisfying m<j > 2 ( 5 n + 7 ) m 1 2 n m + 1 and \B\ > 2<-3n+Vnm2+2l2nm+1qn/2. (13) Then B contains a nonzero solution ofQ(x) = 0. Proposition 1. Let S = B be a box of the type (8) and T be any translation of S. Suppose that p > l 2 2 " m 2 1 0 n m + 1 2 m , my > 2 ( 5 n + 6 ) m 1 2 n m + 1
(14)
2(3n+3)nm 3 +222 nm + 1 gf < \g\ < 2 ( 3 n + 3 ) n m J + 3 1 2 n m + 1 g *
(15)
and
Then the set S + T contains a nontrivial zero ofQ(g). We first derive Lemma 5 from Proposition 1 and then prove Proposition 1. Proof of Lemma 5. Let
5 = U\g €Fn,0< xtj < [m''2+ *] , 1 < i < n, 1 < j < m\ and T = S + g with g = (an, • •• , a n m ) . Then \S\ > 2-nm\B\ > 2(3n+V"m*+212nm+1qn'2 and each side of 5 has length exceeding 2( 5 n + 6 ) m 12 n m + 1 . We may assume that S satisfies (14) and (15), since we may shorten the sides of S if | 5 | is large. By our assumption
p > 122»"»210nm+12m
we
obta
jn
fn(5n+6)m-innm + 1\nm
<- o( 3n + 3 )" m + 2 12 nm+1 n' !! 3 !l
Hence there exists a subset of S containing a cube with edge length 2( 5 n + 6 ) m 12 n m + 1 and satisfying (14) and (15). Since 0 < zy < f ^ i ^ J , we have ay < 2xy + ay < ay +
327 388
Vol. 9 No. 4
Acta Mathematics Sinica, New Series
2
( m ' J 2 + 1 ) ~ 2 ^ aH •+ m 'J ~ 1» 1 < » ' < " , ! < i < m - Therefore 5 + T is a subset of B and S + T contains a nontrivial zero of Q{x) by Proposition 1. The lemma is proved. Proof of Proposition 1. Let xs and XT be the characteristic functions of 5 and T with Fourier coefficients cs(y) and cr(y) respectively. Let a(g) = xs *XT be the convolution of Xs and XT • Then similar to (9) and (10) we derive that the Fourier coefficient c(y) of a(g) -atisfies n
m
I
i
m
\
Wi^r-IHI"-\ "vh • Let 1 < j < n and 1 < / < m and let TT and r be injections of {1, • • • ,j} into {1, • • •, n} and {1 , • • • , / } into {1, • • •, m} respectively. Let k\, • • •, kj, £i, • • •, d be nonnegative integers and
B' |yu«( If
={g\yeFnAy«i)T(,)\<m£^,i
i<«
(16)
otherwise j .
m l > ' ( 0 < 2-( 3fl+3 ) m p, 1 < » < j , 1 < « < / and 2 ^ — < 2-( 3 "+ 3 ) m p, then Lemma 4
applies to B' with t = In . Otherwise 2ki+(' > 2-( 3n+3 ) m m T ( j)r (,) ) for some i,s, then
i^i ^ i=i,=i n n ^m*(«x») — L n...
2i>+c> dr^MHTIIII «« 2 i 5 i,- = i,_i
2m
otherwise
v,
P
3 ;< 3n 3 m:
nm
— 2 . 2( " + + ) 'i2" _2-( 3 n+2)nm 3 9 f )
m+1 i a
p^
2~(3"+3)mo(5n+6)">12nm+l
and so Lemma 4 again applies to B' with < = 2n . Let V* be the set of zeros of Q*(x) in Fn . Then by Lemma 4 | B ' n y | < c i g - 1 | 5 / | + c29?-1, where c t = 2(3"+3).?m:>+1 and c2 = 2~ 2 n m . Set 2 J = 2_J = \ J *. Let 7T and r run through the sets of all injections of {1, • • •, j} Q'G)=0
into {1, • • •, n} and {1, • • •, 1} into {1, • • •, m} respectively such that
TT(/»)
< ;r(») and r(h) <
r(i) for h < i. Then n
m
rKE)i= E E E E i=0l=0 n
r
m
*
£
,,,t(.)r l»««li5^T oo
< ^ZZZZZ-Z i=0 1=0 i
r t,=o
K_>I
_j__ otherwise oo
W l i
0=0
2kt+i'v
•
E Sl9
1 l
2* i+C> + 1 p
2m, (i ) r J ) "-)" » ^m l(l>( ,)
328 Wang Yuan
On Small Zeros of Quadratic Forms over Finite Fields (II)
TT A m ' ( ' X » ) 1 1 1 1 o2(fci+<,) 1 = 1 »= 1 ^
TT 11
389
m m2
««
••• .
otherwise j=0 (=0 T
n^i n
T
fc1=O
(,=0i=l»=l *
i=l
n -*-+«•-)
m
otherwise
o o o o /
1
1
i
+
'
1
= «-"W'EEEEE-Eh«"- iB|- 2™- "nn?i?7 i = o i=o »
T jfc1=o
i=i«=i^
/
(,=o \
t=i »=i
= c12*-g-Maif:(?)a'f:(7)2'+cM-t-wE(;) j=o
w
y
/=o
( |)
v
F7\ n m + 1
'
i=o
Vi/
'
r'-'IBI2-
Thus by Lemma 3 we obtain Y^<*(g) > 1~X\B\2 - 2( 3 "+ 3 )" m2 + 1 12" m + 1 g t-i| J B| - 4 ( ^ j ) " m + 1 q~x\B\2 v >0.6g-1|5|2-2(3n+3>nm:>+112"m+19t-1|B| = g- 1 !^! (o.6|S| - 2( 3 n + 3 )" m 3 + 1 12 n m + 1 gt) > 1. Therefore (5 + T) (~\ V contains a nonzero solution of Q(&) = 0. The proposition is proved. Acknowlegment. This work was supported by the NSF of P.R. China and by a Glorious Sun Fellowship through the •committee for Educational Exchange with China for financial Support.
References [1] Todd Cochrane, Small zeros of quadratic forms modulo p, HI, J. of Number Theory, 37 (1991), 92-99. [2] Heath-Brown, D.R., Small solutions of quadratic congruences, Glasgov Math. J., 27 (1985), 87-93. [3] Wang Yuan, On small zeros of quadratic forms over finite fields, J. of Number Theory, 31 (1989), 272-284.
329
Q) cien ce cK ecord New
Str. Vol. 4, No. 1, I960
Mathematics
Remarks Concerning Numerical Integration* HUA LOO-KENG** ($
% JO AND WANG YUAN ( I
%)
Institute of Mathematics, Academia Sinica
Let f(xly • • •, xs) be a function of .r1? • • *, .r, with period 1; it has a Fourier expansion /(*„•••
r
*,) =
2
•••
S
C ( W l , • • • , 7 » , V " < » " "+ -+»-'->>
(1)
• • • , * / ) < . - 2 ' « - i ' i + " - + - A ) ^ 1 - • -dxs,
(2)
where C(.ml}
• • • , m s)
=
I"' . . . ( " ' f(Xl, Jo Jo
For a given set of natural numbers q and a; (1 ^ a: < q) J — 1, 2, •••,.?, we have
£ / ( ¥ - • • • ' ? ) = * s ••• i
c(»,,...,»,).
fl1rni+—+aj.mJ.=0(nio
Therefore, we have Jo =
Jo
<7 , = ,
2 C(w»i, • • • , m,) *imi+—+asms = U(mod q)
=
\<7
R,
?/l . , \3)
where 2" denotes the omitting of the term with mx = m2 = — = m, = 0. We assume that K M +l)---(l«.| +1)]- ' * Received Oct. 27, 1959. ** Member of Academia Sinica.
^
330
9 where « > 1 and C > 0 are constants.
*< <
2'
It can be easily shown that
-
.
£' +o(-\ V w ... +a ir SU(mod ,j(i-.i + !)•••( w + DJ-
The last sum is equivalent to c/~J°.Q with
(5)
where <(f/ denotes the distance from f to its nearest integer. It was proved by Kopo6oBIIJ that for q being a prime p, there exist integers aiy • • •, as such that R = O(12£L£)
(6)
The constant implied by O in his paper can all be made explicit. He also pointed out the counter part that there exist functions satisfying (1) and (4) with (7)
R>£for any choice of au • • •, a., and q.
Therefore the explicit choice of ax, • • • , a,, q seems to be the central problem in the numerical integration. We prove Theorem 1.
For ax = 1, a2 = pn = \ [ (1 +• V T ) " + (1 - v ' T ) " ]
and q = q* = ^ 7 ^ [ (1 + v ' T ) " - (1 - v ^ T ) " ] , we have
/? = 0(i5i_2-Y Proof.
(8)
It is sufficient to prove that Q=2
2
TT-V1
^v"
=
O(?
"log
331 10
We divide Q into n subsums: 1
=
l
y
.
,„ = 2 . • • • , « ;
however, for the last one, /„, we replace q,, by '-y- • Let y be the integer satisfying |v _ £ • * ! < . I .
(9)
The system of equations
y
=
pm« + 2^mr^
(10)
[
has integral solution in u and v, since its determinant -pi, + 2^2ro = (— 1 )*" + 1. In case qm_x < .v < qm, we have «i/ < 0.
(11)
From pn + V 2 qn = (1 + v 7 2 )", we deduce 9VV — pnX = (qnpm
— pnq,n)u
— Cpnprn ~~
= ( - l ) - ( - pn-mu +
2.qnqrn)v
qn-mv).
Then, we have, by (9) and (11), < | -pn-m't + qn-m"\ < ~ q«.
\qa-mv\
(12)
To each x satisfying qm-\ ^ x < qm, we have a unique v satisfying (9) and (10). Therefore JM < ?»
Z
-.-;
<-2=_ V
-
= o( x / " \
)=
qm—\qn-m'
rr;
O(q'\
Consequently, we have Q = O(nq°) = O(qajoxqn) ;
332
11 the theorem is now proved. Remari^. The constant 1 + v' 2 is used merely for the sake of simplicity. In fact, the unit of any real quadratic field can be used instead. For example, the field R(\/ 5) suggests the use of Fibonacci numbers as well. It suggests also the possibility to treat the high dimensional case by means of the units of a totally real field of degree s. For the counter part, we have the following improvement of (7) Theorem 2. There exist functions satisfying (1) and (4); for any q and aly • • •, a, we have R> ^ M , q
(6')
where C' > 0 is a constant. Therefore, Theorem 1 gives a best possible choice for s = 2. Besides, under the condition that (4')
dx[l • • • dxs
for a fixed r > s, we have Theorem 3. Let q be a positive integer and px < • • • < p, be s pairwise prime integers lying between qv' and 1!qVs. Then we have the following formula for approximating the s-ple integral by a single integral
where at = px • • • pjp-,. Consequently, we have Theorem 4. Under the same assumption of Theorem 3, we have
ir.-r.^.-.*)*.-*-ii:i/(f.-.^)i-o(^ REFERENCE
[ l ]
KopodoB, H. M. 1959 fiAH OGGP, 124(6), 1207—1210-
333
SCIENTIA SINICA Vol. X, No. 6, 1961
MATHEMATICS
A NOTE ON INTERPOLATION OF A CERTAIN CLASS OF FUNCTIONS* WANG YUAN (31 (Institute
5£)
of Mathematics, Academia
Sinica)
Let E" be the class of functions tixi, • • •, xj = 2 •_;• 2
C(«i, • • •, «, V
^ V ^ - A ) ,
(1)
in which the Fourier coefficients satisfy \C(m1,---,m1)\
1
<
(2)
_
Km!- • • T W J
where m = max (1, |ra| ) and a > l is an absolute constant. Let N be an odd prime number and A/] be an integer satisfying
C(mi, • • - , » , ) ~ £ Z / ( - ^ , • • ', ^ ) ^ " ' ^
*>
S 1 -H,
(3) ^^^
and A=minsup
•••
\f(Xi,
• • •, * , ) — P(xu
• • •, * , ) \2dxl}
• • • ,dxs,
(5)
where au • • •, a, are integers. PfldeHhKHH111 first proved that A < Aa(Nf+1N-2'ln2"+J-lN
+ 2VT»—"/n'-Wi)0
(Cf. also [ 2 ] ) . * Received June 5, 1961. 1) In this paper, A* and B» denote positive constants, depending on a and / only.
(6>
334
633 The aim of the present note is to give the following improvement: Theorem 1. A1(N?N-*'+NrOtt-v'ln-1Ni)
< A < A2(N?N-2°ln2aU-l)N + N^2a~olns-lNd.
(7)
Evidently, the theorem given above does not permit further essential improvements. There follows immediately 2a
(2Q-1KJ-11
Theorem 2. Let N\ = [N4a-' In 4a-] N] . Then there exist integers ai = a;(N) ( K z ' O ) such that the inequality P . - - T |K*,, • • • , * , ) - P ( * i , •••,*,)|2<**1 •••<**,<
Jo
Jo
4°2 f r _ , ?
2°(2°-O
* - 1 /•;;•—L
iV
(8)
holds for any /(.r,, • • •, xs) e E" .
Proof of Theorem 1. 1) The right-hand side of the inequality (7) may be deduced from the following two Lemmas. Lemma 1. (BaxBajW31) There exist a; = a;(N), (Ki<s), such that 2---Z'
-
c ,l 1 +-..+a J / / =»(D 1 odN)
where
2 ' denotes
the omitting
( 1
'' '
lsJ
of the term with
L e m m a 2 . Let ls, • • • , 1 , be non-negative l
—11^1-
(9)
/ , = ••• = / , = 0. integers
and
\
Then l
2
"'*
.
(10)
Proof. For ^ = 1, we have
Assume that the lemma holds for s4,{. 2 m ; >0
_ _ -
X
Then < I i + S2 + • • • + ^*+i,
335
634
where *'
(i)
=
—zr=
Za
Suppose N, < ^ ~ ^ - .
£i<
1
2
Then
2
Suppose N, > ^ ~ ^
£t<
l
2
+
2
^i_<
<(3f(a))* +
2
< 3-+*f*(«)
x
2
1
+ Btivr
2
3
*^ + 2'BkNtga) < h'-'U+Jt (/i • • • / t + i ) " " "
< (3-+*+"*C*C«) + 2"S4C(«))
Hence Sl
-+
% [(/2 - « , ) • • -c/t+l - « t+1 )]"
i 1
1
- =
Oi - my
2 h -?^+i
. Then
-
2
<
^ i [ Oi - mi) " • • ('t+i ~ /»*+i) ]°
-==roi4
1
- =
a<mi
(ii)
-—.
^L
.
c?i • • • h+ly
< ( JS±L)
—^r—.
H +1/ a • • • h+ly
<
336
^ ^ ^ ^ ^
635
Similarly, we have H + v (h • • • h+ly Therefore ZJ
— = m
^ #*+i —
—
:——.
my>0
Hence the lemma follows by mathematical induction. 2) Take /Oi, • • •, -v.) = 2 - : - 2 .-
-,.^'>-+-+'"^.
1
(»r"»»,) Then we have, for any «i, • • •, «J3 Jo
Jo
-
2 ( 2- : -2' - = = — * Sir-55j>:Vl ^ " ' l ' ' *
> 2
" I " ! < A'l
7W
.Y +
^-)
l
2'
—
2
«i/i+«2/2 S ofmod A') [ ( / t — Wi)(/ 2 — m 2 ) ] °
Suppose first t h a t
(«,-, N ) = l (/ = 1,2).
+AiNri2a-1)l»'-1Nl.
L e t the solution of the
congruence at ^ = — 0 2 (mod N) be r = «( \a\
4n
M
< qrJrX < •••
Since for any integer b, \pnqt— q»b\ <-22-,
(0<^
therefore Jo
Jo
> 2
2
x
——+/* 5 2vr (2 - 1) /«'- 1 tt 1 >
337 636 >
2
2
* Uq, -mi)
+ AiNT^ln^Nr > m
p-
~ A\
> A2(N\aN-2a + iVf <*-1>/»'-1N,). Evidently, this is true also for the case ( a , , i V ) > l or (a 2 , A ' r ) > l . Hence we have the left-hand side of the inequality ( 7 ) . Similarly, we have a
Theorem 3. Let Nt= [N2"'' In a;(N) (Ki<s) such that /(*],
' ' ' , *s)
holds for all f(xu Kemar\.
2a l
~
N]. Then there exist a,—
~
/«
<^6N
U-lUl—a)
iv
(11)
• •, x,) e E" .
This method may be used
LUaxoBw (i.e., the error term 0 (N-^r1*')
to improve
t h e theorem
m a y be repaced by O
of
(ATT+£)).
REFERENCES
[ l ] [2]
PaSeHbKHH, B. C. I960 O Ta6jiHuax H HHTepnojiHunn (JjyHKUHft H3 HeKOToporo Kjiacca, flAH GGGP, 131, NQ 5, 1025—1027. CMOJIHK, C..A. i960 HHTepnojiHUHOHHbie n KBaflpaTypHbie (^opMyjiw Ha KJiaccax w° n E", RAH CGGP, 131, Ns 5, 1028—1031.
[ 3 ]
BaXBaJIOB, H . C . 1 9 5 9 O IipHgjIHJKeHHOM BblWHCJieHHH KpaTHblX
[ 4 ]
pajioB, BeemuuK FMY, 4 , 3 — 1 8 . LLIaxOB, K3. H . 1961 O npudjiHJKeHHOM peiueHHH ypaBHeHHH BojibTeppa II pofla MeTOAOM HTepaunH, fl,AH GGGP, 136, Mb 6, 1 3 0 2 — 1 3 0 5 .
HHTer-
338 Vol. XIII, No. 6
SCIENTIA SINICA (NOTES)
1007—8 E,,-,E,
MATHPMATICS
On Diophanfcine Approximations and Numerical Integrations ( 1 )
be a set of independent units, which are the solution of Pcll s
'
ec lmtion
l
x ! -p. Let / ( x , , • • - , x J )
be a function
with absolute
• • • s>. j
convergent Fourier series
= ± 4
( x > 0, y > 0 )
v V ---p.y a
/(•t,.-,»<) = S - S C ( » . 1 ) - , » . ; ) ) ( y
2
'
e!»l(«I,I,+
...+«>(IJ^
with least y H —j—"^-^> where k~^\ and l < / , < . • - < « > < < is any choice of l , 2 , . . . , r . \Y/£
may t a
|;
e
where
|CK
m = m a x ( l , ( m l ) , and a > l is a constant. T h e class of the functions with this property is denoted by £ ; . For a = 2 and for a set of integers
we have
"i, •••,"!,
| x;
•,
-'B')l<(»,...»,r
sup I - • • f / ( x , , . . . . . v j r f . v , . . . r f x , Jo
£
' ~ 1
~ ^
=.4i + n( 1 -2{fH}v + ^L w=i 12J/ ^—1 , + 22 TT (1 ~ ? | a " | ) 1 — 1, if 2\g. The main problem in the study of numerical integrations is to find a set. of integers q;au •••,aJ such that the error H(g\al3 •••ta1~) is comparatively small. Kopo6oBcn proved that for q = p being a prime, there exists an integer a such that
'
+
( 1 ^ i sg m~);
(m + 1 ^ i ^ 2 ' — 1 ) .
vcli y;
W e take positive
integers
/ any i ^ ;, we have
n., • * * , " /
"
"
such that for
'
„ „ <£,'
C l e f .'
where c, and cz are two positive constants. i the expression
Consider
+?i,"f,Vrf n , +1 + ...+?«7j 1 v/
(1)
where H0?i a,, • • . , » , ) = q
x; = >•; = 1 (mod 2 ) x
/ < E £ jlJo
— _ 5 ] / ( °' •-•,"J ) < 1 t -1 ^ 1 1 l\ *•"• < ( ^ ) ' /•/(?;«„ • . . , * , ) .
Vd; y;
_ JT + - 2 ^ '
and 7} — 1 transformations:
(CT,V..,-P V f t , - ]
u
,_
-
a
' -
i
'
(2)
( I < w
Applying (2) to Eq. (1), we have 2' — 1 equations. Combining them with (1), we have a system of 2' equations, of which we have the solutions: 2^"' + ?!"' + • • • + + '/'„' = «"' " • E 2 /_,'/ 2 '~" + O ( | e , | - * i ) , qf = e"' • •• s " * ^ 1 / 2 ' - ' A/^7 + O(|fi,|- n i) (1 sg i < «),
HCPI i. -. • • •. -*•') = H(P, -) « - ^ 2 - . For a given p, in order to find the integer a such that H(p, a) takes its minimum, we have to t r>, \\. i • , JJ perform O(p 2 ) elementary operations (add., sub., mult., and div.)- It was improved later to O ( * V ) . The aim of this note and the succeeding one is to give two direct methods. From the numerical view of consideration, the results are as sharp as those obtained by the method of KoporjOB. (They have the same significant figures.)" 1 Let p,, •••,pt be / distinct primes. We consider the algebraic field R(v^pj, •••,Vpt~). Let
,}-»= „;.... £ - y
2
' vQ? + oci«,|-o
0" + 1 < ' < 2 ' — 1). (3) „ _ , , , , , „ . ,. n> « 2 (^S"', ••-, ?L ). we define ? = ?<"> + i '"' ' »»«-•) > T C ? l " + " - + « t f ) . «. = 1, «/+,=«i«}«>/2(niod«) (1 < , < » » ) , a,-+l = d{q\»\mo& q) ( O T + 1 ^ I < 2 ' - 1 ) . For 2|(«5"', ••-, jJfOt w e define ? = 24}"'+ (^<"> + . . . + ,j{£'), a, = 1, «,•+, = d;«'."' (mod a ) (1 < i < m), ai+l = Ufqf (mod q~) (m + 1 < i < 2' — 1). From (3) we have the simultaneous diophantine approximations: F
339 Vol. XIII, No. 6
SCIENTIA SINICA (NOTES)
We take these ?; a,, • • -, «,' to evaluate the value of H(q;a>t •••,ali).
~ (2q+iy
Example X. The field R ( v T ) has s = l + i / T as the fundamental unit.
where KE',^,
Let e" = q'a"> + Vlq["\
i
1 +
2
/ 5
,
_ _ We take R ( \ / 2 , \ / 5 ) and
(
( 2 «(?{"'; 1. 2?!"') « - ^ ^ - (rf- [3])-
*') = 2 Vv.l.iZ',
and £ „ • • • , 6,,^ is a set of independent units of R(^pT. • • • . V p 7 ) ( c f . [ 4 ] ) . Hua Loo Keng <&&&)
£2 = 1 + V^T,
8j
= 3 + V'lO.
prOfn SS
I is the least integer > a, p,,,;,,- is
defined by
Then
Example 2. _
f*?.<>> KEi>. • • •> e2<_, O>
.~^
The effectiveness of the present
method is illustrated by the following examples:
8, =
1009
, e5 ^ - (9 + 4 • T ) ( I 7 + 12 v^ 2 )(19 + 6 ^ 1 0 )
Wang Yuan (2E
= 5787 + 4092 • / 2 + 2588 V 5 + 1830 v'lO,
Academia Sinica; April 14, 1964
we have <7 = 5787, fl, = 1,
a, = 2397,
a, = 1366, a, = 939.
References f }
^
By numerical calculations, we have «CS787, 1, 2397, 1366, 9 3 9 ) ^ 0 . 0 0 1 4 9 8 . Remark. Here we suggest the following method to calculate numerically the integration of 2' - 1dimensional space
ft)
University of Science mid Technology of China; /w|ftj#te o / A,a(Aema/ic5,
Kopo6oB>
H
M
1 % 0
^AH
cccp>
132
( 5 )_
1009—1012. [ 2 ]
[3] [4]
^ ™ K O B , A. H
1963 Xyp.
BUH. Mar. u
Mar.
340 Vol. XIII, No. 6
SCIENT1A SINICA (NOTES)
1009
^
MATHEMAT,CS
On Diophantine Approximations and
We use the same notations as those in the previous note. Let p be a prime >: 5 and r = J—_—.. \
following r + 1 units:
,
o g
| | | | V
I^FO-
. a\'+» log - ^ p r - . •••• t h e s y s t e m o f equations „ „
Hence
2* \
For the algebraic field Klcos
...
det
Numerical Integrations ( I I )
,
^
lo
I sj'Jj" I s|-7>!r / „
„
' * ' • ' ' • ' • ' « f " l = l « i " • ••••
) , we take the
p /
.,
£
"> I
..
{o ^ l ^. 7 -r- 1^
2cos^L, 2cos±L, . . - , 2cos <> > . (1) P P P They are arranged according to their absolute values as follows:
. has a unique nonzero solution /«, .. « M \ W ' ~^~/' From det A Js; 0, we may assume that a, a
For l ^ w ^ r + 1, we introduce the transformation _ ,
Take positive integer n, to be sufficiently large and the system of integers «,, - • • , « r _ . such that
2
+ 1
|el»l>l«i"l>->l«J¥,].
|6i
2ccs^i->2cOsi^(l
1
' - • ^"K
-
( O
itself. W e use the notation (O .».-..{.« 0 < ^ < r + l ) .
and
!!L_2i|
(I
^ I < 1
< 2 < i < , + l>.
Therefore we have
/;•:•;•
•-:!;: \
^= i 1, e S r + " ,
/2- E
w . - . .^i-o(,.^..... ^ v s '
(2
— , t'r'+" '
2-«}l'..
- •
T h e system of equations
2-.«t»
pi
\
.,-, . .
/'
««'>"'= 4«->(2-e
+ 2
<•"'(£}'•' - ^ i O / P
gives
We have the following properties:
_
?{"' = « » " • .
(0
S* «{."=-'•
(ii)
2 «j!» = (-l)H[y].
(iii)
"=1 , ldetA| = P ? ,
(iv)
A S = /.
„ ••••
e<
r" ' +
+o(|.!»"\ . . . . .j>'"'f').
''
'
1
+OCIS!"*1, ••-, a'""'!"') ( 1 < « < O - (2) Let «-W"l. -,-1. «.-+• S I«J"' I (mod q) (1 < i < r).
Let
/ log|ei"|. A ^ l ••
•••. log|«"'l
\log|er»|.
We have det /lifO 11 .
(l«i
\ )•
• • • . log|8<'+"|/
We may take
" 1) For the case det A = 0, we may use the following set of independent units: , , = sin JL iZ+ysin i g> (I = 0, 1, • • -, r - 1), where g is a primitive root mod p.
From (2) we have the following simultaneous diophantine approximations: ,a.
I
j
fiL-sJi1, «—V I1 I 9<+-J-
(2^i
The practical numerical method for finding q\ny(fi < i < r ) can be described as follows: From the expression g , , , ^ . . ^ gin"' = A<»,?(,, + . . . + *(«) e m , we have
341 1010
SCIENTIA SINICA (NOTES)
,„, ^ ,,„) 'h — ~ 2LJ " I >
Vol. XIII, No. 6
Example 3. Considering the field and R(cos—iM respectively, we have
R ( c o s ——) V 11 /
9j.»> = pA|»> - 2 2 AJ"» ( l < i s £ r ) .
> =J
#(9389; 1, 8628, 6408, 2908, 7800) s£
Example 1. Take R (cos ^ L ) and 2-r
4ff 5
< 0.0081175, tt(41204;
5
From 2 - £ ; = -v/ 5 e 2 , e, - e 2 = V 5 , and the expression sf =tfj"'e z - q\"\ we have
29223) < 0.009425. Further numerical examples will be given elselater_
where
lnol/iOUJ
" ( k i " ' l . 1. k i " l ) « ^ f e ^ Example 2. Take R (c°s~^
ei =
2cos^, S!
1, 38810, 31766, 20480, 5610,
Remarks.
Ccf. [I])-
The following numerical
e v a l u a t i n g t h e r . p , e i n t e g r a , is s u g g e s t e d
and
f' . . . P f(Xu ...,Xj-)dx,
= 2cos±L. 7
Hua Loo Keng ( ^ ^ ^ ) Wang Yuan (3E 7 t ) l i ^ . i l j «/ &,™« ^ 7 echnology of China;
s\ s\ = - 227 e, - 45 e2 - 146 s3>
/nif»/«/e of Mathematics, Academia Sinica; . .. - . . n ^ . April 14, 1964
we have g = 418; a. = 1, a, = 335, a, = 103. By numerical calculation, we have H(418; 1, 335, 103X0.0108146.
••• dxs x
~__^__.^^H..*.//W 1 > /.-.«f»;).
8,-2co.^.,
From the equation lefsf| = | e ° e f | , we have - i = 1.356 = 1 . From the expression 3
method of
Reference , , tI „ „ ,„, ,„,„ c . „ r v [ 1 ] Hua, Loo Keng & Wang, Yuan l%0 bci. Rec, New Ser., 4(1), 8—11.
r
342 SCIENTIA
SINICA
Vol. XIV, No. 7, 1965
MATHEMATICS
ON NUMERICAL INTEGRATION OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES* HUA Loo KENG (4£^|?) ANO WANG YUAN (5.
76)
{University of Science and Technology of China; Institute of Mathematics, Acadcmia Sinica) ABSTRACT
By means of the algebraic theory of numbers the authors suggest a method for evaluating multiple integrals. A numerical example of eleven dimensions shows the advantage of the present method. I.
INTRODUCTION
It is the Monte Carlo method to approximate a multiple integral by a single over a random variable. More precisely, let /(.Xjj • • •, x,) be a continuous periodic tion of ^-variables, each with period 1. If we identify the opposite faces of the unit O ^ x . , ^ 1 ( i » = 1,2, • • - , * ) as an .r-ring R, a periodic continuous function may be sidered as a continuous function defined on the ,r-ring. Let (x| ; ) , • • • , x j ' ) ) , be a variable
random
i = 1,2,3, • • •
sum funccube con(1.1)
uniformly distributed on the ring R; then the average
(1.2) for the sequence of numbers (1.1) is approximately equal
of the values of f(.xx, •••,xs) to the value of the integral
[ ••• T / U , •••,x,)dxl---dxl
with a probability error
Jo
Jo
(1-3)
°(T=)The method of theory of numbers is to construct explicitly a sequence (1.1) so that the difference of (1.3) and (1.2) has an absolute error. Some theoretic results in this respect have been collected in a monograph of the authors'21, and we shall not discuss them here. However, it is worth mentioning that both Kopo6oBt5] and Halton161 have made important contributions to the subject considered. * Received Feb. 8, 1965.
343
No. 7 HUA & WANG: ON NUMERICAL INTEGRATION OF FUNCTIONS OF VARIABLES
965
Here, we aim at the specific computational methods and numerical results. In the beginning KopofiOB applied the method of complete exponential sum in the analytic theory of numbers (see, e.g., Hua Loo Keng1'1) and found a method of using a simple sum to approximate a multiple integral with an absolute error which is the same as the probability error Of
z=-f obtained by the Monte Carlo method, where n is the number of points of
division. Later, he proved a theorem which even improves the accuracy; but it is a theorem similar to that of existence, and for computational purpose, it is necessary to compare all "divisions" of a certain type so as to find the one with minimal error. In the light of the theory of algebraic numbers and Diophantine approximations, we suggest a set of points of division. Our division is essentially as good as the best division of KopoSoB (the example given below shows that they have the same significant figures), yet the amount of computation involved is considerably reduced. To be more definite, if his amount is denoted by P, our amount is only O ( A / P )• Naturally, using a computer of the same speed, we may find a method treating the integrals of higher dimensions and obtain results of better accuracy To show the efficiency, we give the following numerical example: For a periodic function / ( x h ••-,.\ n ) satisfying certain conditions mentioned in the text, we have I • • • I /(x b • • -, xn)dx1- • -dxu — I Jo Jo
(jfV1 x 0.233543,
-±±f(jbL,...i*2L)\
a± = 1,
et2 = 685041,
a3 = 646274,
«4 = 582461,
a% = 494796,
a6 = 384914,
a, = 254860,
a, = 642292,
aw = 467527,
au = 284044.
Further examples of 5- and 6-ple integrals shall be given in the text. It is worth while to mention that the division points so obtained have applications to the theory of probability. II.
MULTIPLE INTEGRALS AND SIMPLE SUMS
First of all, let us construct some auxiliary functions: Lemma 2.1. Let {x} be the distance from x to its nearest integer. Then we have
3(l-2{*})>=
±J^L, 6
where
n = max
( 1 , \n\).
Proof. It is sufficient to prove that
344
966
SCIENTIA SINICA
Vol. XIV
f 1 2
cm = T 3(1 - 2{x}ye- """*dx
= j
Ju
for m = 0, 6
-T—2
for /;z ^ 0.
Evidently we have cm = 2 1
Jo
3(1 — 2x) 2 cos 2-nmxdx.
For /« = 0, c0 = 2 ( ' 3(1 - 2x)2rfx = -
(1 - 2x) 3 I* = 1.
Jo
b
For OT==0, integrating by parts twice, we have /• f t i ^2 s ' n 2-rtmx i . - . f3 (1 — 2x) , l sin 2nmx + 24 1 ^ ax = cm = 6(1 — 2x) 2 2n7?z
12 ,. nm
u
Jo
« N cos 2-jtmx * 2-itm »
2nm
s
48 f cos 2nmx Ju (T-Ttnif-
,
Consequently we have Lemma 2.2.
g(x,-,,)-y
:
no-2{X.D - s - - s
»=i
g
!^j-+-^r:).
jr 2
-»
,
—m\- •
6
w2
,
•—m,
6
Let ^ > 0, (?l5 • • •, a, be integers; we have
i^K..,^)=
E-:-2 m1n1+—+m1a1=0(moi q)
-=^1 6
C2.D
r"\ ' " '
6
'"/
The main idea is to find q, alt • • •, a, so that
(2.2) as small as possible. Naturally, the best way is, for a fixed q among all possible 0 ^ a» < q, to find a set of («!, •••,as) for minimizing H^q, a^ • • - , « , ) • However, it is too complicated to do so; hence KopofioB suggested the following method. To find the smallest value among the p — 1 sums HiCfcl.s. •••.«J~1). where p = q Is a prime, and £ runs over 1, 2, • • •, p — 1. This method requires O(
345 No.
7
HUA & WANG: ON NUMERICAL INTEGRATION OF FUNCTIONS OF VARIABLES
967
With the aid of algebraic theory of numbers, we give a set of q, au •••,as, without very complicated computations (i.e., the ordinary calculating machine is good enough for our purpose). To estimate the value of H-i{_q,al>- • • , « , ) , we require only O(q) elementary operations. Let /(xi, • • • , x J ) be a periodic function with period 1 for each variable and having the Fourier expansion /(*!, - - •,*,) = S
c
• •• E
^>
• • •• « , ) * 2 * / ( - " + - + B " I ' ) ,
where I c("h, • • ". »h) I < -= and IT is a positive number.
™—,
For example, the function satisfying
, r , ,/(>=!, •••,x I ) <(2«) 2 ' C
9xf • • • 5.\;
belongs to the class considered. Consider
Consequently
P • • • Jo [KX*
Jo
=
"~
• • ; xM*. • -dXs - ^JL^ ± fV(*£,..., ^ 2
ajm^
- - -
S'
(-flj.mj.S0tmod 4)
±L) =
q ]
c(»«i, • ••,«,),
where 2 - • - S ' indicates the omitting of the term m^ = • • • = /rc, = 0 since c(0, • • •, 0) = P - - - T/Cxi, • • -, x,)dxi- • -dx,. Jo
jo
Consequently, the difference between the multiple integral /(*!, • • •, x , ) a V • -dx, 1 ••• Jo Jo and the simple sum 1 V
/(
ait
a 1
> \
(2.3)
346
968
SCIENTIA SINICA
Vol. XIV
is dominated by
by (2.1) and (2.2). III.
SOME PROPERTIES OF THE TOTALLY REAL ALGEBRAIC NUMBERS
Let 3^ denote a totally real algebraic number field of the ??th degree. For a number r\ of S>~, let ?7 (1) (= r;), T/ 2) , - •-, ^ (o) be its conjugates. Assume that Si,
be an integral basis of $?~.
• • - , & „
Form a matrix (3-D
• s<»>, • • - . s i , - ' / The matrix
S =fl'fl= ( jslt'sjt')
(3.2)
is called the fundamental matrix of the field $1*. Clearly, it is a symmetric matrix with rational integral elements. (The invariants of the fundamental matrix under the modular group are characteristic properties of the algebraic number field ^ " . ) From (3.2), we deduce immediately £-1 = S-IQ\
(3.3)
The determinant of S is the discriminant of the field. Let 7; be a unit of 5?~ with absolute value > 1 and its conjugates (beside itself) satisfy
I^Kclil""17. where c is a constant.
/ = 2, ...,»,
(3.4)
We express 17 as follows: ij = AA + • • • + kA,
where A1( •••,hn
(3.5)
are rational integers. From (3.5) and its conjugates we have (,W ...,,<•>) ~ O k — , A . ) f l ' .
(3-6)
Hence (j? (»
. . . , ,(->)£ = ( ^ . . . , A,)S = (*„ • • •, *.) (by definition).
347
No. 7
HUA & WANG: ON NUMERICAL INTEGRATION OF FUNCTIONS OF VARIABLES
969
Or i — 1
Therefore, we have
| >;S, - k,, | < _ | , « | | S«-rt | < (n - 1) cS h |" A
(3.7 )
where S = max | fb'-p | . Let 1 = «!©! + • • • + a , S ,
(3.8)
and & = tfi&i +
• • • + (7nA,,.
Then, from (3.7), we deduce that
h~*l
(3.9)
> = i
where the constant A depends only on JF~ and C. (A will appear later but may be different in each occurrence.) Hence, we obtain the simultaneous Diophantine approximation
3, - -f < -TT-?
1 < i < »•
(3-10)
This is not new. Our purpose is to suggest a computational method for obtaining k'jS. For n — 2, we can use continued fractions to treat the present problem. In case of n > 2, the situation is entirely different. Various classical methods can only prove the existence of an infinitely many sets of k, k^, • • •, kn satisfying (3.10), but this does not suggest any effective way for finding k, kx, ••-, kn. It is shown in this section that the problem for finding k, kv ••-, kn is equivalent to the problem for finding an 7; in the totally real algebraic field so that (3.4) is satisfied. If we know a set of independent units, then by the following lemma, we can find a sequence of units {77/} satisfying (3.4) and I rjt; I —*• 00, as ; —» 00. Consequently, we have infinitely many sets k, kt, • • •, kn
satisfying (3.10). (it is practically known.) Lemma 3.1. Let Ljixx, • ••, x m ) = aixxx + • • • + aimxm,
1 < i < m.
348 SCIENTIA SINICA
970
Vol. XIV
// det (a,-/) ^v 0, there is a -point (x 1 ; - • •, x,,,) j«cA that L, = L, < K, where K is any given real number. N = max(Uu|
1 < z", / < /»,
(3.11)
Further let
+ ••• + \ a l m \ , •••, \ a m l \ + ••• + \ a
m m
\).
(3.12)
:
Then there is a set of integers (y[ \ • • •, y^~) such that Li(y['\---,yi;))<-(Zt-l)N.
(3.13)
and U,-(yi. • " •. yJ - L,{yi, • • -. y ra )| < 2N. Proof.
(3.14)
We solve the system of equations i-i ~ -^2 == ' ' '
=
i«
==
K— 1
for .\-j, • • •, x,,,. Obviously, the solution will satisfy (3.11). Now, we take K = — 2Nt and y\l) = [ x , ] ; then L,(y\'\
• • ; >tf) < L.-Cx,, • • -. x,,,) + ^\aik\
< - 2tN + N = - (.2t - 1)N
and I L,(y['\
• • ., y£) - L,{y\-\
• • -, >•<;>) | =
f]
(aik - « / t ) j ^ <
= si««-«rti <2N. The lemma follows. By means of the lemma, we are going to find the unit rr Theorem 3 . 1 . In the totally real algebraic number field £F~, we have a sequence of units i?,( = '?J1)) t = 1, 2, 3, • • •, whose conjugates rj[2\ • • •, J?(
2< f< »
(3.15)
and e~2' | i?J" K . | n ? I < e2° I i?,u)|,
2 < /, / < »,
where a is a constant depending only on the field &~. Proof. Let eu
•••,Br
be a set of independent units of gi~. Let
(r =
n — 1)
(3.16)
349
No. 7 HUA & WANG: ON NUMERICAL INTEGRATION OF FUNCTIONS OF VARIABLES
971
Taking the logarithm of the absolute value of £(<>, we have r
log I ^ I = 2*>Iog|e<-<>|,
2
i = 1
Let max ( 2 log j ej'11 ).
a= Since
det(log|e}'-»|)^0, the solution of log|£<»| = • • • = log|£ ( ">| = - l a t - 1 is denoted by lu • • • ,lr. Let = a\'\
[li]
1^
t
< r.
By Lemma 3, the units
satisfy the requirement of our theorem. IV.
THE CYCLOTOMIC FIELD
Let p be a prime ^ 5 . n = — (p — 1 ) , and r = — (p — 3) =
real cyclotomic field R
? \ (2cos—I P/
is an aigebraic fieid of degree n.
n
— 1. The
The field has a set
of integral bases 2coS^, p
2cos^,---, 2 c o s ^ . p p
(4.O '
It is well known that n
22cos^-=-l /=i P
(4.2)
and Ij2cosM=(_1)L
2J = ( - l )
« .
(4.3)
We use g to denote a primitive root, mod p. Since 2 c o s 2jr g / ± B
P
_
2cos
/ _ 2 * A= \ p /
2cos2«
p
^
( 4 4 )
350 972
SCIENTIA SINICA
Vol. XIV
the integral base (4.1) can be expressed as 3, = 2 cos — gl+'" P
1 < I < i7,
(4.5)
where |3,| > |S,|
(1 < I < « — 1).
From (4.4), we define 3,±1> = S,. The transformation cr:
S; —»• S, +1
is an automorphism of the cyclotomic field, and n automorphisms a, a2, • • •, a"~\ a" ( =
form the group of automorphism of the field. conjugates under the automorphisms. Since
Z 2- cos licitP
1)
A number r> of the cyclotomic field has n
~ 2-itkt sr f 2n(l + k)t , _ 27t(7 — ' — 1 - 2 cos — - 2 cos = > , n2 cos — J fr l\ P P =
_
k)t\ — = VI
rp, for I = k,
2 +
lO, for Z % k, we have
(
\2
S1( S2, • • - , © „ S2
'
S3>
"•'
S l
= p / -2A/,
(4.6)
S,,©!, • • • , a n _ 1 /
where Af = (»»,-;), w,-,- — 1.
Taking a set of indepent units
e{'---ej
(
9 \
of it 2cos— ), then from the method
of § III, we take ij = e{'---ej'
(4.7)
so that the magnitudes of the absolute values of its w — 1 conjugates are about the same, and all less than 1. If we let
V= 2
A,®/.
(4-8)
then by § III, we have
|f-*<|-of]IiM-
1
"'
(4 9)
'
351 No. 7 HUA & WANG: ON NUMERICAL INTEGRATION OF FUNCTIONS OF VARIABLES
973
where «
k = - g Ay. '=1
„
(4.10)
£. = pA; — 2 2
*;•'
K
' < »•
• = i
It is known (Fricke11) that sinyg< + 1 Pi =
,
K*
(4.11)
sin— g' P is a set of independent units of the cyclotomic field to use the set of units 2 cos—, P
i?(2cos — ). V pJ
It is more convenient
1<2<7-.
(4.12)
However, they do not always form an independent set. In the next section, we shall find a necessary and sufficient condition for which (4.12) are independent units. V.
UNITS OF THE TOTALLY REAL CYCLOTOMIC FIELD
Theorem 5.1. Let f, = 2 c o s ^ g ' f P A necessary and sufficient
l = \ , ...,r.
condition that t,x, • • - , £ , . form a set of independent units is that
(i)
2 is a primitive root, mod p, or
(ii)
2 belongs to the exponent — ( p — 1) = n, mod p and p = 1 (mod 8 ) .
Proof.
If 2 = gl ( m o d p ) , then
,
^ =_ 2 cos £ ? g> = 2 cos ^ g^
1)
sia^-g^ ^
=
.
If 2 is a primitive root, mod p , we take g = 2, so that 1 = 1, and hence
Therefore fu • • - , ? , are independent. 2) If 2 is not a primitive root and belongs to the exponent S, then let p — 1 = £r
352
974
Vol. XIV
SCJENTIA SINICA
and take the primitive root g, mod p such that g' = 2(modp). (i)
/ / l\n, then let n — It, hence
? i ? / + r • ' ?
Since I > 1, therefore f1; • ' • • , £ . are not independent, (ii) Let I | 2w, but l\n, i.e., Z = 2m and 7Z = mt. For w ^ r l , we have Clfm + l" ' •f(;-l),.i-H
=
(P2ra+r • •Pta)(.°3ra + r " " Pan.) " " "(Pm + l" " "P3m) = J_J_ P/ = 1i- 1
Therefore £lt • • •, fr are not independent. For »z = 1, if we have integers / j , • • •, /„, not all zero, such that ± 1 = ?'••• •£'„" = (P3ft)''(ft • P>)'J- • •(p,,P 1 )'»- ! (p l P2)'- i (p2P3)'" = _
p;B-j+;n_,
p'n-l + ln p'n+h . . .
p'n-i^'n-1^
owing to the independence of p b • • -, p,M it follows that l»-2 + Zn_j = Zn_j + Zn = L + h = • • • = Zn_3 + Zn_, . Consequently we have h = ^2 = ' • • = InIn other words, fi, •• •, t,, are independent, and 2 belongs to the exponent n, where 2\n. From p1-1
2 - s (!-) = ( - l ) ~ i ~ we have p = l VI.
(mod?),
(mod 8 ) . This completes the proof.
SELECTION OF THE SET OF POINTS OF DIVISIONS AND NUMERICAL RESULTS
Let Sj, ••-,'&„ be an integral basis of the totally real field i?(2cos —j
|S;| > IS. |
satisfying
(1
We define k, ku • • •, kn as in (4.10). Define « = |*|, ax=\, «,+1'= \k,\
(K;
(6.1)
353 No. 7
HUA & WANG: ON NUMERICAL INTEGRATION OF FUNCTIONS OF VARIABLES
975
We use the following approximation formula for the multiple integral:
j;... £ /(,,... ,,),„... * . - • i ± / ( - ^ - . ^ ) .
c«)
Our method is summarized as follows: Take
a
set of independent
units
sx,••-,e
r
of R
I \
2 \
2 cos — 1 a n d t a k e pI
lx,---,l,
so that the absolute values of the ;z — 1 conjugates of •rj =
are about the same, and less than 1.
£(i.-
•' s ' /
Express ?; in terms of the integral basis
•>! = AiS*! + • • • + }!„•&„.
(6.3)
Then we take q =
2
^ '
«i = L
«/+i =
Vh,••- 2 2
A/ ,
1 < / < r,
as the set of points of division. Now, let us give some numerical examples: Example 1.
Take R\Zcos — j and
6! = 2 cos —, 5
62 = 2 cos —. 5
From
we have
Hi( I ^'I; 1, ki"> I) = 1 + O ( i f i & ) . This is the result of [2]. Example 2.
Take R(2COS — J and
gj = 2 cos — ,
s 2 = 2 cos — ,
e3 = 2 cos — .
From the equation lej-e/l =
|efef|,
we have approximately ^=1.356-.-=- ± . P 3
(6.4)
354 976
SCIENTIA SINICA
Vol. XIV
From the expansion e$e\ = — 227 s1 — 45 e2 — 146 e3, we have q = 418, ax = 1 , a2 = 335, <73 = 103. After numerical computation, we have ^ ( 4 1 8 ; 1, 335, 103) < 1.0108146. I 7 \ Example 3. Take K ( 2 c o s — J and ei
= 2cos^, 11
g,
= 2cos^, 11
e3 = 2 c o s ^ . e, = 2 cos *?. es = 2 cos ^ . 11" 11 11
Solving the system of equations 162646563!
—
1636562621
—
164636,651
—
1 e 3 Si 64 e 2
,
we have JL== 1.412 = ^-, 5 5
-^- = 1.584 = —, 5 5
— = 0.944 = - ! . 5 5
From the expansion els|e?e? = - 3345 gj - 271 e2 — 2825 e3 - 998 e4 - 1950 e5, we have q = 9389, a1 = 1 , a2 = 8628, a3 = 6408, a4 = 2908, a, = 7800. After numerical computation, we have #i(«;«i. •••, *s) < 1.0081175. /
p \
Example 4. Take R 2cos— and V 23/ o
8, = 2 cos
12jt
,
o
„
2JT
lO^r
23
s, = 2 cos —, 23
e3 = 2 cos
An o £4 = 2 cos —,
„ 8it £5 = 2 cos —,
„ 6jr £6 = 2 cos — .
,
23
From the expansion e\s\ele^&\ = — 12494 e2 — 726 s2 — 11084 £3 — 2738 e4 — 8587 e5 — 5575 e 6 , we have q = 41204,
*! = 1,
a, = 20480,
a2 = 33810,
a5 = 5610,
a3 = 31766,
ab = 29223.
355 No. 7 HUA & WANG: ON NUMERICAL INTEGRATION OF FUNCTIONS OF VARIABLES
977
After computation, we have
HiC-7;*!, • • •, ad < 1.0094250. /
?
\
Example 5. Take R 2 c o s — and V 23/ o
22TT
£i = 2 cos
23
£ 9 = 2 cos
\An
23
e3 = 2 cos
23
„ 6jr e6 = 2 cos — ,
23
,
s10 = 2 cos
IOJT
23
23
,
23
,
_
e u = 2 cos
12ir
23
O
Ait
O
8W
s 4 = 2 cos — ,
23
16:7r e7 = 2 cos , o
23
-
20TT
o
e2 = 2 cos — ,
~ 18jr 65 = 2 cos , -,
2?r
O
,
e8 = 2 cos — ,
23
.
From the expansion eje*e5e?ejejejeje|ejo = - 120834 sy - 2251 s2 - 116374 e3 - 8837 e4 - 107785 e5 - 19269 e6 — 95704 e7 — 32774 e8 - 81027 e9 — 48350 e10 - 64842 e u , we have q = 698047,
ax = 1 ,
a2 = 685041,
«3 = 646274,
a4 = 582461,
d5 = 494796,
a6 = 384914,
a7 = 254860,
a1(J = 467527,
au = 284044.
ff8 = 107051,
a9 = 642292,
After computation, we have Hi(«;l,«i. •••, «n) < 1.2333543. Remark. We can also use as- well other totally real fields for finding the division points for evaluating multiple integrals, for example, the Dirichlet field, i.e., the field obtained by successive quadratic extensions to the rational field (Hua and Wang)'3'41. However, we conjecture that the cyclotomic field gives the best result among the fields of the same degree. Another advantage of the cyclotomic field is the convenience for calculation. Example.
We use the Dirichlet field R(*J 2, V 5 "), and take £l
=
l
+ V5,
e2 = 1 + 4~2,
63 = 3 + VTO.
Expanding efeje? = (9 + 4 V T )(17 + 1 2 V T ) ( 1 9 + 6 ^ 1 0 ) , = 5787 + 4092y^2~ + 2588-
356 978
SCIENTIA SINICA
Vol. XIV
REFERENCES
[1 ] Hun, Loo Keng 1957 Additive Theory of Numbers, Peking, China: Science Press. [ 2 ] Mua, Loo Keng & Wang, Yuan 1963 Theory of Numerical Integrations and Their Applications, Peking, China: Science Press. [ 3 ] Hua, Loo Keng & Wang, Yuan 1964 On diophantine approximations and numerical integrations (I), Scientia Sinica, 13, 1007—1009. [ 4 ] Hua, Loo Keng &: Wang, Yuan 1964 On diophantine approximations and numerical integrations (II), Scientia Sinica, 13, 1009—1010. [ 5]
[6]
KopofiOB, H. M. 1963 TeoperuKO'iucAOBbie merodbi 3 npu6AuxeHHo.n
anaMU3e, THOMJI.
Halton, J. H. 1960 On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Num. Math., 27(2), 73—79. [ 7 ] Fricke, R. 1928 Lehrbuch der Algebra, Bd. Ill, s. 225.
357
ON INTERPOLATION OF A CERTAIN CLASS OF FUNCTIONS*
WANG YUAN Institute of Mathematics, Academia Sinica
I. Let E" (C) be a class of functions: oo
— oo
where |C(mi,...,ms)| <
rj
——,
(mi---ms)a
in which fh = max(l, |m|),a > 1 and C > 0 are absolute constants. If / € Ef(C), then we define v by oo
V(xlt ...,Xa) = J2---Y,
S
( m i ' • • • ' m,)C{mi,. . . , m ,) c Mm.*.+-+m 1 x.) )
— oo
where C(mi,..., ms) is the Fourier coefficient of / and B(mi,...,
ms) satisfies
0<W
\B(mi,...,rns)\<J^—^,
Let N be a prime number and Nr = [^^^(lnJV)"^--?"1'] +1. Let rii
JV / , n -1- V ^ / . / a l ' c
\
, -. a«K\
C(m 1 ) ...,m,) = - g / ^ — , . . . , — J e Q(x 1 ,...,a; fl )=
2 T r i a i m i + '•+"«"*»;-
-
fc
,
B( m i ,...,m s )C(m 1 ,...,m J )e "< m i I >+'- + r a ' 1 ')
^
2
rni---m,<JVi
and A = min sup a
fSEf{C)
where a = (ai,..., aa) is an integral vector. *Kexue Tongbao, 9 (1996) 389-391.
\v{x\,...,xa)-Q{x\,...,xa)\,
358 358
Y. Wang
The aim of this paper is to prove the following two theorems:
Theorem 1.
The estimation (s-l)(q2 +aui-ui)
a(a+u-l)
2a-i (inAT)
A
2a-i
is a constant depending on a and s.
Suppose that / € Ef (C) and / is an odd function with respect to each variable, and that u(x\,... ,xa) satisfies Poisson equation in unit cube Gs and takes value 0 on its boundary. Then from Theorem 1, it follows that Theorem 2. Let s > 2. Then for any given prime number N, there exists an integral vector a = ( a i , . . . , as) such that U(XI, ... ,XS) - ^ f [—,...,
-j^r- j 1pk(Xl, ... ,XS)
-a(as+2-s) (a-l)(a28+2a-2) < C2(a, S)N (2a-l)» (In N) \2a-l)s f
where 1
M*u...,xa) = -—
s
s e2*i(mi{xi- £)
,
+ -+Tn.(x.-Sf/L))
m; + - + m;
m1---ms
l
'
"
in which ^2 denotes that the term mi = • • • = ms = 0 is omitted. These two theorems give improvements for the previous results due to Korobov [1, 2] and the author [3], for example, the error term in Theorem 2 is O(N~4^3+e) for the case a = s = 2, while the original results of Korobov and the author are O(N~1+E) and O(7V- 6 / 5 + e ) respectively. II. To prove Theorem 1 we need the following two lemmas. Lemma 1. [4] Let k(l < i < s) and n\ be integers satisfying h- • -ls > 3 s and 1 < n\ < l\ • • • ls/y and let a be a real number > 1. Then
Lemma 2. [5] Let N be a prime number and a > 1. Then there is an integral vector a = ( o i , . . . ,as) such that r-^' ajmH
\-aBms=0(mod N)
1
.
. (lniV)"^- 1 '
359
359
On Interpolation of a Certain Class of Functions
Proof of Theorem 1. 1) 5Z
\v-Q\< +
\B(mi,...,ma\\C(mi,...,ma)-C(mi,...,ms)\ |5(mi,...,m s )||C(mi,...,m,)| = £i + E 2 .
Y.
2) Since C(mi,...,ma)-C{mi,...,ma)
^
=ai/H
C{li+mi,...,la+ma),
\-aJa=0(modN)
we have
iEli<
i
y
c
r
It follows from Lemma 2 that the congruence aih -\
h asls = 0 (mod N)
has a nonzero solution satisfying
whenever iV > cs(a, s). Note that the theorem holds evidently if N < cs(a, s). Therefore by Lemmas 1 and 2, we obtain [log 2 NX]
iExi
-,—1—~
y
{ m i
fc=0 2-*-iJV1<*1...*RJ<2-*JV1
m s )
l
y [log2 iVi]
< C Y (^^Ni)-" fc=0
X
v^;
0lll+...+0Xio<modAO
y> rni-^2-"*
i (Cl + " » l ) - " ( U m , ) ) »
2'
T
F
^
< c 7 (a,s)CiV 1 - u;+Q Ar-«(i n iV)^-i) °1
x J](2 Q - 1 )- f c < c8(a,s)CJV fc=0
-a(a+a;-l)
2^=1 (l n AT)
(s-l)(a2+ctu>-u>)
55=1
.
360 360
Y. Wang
3) |E2
'-_
?
„ (rni-fl,)^
mi---m s >iVi
v
< c 9 (a,s)CAr i - a - a ' + 1 (lnAr 1 ) s - 1 -a(a+a;-l)
2a-1 (In AT)
(3-l)(a2+aii)-aj)
2a-l
The theorem follows from 1), 2) and 3).
References [1] [2] [3] [4] [5]
N. M. Korobov, Problems in Comp. Math, and Comp. Tech., MASGES (1963). N. M. Korobov, Number theoretic methods in asymptotic analysis, GEFML (1963). Wang Yuan, Set. Sinica 14[4] (1965) 629-631. Wang Yuan, Sci. Sinica 10[6] (1961) 632-636. N. S. Bachvalov, Vestnik Moscow Univ. 4 (1959) 3-18.
361 Vol. XVI No. 4
SCIENTIA
SINICA
November 1973
ON UNIFORM DISTRIBUTION AND NUMERICAL ANALYSIS (I) (NUMBER-THEORETIC METHOD) HUA LOO-KENG (*£Jf @£) AND WANG YUAN (=E
%)
Received June 13, 1973.
ABSTRACT
In this paper, the uniformly distributed sequences of sets defined by means of a real eyelotomic field have been dealt with. We have obtained the estimations of their discrepancies and applied them to the problem of numerical integration. I.
INTRODUCTION
Let Os be a unit cube 0
P.,0") = (&>K», • • ; *?>Ki», (1 < i < »,) be a set of points in Gs. For any ( n , • • •, r*) £ &*, let iV^Cn,''", r*) denote the number of points of P O / (i)(l < j < «,) satisfying the inequalities If
0 < x["iKj) < n , • • •> 0 < ai"/'(i) < .. Nn(n,---, lun ^
rs)
=
r
-
rr..Ts)
then the sequence of sets ( P n ; ( i ) ) ( w ! < n2 < • • • ) is called uniformly distributed on
<
n, where (p(%) = o(l), then the sequence of sets (,P«I(j')')(nl
a,2 = 2 o o B - ^ - , ••-,
co ? = 2 eos ^ 2 -
The
362 484
SCIENTIA SINICA
Vol. XVI
and let -^
co,- < c (9t r )wr l ~ ? ,
( K i < q)
be the simultaneous approximations of co's by rationals, where we use c ( / , • • •, gr) to denote the positive constants depending on / , • • • , fir only, but not always with the same value.
Theorem 1. Let P ( i ) = ( W } , •••Acoj})
( j = i,2, •••),
(i.i)
where {x} denotes the fractional part of x. Then the sequence P ( j ) O ' = 1, 2, • • •) hasdiscrepancy cp(n) = c ( * r , e>-' + % g Semgr any pre-assigned positive number. (The convention will be adopted throughout the present paper). Theorem 2. Set
p.,o)-(U| p - } , . . . m } ) (i<,•<„,).
(i.2)
T/ien #/ie segwence o/ sefe ( f n ; ( j ) ) ^.as discrepancy (p(w) = c(5Rr, e)n 2 2«+£ . Let EsCC) be the class of functions f(xu
• • ; Xs-) = 2
- •S
C
(
W
" * • •« m,')6M{-m'*>+™+m'*'\
in which the Fourier coefficients satisfy
ICOn,,---, m , ) K .
C
_
(mt • • -m,)
,
where a and C are positive constants and m = max (1, | m | ) . Suppose a > 1 throughout the present paper. Applying the sequences of sets (1.1) and (1.2) to the problem of numerical integration, we have Theorem 3. Let I he the least integer ^ a and let ;in, Lj he a set of integers defined by
Then Sup
f(Eam
• • • f(xu Jo 3o
• • • , xq)dx,
•••dxq
— ————- 2 P«, 1,i f(wih (2M + 1)' jtr!,,
< C • c(9t,,a, s)w-a+£.
'•'»««i) (1.3)
363 No. 4
HUA & WANG:
ON UNIFORM DISTE. & NUMERICAL ANAL.
485
Theorem 4. We have
Sup ['...['/(^....^^...^-iS/^.-^.-.-^) < C • c(9tr, a, e^H^"*.
(1.4)
It is well known that under certain conditions, the integral of non-periodic function may be calculated by an integral of periodic function1141. We may also use the following simple formula instead of formula (1.3) for the case a = 2. Sup I [• • • (' f^,
• • ; *,) dxr • -dxq - - 1 S
( l - i l l ) f(wj, • • -, co 9 j)
< C • c(9l r ,e>- 2 + £ .
(1.5)
It is followed immediately by the ^-result of Eoth[91 on the discrepancy of uniformly distributed sequence of sets that the estimation given in Theorem 1 is the best possible one of its kinds apart from some possible improvements about the order nc. This type of distribution may be recognized as the best distributed sequence of sets. The other known best distributed sequences of sets are those proposed by KopofioB1141 and Halton121 in 1959 and 1960 respectively. We can prove also that the error term given in Theorem 3 does not allow essential improvements1151. Perhaps, it is worth mentioning that only c(9t r ) log w; elementary operations are required for obtaining the sequence of integers (fc{» •••, *i»;n,) in (1.2). The classical method in numerical integration is established by means of the sequence of sets (Jl,---,Jl)
( 0 < i n ••-, j , < m - l ) .
(1.6)
That is, the integral over &s is calculated approximately by the sum
JL
S
f(M,...,
M\.
(1 . 7)
The number of points of (1.6) is n = ms. We can prove easily that the discrepancy of (1.6) is > n~l/s and the error term in classical quadrature formula of the functions belonging to E?(,C) over Gs is > 2Cn~a/s. 2
^Take / ( x , , • • •, xs) =
2 im
OCe *'".'. + e- * >*>')/m°. Then we have [ • • • [ /(*„ • • •, xs)dxl • • • dxs = 0 and
-1 y 1 f(Ii ... ^ = 2 0 \ n ,^ijt ' \m' ' m) n"/!')
Jo
Jo
In 1959, BaxBajioB1121 and the authors161 proved independently the formula (1.4) for the case q = 1 with the error term 0{Ff" log 3.F;) by means of the Fibonacci sequence (F,), where
''-^((^r-e-^n <^-
364 486
SCIBNTIA SINTCA
Vol. XVT
Since
f ^
i] O- - 1, 2, • • •) and ({jr}, {%^}) (1 < i < *,)
(1.8)
obtained by Golden section and Fibonacci sequence respectively. Hence we suggested also the possibility to treat the high dimensional case of numerical integration by means of a real cyclotomic field and gave some numerical examples of the formula (1.4) for the case 2 =£j # =£j 10[6~sl. Indeed, other totally real fields can be used as well instead of the cyclotomie field, for example, the Dirichlet field B(V-P^, • "", V-P*), where P's are h distinct primes, but it seems that the cyclotomic field often gives best result among the fields of the same degree. Another advantage of cyclotomie field is convenience for calculation. The proofs of the above theorems depend on the following important result of Wolfgang M. Schmidt1101 concerning the simultaneous approximations of algebraic numbers by rationals: Lemma 1.1. Let au •••, as be a set of real algebraic numbers such that 1, « , , • • •, as are linearly independent over rational field B. Then we obtain (ojm, + • • • + cs,m,) > c(«1; • • •, a,, e)(«v • -w,)"1"1,
ml! • • •, m, denoting any set of integers which are not all equal to zeros and (x) = mm({x}, 1-{*}). We have also used the number-theoretic methods in numerical analysis introduced by BaxBajioB[12], Kopo6oB[14], HaselgroveB1 and Hlawka[51. If a similar result of A. Baker[I1 is used instead of Lemma 1.1, we may also obtain the corresponding results on numerical analysis. For example, we may prove that the sequence PU)
= ({ej},
{e2j},---,{esj}),
( i= l , 2 , • • • )
(1.9)
has discrepancy q>(n) = c(s, e)n~i+e. By Wang 520-calculator, we obtain the following example of the formula (1.5) ••• ,._4_
o
f(x
V
u
• • -, x ^ d x l • • • d x <
v
—
S
3000 j±=jm \
X / focos^-, 2j cos — , 2; cos -^-, 2; cos -^j II.
S?"s,
•A1L) 3000/
< 0.065C.
T H E TOTALLY REAL ALGEBRAIC FIELD
Let &r, denote a totally real algebraic field of the degree s. let r/°( = 7j), rf2\ • • •, rj(s') be its conjugates. Assume that cou
is an integral basis of £f,.
(! -
• • •, ws
Form a matrix
£ = (wf>) (1<», i < « ) .
For a number »/ of
365
No. 4
HUA & WANG:
ON TJNIFOEM DISTE. & NUMEEICAL ANAL.
487
The matrix
8 = Q'Q = ( 2 to?>u>?>), (1 < *, j < s) is called the fundamental matrix of J^~',. Clearly, it is a symmetric matrix with rational integer elements. The invariants of the fundamental matrix under the modular groups are characteristic properties of the algebraic number field. The determinant det 8 of 8 is called the discriminant of the field. Let r] be a unit of &~', satisfying M>1,
\ff»\
(2-1)
We express i\ as follows:
(2.2)
<=i
where k's are rational integers. (,«
Prom (2.2) and its conjugates, it appears •• • , , « ) = (*„ • • • , * , ) # •
(2.3)
Hence (,
(by definition)
or < = i
Therefore, we have
1=2
by (2.1).
Suppose s
1 = X J a'a3'' < = 1
and s 1 = 1
Then we deduce that J
s
<• = i
< = i
^
s
< = i
Hence to, ~ - ^ < c(J^,) | n 1 "'"^I, n
(1 < j < «).
(2.4)
In the case of s > 2, various classical methods can only prove the existence of an infinitely many sets of integers (hi, • • •, hs; w) satisfying (2.4), but this does not suggest effective way to finding (hx, • • •, hs;n). It is shown in this section that the
366 488
SCIENTIA SINICA
Vol. XVI
problem for finding (ht, • • •, hs; n) is equivalent to the problem for finding a unit JJ in the totally real algebraic field 3?", so that (2.1) is satisfied. If a complete set of independent units of &"'s is known, then by the following theorem, we can find a sequence of units (•>?;) satisfying (2.1) and |^/| -»• oo
(as !-»• oo).
As a result, we have infinitely many sets of integers • • • , h s;
(hu
n)
satisfying (2.4). Theorem 2 . 1 . In the totally real algebraic field £P~'„ we have a sequence of units Vi (.= »/i1}) 0 = 1) 2, • • •)» whose conjugates satisfy and
e'2c I vP I < I ifj01 < e" | nf |
(2 < i, j < 0 ,
M)/i«re c = c(&~s~) > 0 .
Proo/. Let e
u
' ' ' ) £^—i
be a complete set of independent units of &~ s. Set |(») = gC.-w,... e (»/,_,
(2 < * < s)
and
c=max(2|log|ej')ll). 2<|
/
Since det(log|£J;)|) ^ 0 , ( 2 < » < s , K j < « - 1 ) , •we denote the solution of the system of linear equations log||M| = . . . - l o g | | « | = - 2 c Z - l by 7 = 7«>
. . . 7
= 7«)
Set where [x] denotes the integer part of x.
Define
Then it follows log I ^ I = S < l o g |e}»| < S ^'>log|6}'>| + 2 ;=1
/=1
/=1
|log|6}»| |
J-l
= log|^(<)| + 2
Uogle^ll < ~2cZ + c = - ( 2 Z - l ) c
(2 < i < s)
367 No 4
HTTA & WANG:
ON UNIFORM DISTR. & NUMERICAL ANAL.
489
and
I log hi 0 1 - loghi'M | < | log | £«> | - fog|^>| | + _ ] ( l l 0 g l 4 " l l + |log|e£»||)<2c ( 2 < » , j < « ) . The theorem is proved. Remark. It follows immediately from Theorem 2.1 that there requires only c(^~s)log \n\ elementary operations for finding the set of integers (7^, • • •, hs; n) satisfying (2.4). III.
SOME EXAMPLES
1) Let p be a prime 3* 5, r = — (p — 1) and g = r — 1 = — (p — 3). 2 2 real cyclotomic field 9tr = 2? (2 cos — J is an algebraic field of degree r.
The
The field has
a set of integral basis co, = 2cos — ,
W2
= 2cos — , ••-,
Wf
= 2cos—.
(3.1)
Let a be a cyclic permutation of (a>,, • • •, cor~) («! -> co2, • • •, tor -> w,). Then from a number 97 ( = J?CI>) of 9?r, we have its q conjugates rj(2\ • • •, j / w under the g transformations a,ar2, • • -, aq. Hence 8 = Q'Q = pI — 2M, where I is the identical matrix and M = (m^-), in which m,,- = 1. Taking a set of complete independent units eu • • •, eq of 9tr, then by Theorem 2.1, we construct a unit satisfying h|>l,
c-*|i? y> | < I^'M < e f c | i 7 w l ,
(2<»,
;
where c = c(9l r ) > 0. If r < = 1
then from
(fc,, • • •, hr) = (fcI; • • •, ftOS = (ft,, • • •, kr)(pl we have r
hi = pft,- - 2 _ > ] fcj, ( 1 < i < r) . y= i
Since r
< = i
M),
368 490
SCIENTIA SINICA
Vol. XVI
hence it is found that r
n= — 2
r
h =
~ 2 ki-
i
Therefore, we have the simultaneous Diophantine approximations -^i - co,. < c(9tr)Tf'""«, n It is well-known that ,
Pi-
(1 < » < r ) .
(3.2)
U
an-g is a set of complete independent units of the cyclotomic field 9tr, where g denotes a primitive root mod p. It is more convenient to use the set of units «, = 2cos — , P
( K K g ) .
However, they do not always form an independent set. In order that they form a set of complete independent units, it is necessary and sufficient that ( i ) 2 is a primitive root mod p, or (ii) 2 belongs to the exponent r mod p and p s i (mod 8) [sl . 2) Let ply • • •, Pi, be h distinct primes, m — 2h and I = in — 1. The real Dirichlet field <&m = R(*s/~pu • • •, */p~h') is an algebraic field of degree m. Let Ei,
• • • , £ /
be a set of complete independent units of S)m. Take e's as the solutions of Pell's equations 2 =± 4 X2 _ Vi---piky r
W Pi ' ' ' Pit
2
Li
with the least of — A
—
y, where k>l and 1 < t, < • • • < ik < h is any
choice of 1, 2, • • -, h. Suppose — g =
j f + ^ t , x; = yt = l (mod 2) ( K » < r), (r + l < » < 0 ,
\xi + -/diyi, where xt and j/,- are rational integers. The integral basis of <£):m is
COj = 1, U>2 — 6t, • • '•, COr-H = 6rj
»'+2 =
V^r+U
" * ">
M
«
Consider Z transformations
/—
f — */~P» for v = ij l y PK> otherwise,
(K:}^k)
=
V 4
369
No. 4
HUA & WANG:
ON UNIFORM DISTE. & NUMERICAL ANAL.
491
where k > 1 and 1 < ^ < • • • < ifc < h is any choice of 1, 2, •••, h. Applying these transformations to any element rj ( = ^ Cl) ) of *2)m, we have its Z conjugates rf2), — , rfm'>. Form a matrix Q = (»?0
( 1 < t, i < m),
then we obtain
s-ffa-(A °Y
VO B / where A = (c,j) (1 =Sj i, j ^ r + 1) and B = (&,,„) (1 ^ ,«, v ^ Z — r) in which fln = 2A, OlV = 2 M ( ^ _ , + d.--!^-,) ( 2 < » < r + l), a,,-= »;, = 2 * - ' ^ ( 2 < j < r + 1), a,-, = 2h-2xi.lxj-1(.2 < i, i < r + 1, i *? j ) , 6^ = 2*dr+B ( K y. < I - r) and V = 0 (^ ^ v). Construct a unit of ^
m
according to Theorem 2.1 such that hl>l,
e-^h^l < \riw\ <e 2c |7 ? « ) |
(2
where c = c ( ^ M ) > 0. If
then from
(fc lf •••,ftJ » - ( * ! , •••, O
we have
n = *, = 2*-' (2ft, + S *.*'+i). \
<• = i
/
hj = 2*"2 f 2xHl h + dHl y)-,ft,+ 2 *.--i a^-i *i) ^
• = 2
Ay = 2 * ^ ^
(2 < j < r + 1),
'
(r + 2 < y < m).
Hence the simultaneous Diophantine approximations to,--A < c(^,)»"^,
IV.
(Kt<m).
UNIFORM DISTRIBUTION
In this paper, we use y = ( 7 , , • • •, y , ) to denote the vector with real components and m = (m,, • • •, m,) the vector with integral components. We also use the notations llrll = fr--fs,
\Y\ = \rr--r,\
and («, ^ ) = 2 J ""*•• ^ t h e
scalar
p r o d ^ t of a and
<= i
/?).
In this section, we shall prove the formula of Erdbs-Turan-Koksma.
370 SCIENTIA SINICA
492
Vol. XVI
Theorem 4.1. Let rj be a number satisfying — > JJ > 0 and let h be an integer 6 > —. Then for any y € Qs, we have V -Nnl(Y)nt
where
\y\ < 2 ' + V ( » l ) ,
K%)= s'xV-S e2 "' (p "' y) ' m) + ^ + 2 ^ , , ^
A < / ? - a < l —A.
Then there exists a function
£
(ii) 0 < 0(x) < 1, for a — — A < x < a + — A and § 2 2 (iii) <j}(x) = 0 ,
^-A<x< 2
for p +— A < z < l + a - — A, 2 2
(iv)
If a uniformly distributed sequence of sets
— ^ . , ( y ) - Irl < < P ( » I ) /or 5 < r< < 1 — s (.Ki^s),
then
-Nnl(.Y)«/
Irl < 2 - x » / )
/wMs /or aK y f (?,. Proo/. For simplicity, hereafter we omit the index I of «; and hU). 1) Set 5 < «,• < /?,- < 1 - d (Ki^s) and let Nn(a, /?) be the number of points P»(j) ( K i < ») satisfying
371 No. 4
HUA & WANG: ON UNIFORM DI8TB. & NUMERICAL ANAL.
493
We can prove easily that
— #„(«,
fi)-\0-a\
n
<2V(«).
2) Let &S be the domain 5 < Xi < 1 — 5
( K i < s)
and &) denote the complementary set of S3 in (?,. The measure of <£) is ^ 2'd < 2Jqo(?i). By 1), the number of points Pn(j) belonging to <£) equals (1 — 25)% + 92'nq>(n), where 9 and d's that will appear later are not the same, but with absolute value ^ 1. Therefore, the number of points of P B (j) belonging to 3D does not exceed (1 — (1 - 2 S ) ' > + 2'
Irl
< (2' + 2' + 2'+1)
2'+29(M).
The lemma is proved. The Proof of Theorem 4.1. By Lemma 4.2, it is sufficient to prove that — tf.(y) -
|y|
<
under the conditions 3»? < r< < 1 — 3>7 (1 < i < s) . Introducing function fl, for 0 < y < x, [0, for x < j/ < 1, we have evidently n
n
(4.1)
/=1
For 3»/ ^ a; ^ 1 — 377, we construct according to Lemma 4.1 two auxiliary functions G?Ky) and G?\y). Function GtKy) satisfies ( i ) G«Ky) = 1, for 2TJ ^ y ^ x - V, ( i i ) 0 < Gi'Ky) < 1, for v < 3/ < 2V and a; — i\ < j/ < x, (iii) (?Sl)(2/) = 0, for x < y < 1 + 77, (iv) G^Xy) has a Fourier expansion where
G?Ky) = x-2v
+ E'C1{m)e2"imy,
I O,(m) I < min (a; - 2,,, - 1 - , — M .
372 494
SCIENTIA SINICA
Vol. XVI
Function Cfi2)(j/) satisfies ( i ) ' G™(y) = 1, for - i j < j/ < a;, ( i i ) ' 0 < G^\y~) < 1> for —2»? < y < — r\ and x < y < x + rj, (iii)' G?Ky) = 0, for x + i; < 2/ < 1 - 2*7, (iv)' G{?\y) has a Fourier expansion G£2)(3/) = * + 2J7 +
l'C2(m')e2"imv,
where
| C2(m) | < min (a, + 2V, -^—,
- M .
From (iv) it follows ©£"(») - * - 2^ + 2 ' C1(m)e2"">!' + 9 2
"^L^-
3t 2 7j/l
|m|
•where 5 J — ^ \ — = TiT1. Put a; = &,- and y == jy, in the above formula and write CJt} = a;,- — 2»? for all *'. Then we have s equations. Multiplying separatively the left hand sides and the right hand sides of these equations, we have ©?,'(»,) • • • #?,'(»,) =
+ -?7(1+ S +
2 '
I»,-KA
C
2
e.Cm,) • • • C.CmO e'-'C-.ir.-H-*-^)
I C(m)lY =0^-2,7) •••(*,-2,)
' ( m i ) • • • C10nI)e2"«».».+-*11/«''' +
gl
°r9fe
»7^
• ( l + Y) l c i ( m ) l < 2 + — + — (* — < 2 + logfe < log9fcV Consequently, it is found that ©2^(»i) • • - ^ ( ^ ) = *i• • • x, + +
&
/log^
2 ' dOn.)• • •_01(mJ)e1"l(-*+"'v"'1'')
|m,-|<*
+ 2I+1\
(4.2)
Similarly, we find ©?>(»,) • • • ^ ( y O = ast • • • x, + X T ^ ( m , ) • • • C2(ms)ew(-.v,+-+-,»,> |ra,-|
+ afkg*9* +2 f+ ' ? y
(4.3)
From the definition of
-^(J/,)
< ff^Cy.)• • -e«;(y,).
Hence the theorem follows from (4.1), (4.2), (4.3) and (4.4).
(4.4)
373 No. 4
HUA & WANG: ON UNIFORM DI8TE. & NUMERICAL ANAL. V.
495
T H E PROOF OP THEOREM 1
Since 1, wlt • • •, coq are linearly independent over rational field B, Theorem 1 follows from Lemma 1.1 and the following Theorem 5.1. / / for any vector m ^ o, the inequality
<(m, y)> > 6||m||-«
(5.1)
holds, where a and b are constants satisfying s + 1 ^ a > 1 and 1 ^ 6 > 0, then the sequence PCi)-({r.;},
•••, ( r , i » G = l,2---)
has discrepancy
±e^\<mrn(n,^). Proof. If 8 is not an integer, then we have JL.
p2xiS(.n+l)
tf2»<«
-I
1
V 1 e2'W = t—, f__ < 1 < 1 ^ e2"iS-l |sinrf| 2 ( 5 ) ' Hence we have the lemma. Lemma 5.2. Let g(rn) be a non-negative function of m. Then y'g(ni)_
i f e llmll ^ ^
yi
ifc'.^f-
yv
& U+i- • •<
x S ••• S Ifc: K A + 1
w/iere 2
Ifc,- K A + 1
1 m
••• £ ' Kfc),
2 lie,-
Km,-
lie,- K m ;
denotes a sum in which i = (%, • • •, i,) rMws ewer aM permutations of (1, 2,
i
Proof.
Since
S - ^ = »C0) + E ^ (Km) + ff(-m))
«• "I" J- I * ! < * + !
™ = l Wl
lit K m
/ I I A: K A + 1
374 496
SCIENTIA SINTCA
Vol. XVI
we have |m,-|
||m||
|m,-:<* »»V • • » l - l
N
m;=l * " , lfc,|<m,
y
h I*,[
/
A
iffi,-i<*
*
1
< ... < y V — s^ ... y> x
S ••• S
Ifc,- KA+l 'l
S
I*,- I<*+1 lie,- Km,I l+l l+l
«vwi,_,
1
••• 2 '
3(k)
-
Ife/ Km,s s
The lemma is proved. Lemma 5.3. Set m ^ ? o and # = [2'"||m||fl6"1] + 1 . If (5.1) holds for any m =N= o, taen «m awj/ interval (P, P + Q""1], there contains at most a point (k, y ) s
= ^f^kiYi,
where k is a vector with integral components which satisfy |fc,-| <J |rw,|
< = i
( K i < s). Proof. If there contain two points (k', y) and (k", y) in the interval (P, P + Q-1], where k'^= k", |fei| < |m,| and |fc"| < |m,| ( K i < s), then we obtain < ( k ' - k " , y ) > < > 6||k' - k"||"' > 2-»&||m||-a > Q~\ which leads to a contradiction. Hence we have the lemma. Lemma 5.4. Set m ^= o and Q = [ 2 " II™!!"*"1! + 1. m ^ o, then
If (5.1) holds for any
* . . <4Qlog3Q.
y '
l A i <(k' y)>
Proof. Divide the interval (0,1] into Q subintervals
'' = ({• ^ 1 a = ».v-,«-o. By (5.1), we deduce that none of the points (k, y ) lies in the interval Io, where k =^ o and |fc,-| < |m,| (1 < i < s ) . It follows by Lemma 5.3 that there contains at most a point (k, y ) in any interval I,-, where j ^ 1. Hence
£' The lemma follows.
77r
L
^r<4 2-<4Qlog3Q.
375
No. 4
HUA & WANG:
Lemma 5.5.
ON UNIFORM DISTfi. & NUMERICAL ANAL.
If (5.1) holds for any m =N= o, then
Let h be an integer ^ 2 .
^V ^ c ( a ' b> s ) / l K a ~ 1 ) 2 ' ij-iiTF K-K* ||m||<(m,y)> Proof. From Lemmas 5.2 and 5.4, we have J
, ^ , | | m | | < ( m , y)> " ^
x
••• V
V Ifc,-I
h i
"- m,-
_
A
A; .
^
( l o g /l)1+
'*" '•
h
^
...
V
lfc,-;K/<+l Ifc,-
s J=0
^
497
m?/+1- • -mj,
V
Km,-
Ifc,- Km,-
1
\\k,Y)/
h =1
m
m,- =1
tl+i'
'mis
'
< c(a, 6, s)hsU-l) (log A)1+"-. «. The lemma is proved. The Proof of Theorem 5.1. "V1'
!
By Lemmas 5.1 and 5.5, we have
J_ "V e2*,(m, r)j
,4i<* l|m|| » ^
<
_!_ -y>'
1
^ n , ^ A 2||m||<(m,y)>
< c(a, b, s^n"1 h'(°-° (log h~)1+'s>, °. Take rj = •— and ft. = 8w2. Then we have the theorem by Theorem 4.1. In VI.
THE PROOF OF THEOREM 2
We use P'M to denote the s-dimensional parallelopiped with edges parallel to coordinate axes and volume < M, and u = (w0, • • • , «,) and h = ( 1 , hlt • • • , fc,) the C* + 1)-dimensional vectors with integral components. Evidently, we may deduce Theorem 2 from Lemma 1.1 and the following two theorems. a =
Theorem 6.1. Let n be an integer > 1 and M be a number > 1. ( a u • • •, a,) be a vector with integral components. If the congruence
Further let
s
(a, m) = 2
a^j == 0
(mod w)
(6.1)
> = i
has no solution in the domain ||m|| < M,
m ^ o,
then the set
({•?}•-•{*})• «<'<•>• /ias discrepancy (p(.n) = c(s, g)M- I + £ .
(6.2)
376 498
SCIENTIA SINICA
Vol. XVT
Theorem 6.2. Let — - Yi < dn~1'^ , (1 < i < s), (6.3) n be the simultaneous approximations of y's by rationals, where d is a positive constant. If (5.1) holds for any m ^= o, then there exists constant c(a, O, d, s) ( < 1) such that the congruence s
< < = ° (mod
h U
(h, U) = Mo + 2
i' = l
OT
)
^6-4)
has no solution in the domain
\\u\\ < c(fi, o, d, 5> ( l + ^ ) / ( 0 + 1 ) ,
a^o.
(6.5)
Lemma 6.1. Let I be an integer ^ 1. Then the s-dimensional domain |jm|| < IM
(6.6)
can be covered by at most c ( e ) ' J 1 + £ M " parallelopipeds of the type P'M. Proof. Take
c(e) = 22+t 2
y= o
(J-(l+" + 2-s0 •
1) For s = 1, since the domain (6.6) is the interval (— IM, IM), it can be covered by at most 21M _ M
2l
intervals of the type [c, c + M], where c is a real number. Hence the lemma is true for s = 1. 2) Suppose k is a positive integer and assume that the lemma holds for s — 1, • • •, k. Now we proceed to prove the validity of the lemma for s — k 4- 1. Divide the domain m1---mk+1
(» = 0,1, • • •, [log, M]) ±2'Z,
(t = 0,1, • • •, [log22lf]).
3) Suppose that mfc+i = j . Then iBl ...m fc
Hence by our inductive hypothesis, we see that the above ^-dimensional domain can be covered by at most
Q = c(e) fc (J-V) + l)1+EME
377 No. 4
HUA & WANG:
ON UNIFORM DISTE. & NUMERICAL ANAL.
499
parallelopipeds of the type PM. Using these P{£, as bases and 1 as height, we construct P« + 1 . Hence the sub-domain «ijt+1 = j can be covered by at most Q parallelopipeds of the type PM+1- Consequently, the domain defind by (i) can be covered by at most
2 2 c(s)* ([4-1 + l) 1+ V u
y= o
J
J
(6.8)
'
parallelopipeds of the type PM+14) Consider the sub-domain of (6.7) 2' I < mk+l < 21+II.
(6.9)
Then for mk+1 = 2':l + 1, we have
Hence according to our inductive hypothesis, the above domain can be covered by at most
°00* ( f )E
(6.10)
parallelopipeds of the type Pun* • Using the Pun1 as base and 2' as height, we construct PM + 1 . Since 2'' I + 2'' I + 1 > 2 I+I I, the domain (6.9) can be covered by at most cQe^l ( — J parallelopipeds of the type P&+1. Consequently, the domain defined by (ii) can be covered by at most Cl08 2 An
2c(e)* 2 ;=o
Mfj
(6.1D
\Z /
parallelopipeds of the type Pu+15) It follows by (6.8) and (6.11) that the domain (6.7) can be covered by at most
parallelopipeds of the type Pu+l- Hence the lemma follows by mathematical induction. Bemark. The term c(eyl1+eM* in the above lemma may be replaced by c(s)flogI~1 • ll2: 3lM . Lemma 6.2. Let T'M be the number of solutions of the congruence (6.1) in the domain (6.6). / / the congruence (6.1) has no solution in the domain (6.2), then we have T'M
Hence we have the lemma.
(mod n) .
378 SCIENTIA SINTCA
500
The Proof of Theorem 6.1. Take 38s = e.
y
_J_ J_ -y e2.«..-w. Hm|| » ^
W
Vol. XVI
By Lemma 6.2, we have
y
=
_X_< ^ W
,^»
(a, m)=0 (mod n)
= — TJ Tit1 (— - —^—) + ,
||m|| "" 2 J
T
- r{<)
JM
< c(syM-i+eKs.
»\
(T\t = 0). Take v = ^ and h = 7([M] + I ) 2 . Then the theorem follows by Theorem 4.1. The Proof of Theorem (6.2). Let u ^ o b e a solution of the congruence (6.4). If M, = 0 (1 < i < s), we obtain u0 ^ 0. Prom (6.4), we have u0 = 0 (mod m) . Hence the j|uji ~^n. Consequently, u is not belonging to the domain (6.5). Therefore we may suppose (wi, • ••,«,) ^ (0. • • •, 0) . If Ml
Mj::3
/ & \7PT (i+i)/(«+D w l2ds;
then it appears
hence we have the theorem. Now suppose
Since (a) - <^>, from (5.1) and (6.3), we have
J
^>(?>-e±^><Sr*H;s(i-r,W
Therefore I«Q[SI-
(2d»)^ ^
~**>\
n(l+')/
The theorem follows. VII.
THE PROOF OF THEOREM 3
We use the notations IQ) =
Jo
•••
Jo
^ x i> ' " ' ' x^
dx
i""-d^
379 No. i
HUA & WANG:
ON UNIFORM DISTE. & NUMERICAL ANAL.
501
and
Clearly, Theorem 3 is the consequence of Lemma 1.1 and the following Theorem 7.1. If (5.1) holds for any m =N= o, then Sup
feEfm Proof. sm
n
hsm-nS
| P _ ( / ) | < C • c(a, b, a, s) M<-«+'«<*-i»/Ca-i) ( l o g 3m)«+«•*,.„.
Since
s£C 1 (where h is a positive integer and 3 is a real number but not an
integer),
= c(o) + (^TT72'o(m)(Sse-™.-y = C(o) + 2 ' C(m) (^(2n+lMm,y)Y \ (2w + 1) sin7t(m, y) /
and C(o) = / ( / ) , then we have S
Sup |PO(/) I < C S ' ^
^ n ( ^ + l M " ' y ) , ° = C(Z, + 22);
llmll" (2n+ l)sm3r(m, y)
/«E,«(C)
(7.1)
q
where St denotes a sum in which the m's satisfy the conditions |m, | < « ' " ' (1 ^ i <1 s) and m ^ o, while E2 the remaining part. For a > 0, a, > 0 and 2 j
a
> < °°>
w e
have
i
i
Hence by Lemmas 5.1 and 5.5 it is found that ^ '
2-(2n+l)-
<
i
^_^_ ||m||-<(m iy )>-
(
2°(2n+l)"V
y
i
V
^ J _ s _||m||<(m,v)>/
< c(a, b, a, s) n-'+"i'-^a-D
Q o g 3M)-+«*lf..
(72)
380
502
SCIBNTIA SINICA
Vol. XVI
It is evidently
(7.3) Substituting (7.2) and (7.3) into (7.1), we have the theorem. VIII.
THE PROOF OF THEOREM
4
Use the notation
«.(/) = « « - | S/(4). Theorem 4 follows immediately from Lemma 1.1, Theorem 6.2 and the following Theorem 8.1. If the congruence (6.1) has no solution in the domain (6.2), then we have Sup | Q n ( / ) | < C - c ( « , e ) ' i l f - + < . Proof. Clearly, we may suppose that e < a — 1.
Since
- 2 / (**-) = - 2 2 C(m)e>"<~ -»'= C(o) + S ' C1^) — 2 w
e2 (
;=1
" ""°'/" = c (°) +
2'
(a,m)=0 (mod »)
CCm),
from Lemma 6.2, it is known
Sup \QM)\
/SEjCC)
2'
(a,m)=0 (mod ») ll«n||
i ^ ^ S ^ /=l
1
k'^;
"
= ^^(faTF)
(1 \l"
1 (Z + l )
a
Ti ~^r < c •"(«' e)'^~a+E —a
J'
x~a~ldx < -—\. la+1' IX.
The theorem is proved.
EXAMPLES
Denote
PSCD = K/) - First of all, we prove the following
S
f1 - ^ ) /Civ) •
2
^
381_ No. 4
HUA & WANG:
ON TJNIFOBM DISTR. & NUMERICAL ANAL.
503
Theorem 9.1. Let y,, • • •, y, be a set of real number such that 1, y t , • • •, ys are linearly independent over the rational field R. Then we have
Sup |P*(/) I < C (A' (W(n;
• • •, r.) - D,
Yl,
where
W(n; r i , • • •, y,) = ^ + 2 £ 2 (l - |-) ]I (1 - 2{r»i})2. Lemma 9.1. We have 60
p2itimx
=-=3Q-2{*})2.
S
6 Proof. Since [0, 3 P (1 - 2a;)2 e2"""1 dx = \ 6
J
°
fcs?-.
for m = 0,
for
therefore we have the lemma.
» ^ o.
The Proof of Theorem 9.1. Since
and
1 =
X^
• rnl
. -.N .
y , sin Q2& + smxS
1)JT5
k=0
where S is a real number but not an integer, we find
\
/sintter5\ 2 = I \sin.it8
I, )
2 (i - Jf)/Or) = ^ g (•-liDSc'Cn.v-*-^ = C(o) + -1 2 ' C(m) (™«*(»°.Y)y. m
\sni3i(m, y) /
Hence by Lemma 9.1, we have /(Kjco
«2
l|m||2 \smir(m, y ) /
w2 ^ ^ ||m||2
6 II
382 504
SCIENTIA SINICA
Vol. XVI
=^(f)'s i; (s-n a - 2{r^ -1) = C(TT( 2
(»- \iD-,i[ (l-2{r,i})2-l)
-c(y)'OF(»;ri,
•••,r,)-D.
The theorem is proved. By Wang 520 calculator, we obtain the following two tables: Table 1: s = 3 w(n;
n
V 5
" 1 , v'T, Vio)
Win; e, e\ «')
100
1.08877
1.10689
500
1.01351
1.00914
1000
1.00572
1.00294
Table 2: « = 4 m
JF(rc;
2 C O S - | J , 2 COS ^ J , 2 COS-^-, 2 eos | J )
Win; e, e\ e\ e4}
1000
1.03263
1.13899
15O0
1.02139
1.11848
3000
1.00887
REFERENCES
11]
Baker, A. 1965 On some Diophantine inequalities involving the exponential function, Can. J. Math., 17(4), 616—626. T 2 ] Halton, J . H. 1960 On the efficiency of certain quasirandom sequences of points in evaluating multidimensional integrals, Num. Math., 27(2), 84—90. I 3 ] Haselgrove, C. B. 1961 A method for numerical integration, Math. Comp., 15(76), 323—337. [ 4 ] Hlawka, B. 1962 Zur angenaherten Bereehnung mehrfacher Integrate, Mon. Math., 66(2), 140—151. 15] 1964 Uniform distribution modulo 1 and numerical analysis, Comp. Math., 16(1—2), 92—105. f 6 ] Hua, Loo-keng & Wang Yuan 1960 Remarks concerning numerical integration, Sci. Sec, 4(1), 8—11. 17] 1964 On Diophantine approximations and numerical integrations ( I ) ( I I ) , Sci. Sin., 13(6), 1007—1010. [8] 1965 On numerical integration of periodic functions of several variables, Set. Sin., 14(7), 964—978. I 9 ] Both, K. F. 1954 On irregularities distribution, Math., 1(2), 73—79.
383 No. 4 [10] [11] [12] [13] [14] [15] [16]
HUA & WANG:
ON UNIFOEM DISTE. & NUMERICAL ANAL.
505
Schmidt, Wolfgang M. 1970 Simultaneous approximation to algebraic numbers by rationals, Ada Math., 125, 189—201. Wyel, H. 1913 Uber die Gleichverteilung von Zahlen mod Bins, Math. Ann., 77, 313—352. BaxgajioB H. C. 1959 O npH6jnuKeHHOM BbiiHcjieraffl Kparawx HHTerpajioB, Bee. Moc. yn-ra., 4, 3—18. BHHorpaflOB H . M. 1971 MeTOA TpHroHOMeTpuqecKHx CyMM B TeopaH iHceji,
[17] # [ B # * 1973 £%.&mmm&mmvr, * s ^ t t CHX). Note added on the 19th of May, 1973. A theorem similar to Theorem 5.1 was proved independently by H. Niederreiter and some results given in [8] were improved by S. Haber (See [16]). In the course of publication, we add two more new books, [16] and [17].
384 SCIENTIA
vol. x v n No. 3
SINICA
Jtme
1974
ON UNIFORM DISTRIBUTION AND NUMERICAL ANALYSIS (II) (NUMBER-THEORETIC METHOD) HUA LOO-KENG ( ^ j p g l )
AND WANG Y U A N ( ] £ % )
Received June 13, 1973.
ABSTRACT
In this paper, we shall give some applications of the sequences of sets defined in a previous paper to the problems of numerical integration, interpolation, and the approximate solution of Fredholm integral equation of the second type. We have studied in addition the well-known sequences of sets introduced by Kopo6oB. I.
STATEMENT
OF RESULTS
Let /(JC) = f(%i, • • •, £,) be a periodic function of s-variables, each with periodic 1. Let a = («i, • • •, ffj) be a vector and let py, = fa = 0 for osfc = 0, where p^ represents a non-negative integer and 0 < fa ^ 1 for afc = pk + fa > 0 (1 ^ k < s ) . Define Stf
= ( 2 O ~ ' ( / ( z i > • ••, x* + h , •••, x s ) — f ( x u • • •, x h — h , • • •, a ; , ) ) .
Suppose that the partial derivatives d^-^'f dx[' • • • ax/
=/(jf yr 1 ,-,r / ) >
(o
l
exist, which all are the periodic functions of s-variables, each with periodic 1. Denote
i i / i = sup n ((*r p **j fc )/) (p -••••"'. 0
a
^
o
The class of functions / satisfying the following conditions ||^<W-.<W|| < A is denoted by S'QA), where 6lt • • •, ds = 0 or 1 and A ( > 0) is an absolute constant. For the special case » ! = • • • = « , = «, we denote the class of functions by !!*(£). The inventions and notations introduced in [ 1 ] are also used in this paper. Applying the sequences of sets (1.1) and (1.2) of [1] to the problem of numerical integration, we have Theorem 1. Let a he a number satisfying 1 ^ a > 0. Then
385 332
SCIENTIA SINICA
sup
1tH*tA->\>°
• • • /(a;,, • • •, xq)dxx • • • dxq
n
•>"
Vol. XVH
2 /( w « 3' ' " > w"^
1= 1
(1.1)
Theorem 2. Let a be a number satisfying 1 > a > 0. Then
^jj;-j;fe,-,^,-*,-|s/(t,^-,-,^)| < Ac(%, a, s)n~f ""S"+£ .
(1.2)
Next, we use the sequences of sets (1.1) and (1.2) of [1] to treat the problem of interpolation and the problem of approximation solution of Fredholm integral equation of the second type. Moreover, we have studied the uniformly distributed sequences of sets
({*}.{#•••••{#)• o " " ^
<">
( { * } • # } • - • { ? } ) • <'<>«"•
«•«
and
where p is a prime number and a = a(p) is an integer depending on p . The two sequences of sets were proposed first by Kopo6oB in 1959 and 1960 respectively"'61. Theorem 3. Let p be a prime. Then
d / ^J{;-j:^'--'^--|§ (i'l'--'l)l / Ac(a, s~)p-1/2, for 1 > a > —, 2
<
a
l+J
Ac(a, s)p~ (log p)'- ./'.«, for
1
a
(1.5)
-^> >°-
Besides, we propose the sequence of set
(1.6) and apply it to the problem of numerical integration. Consider the Volterra integral equation of the second type
(1.7)
-i-, for « > 1, »(«) =
_ 2« -• , .
(1-8) -r, for 1 > « > 0,
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9 = [p*'%
(1-9)
0°U>,
1( 2P
L
log2 log2 3p J
and
f
1
(l-io)
X I* [*'••• f*'~' e 2 " ( " I ' J: ' + - +m ' A: ' > ri!2; 1 • • • & , . Jo Jo
(1.11)
Jo
Theorem 4. £e# p be a prime and a be a positive number. Then there exists an integer a such that the solution of Eq. (1.7) can be written as
x
< £ f) • •K (*?• *?)' (?) + 0 ( "-"" ) '
(1 12)
-
where the constant implied by the symbol "0" depends only on a, A and e. In order to obtain the integer a = a(p), we require cCs^p2 elementary operations. For the proof of Theorem 3, we require the well-known result of A. Weilt2] concerning the estimation of exponential sums. Lemma 1.1. Let p be a prime and let m,, • • •, ms be a set of integers, at least one of which is not dividing by p. Then it is seen that p
^
e2*Hmij+~+m,j>)/p
^
(S — 1 ) V P .
)=1
The estimations given in Theorems 1 and 3 for the case 1/2 ^ a > 0 are the best possible kinds, apart from some possible improvements about the order (log p)'~l+s'-« (cf [ 1 ] ) . Previously, the formula (1.5) was proved under more restrictive condition a > 1/2 (see Kopo6oB[51 and llIaxoBc']). The Theorem 4 gives a slight modification to the result obtained by IIIaxoB18'91. In his original result, the error term is O(p 2 an integer ^ 2. II.
2
*), a being
THE CLASSES OF FUNCTIONS
Let c be a constant such that 0 < c < 1. Let pix)=
cos2 (— log2 — IV for — < |g| < 2 c , V
2
c l /
2
0, otherwise and let
toix) = 1 - S thix), » = i
where fit(x) = fj.(2l~'x)
(t > 1).
(2-D
387 SCIENTTA SINICA
334
Vol. XVH
Let / ( * ) be a periodic function of s-variables, each with periodic 1. Suppose that /(je) has the Fourier expansion
For a vector t = (f1} • • •, ts) with non-negative integral components, we define where C f (m) = C O ^ C m , ) • • •
ftl(«,).
(2.2)
Let a = (cBt, • • • , « , ) be a vector with non-negative real components. The class of functions / ( * ) satisfying
IMI = sup|(pt(*) I
is denoted by Q°(B), B(> 0) being an absolute constant; it is so especially, when denoted by QfQZ) for the ease ffj = • • • = as = a.
Lemma 2.1. Let a>0.
Then
fl?U) e QXA • e(«)') C # ? U • c(«)'). Lemma 2.2. Xe* a > 0 and /(x) € QX-B). Tfeew it is inferred fix) = 2
V(*),
wfeere 2 " denotes a sum in which t runs over all vectors with non-negative components. See BaxBajiOBt3>/|].
III.
THE
PROOF OF THEOREM
1
Denote
«.(/)
= «/) - - S /(jy).
Evidently, Theorem 1 is the consequence of Lemma 1.1 of [1], Lemma 2.1 and the following Theorem 3.1. Suppose that 1 > a > 0. 7/ #g. (5.1) of [1] Mds /or awj/
m ^ o, then
sup |i? n (/) I < B c ( a , b, a, s>-"+'(<'-1>(log 3w)'+'*>.«. Proof. By Lemma 2.2, we have BnCn = S " « . ( v i )
(3.1)
for / € Qi(.B~). It follows by the definition of q>, that where
|B.(
(3.2)
*0 = *. + • • • + * , .
(3.3)
388 No. 3
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From Lemma 5.1 of [ 1 ] , we have
|B.(v.)KS'|C.(m)|i S eJ'«-T» j=l
"
^ n - ' S ' l C / m ) ! ..
1
...
(3.4)
<("», y)> Hence from Eqs. (3.1), (3.2), (3.3) and (3.4), we have sup | B . ( / ) | < sup S"|B,(g>,)| < S , + 2L,
(3.5)
where J-" l s > | C « ( m ) |
Si==sup
/«o;w . o ^ .
w
<(»»,y)>
and
S 2 = 2J5 2 " 2-"o. Since it follows from Eq. (2.2) that Ct(m) = 0 for ||m|| ^ 2'», therefore by Lemma 5.5 of [1], Lemma 2.1 and Eq. (2.1), we have
2 L < sup n-1 2 ' .. M
iimiK* \ \
2 m
,.S"|C t (m)l
>y)/
. _! y » |C(m)| < sup n ' 2-i ,( N\
' " ,,Sj|m||«<( 1 m , y) >
'
, , S ' n ll»ll<(Z,r)>
< Bc(a, b, a, synT'^'-VQog
3w)I+5*..«.
(3.6)
Since t h e n u m b e r of non-negative solutions ( i 1 ( • • • , ts) of t h e Diophantine equation ( 3 . 3 ) is
(to+s-l\ \ ,_1 /
jto + s-Dl ^ *o!(*-Dl
c ( 0 < r
therefore 00
S 2
2-"f-I<J3c(a,s>-'"(log3M)J-1.
2 «=[log2»] + l
Substituting (3.6) and (3.7) into (3.5), we have the theorem. IV.
THE PROOF OF THEOREM 2
We use the notation
where a = (Oj, • • •, a,) is a vector with integral components.
(3.7)
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Evidently, Theorem 2 follows immediately from Lemma 1.1 of [1], Theorem 6.2 of [1] and the following Theorem 4.1. Let a be a number satisfying 1 ^ a > 0. Let n be an integer ^ 2 and M be a number Ss 1. If the congruence s
(a, m) = 2
a m
i=l
< < = 0 (mod n)
(4.1)
has no solution in the domain (4.2)
\\m\\ t^M, m*o, then we have
sup | #„(/)!
have
Proo/. Clearly, we may suppose that e < a. In a similar manner to (3.5), we sup |flf,(/)| < S i + S 2 ,
(4.3)
where S i = sup 2 " I £.(9«) I > and
S 2 = sup
2 " I^»(«P«)I-
By (3.2), we have 2-°'o<2B 2 " 2-<"-')'o-»o
S1<2B 2 "
» 0 >log 2 M
r 0 >log 2 M
+£
(4.4)
Sinee C,(/n) = 0 for ||m|| > 2'», and l-8f.(Vt)|= S'C,(m)-l-S« a " ( " l l l ) i / " < W
S'
i =l
(«, /JI)=0 (mod B)
2'
S"|C«(m)|=0.
I
therefore it appears
S 2 < sup
/«5^(B) («, m)=0 (mod ») »0«log2M
Hence the theorem follows from (4.3), (4.4) and (4.5). V.
T H E PROOF OF THEOREM 3
Denote
««-«»-i,±'(i.f-••£)• By Lemmas 2.1 and 2.2, we have
(4.5)
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sup |T,(/)| < S , + 22,
(5.1)
where ffHf(J)
»0>log2p
and
S 2 = sup 2 " |r,(?»t)|. ftHf(A) <0
It is similar to (3.7), that 2 a i
Sx<2 sup 2 " IWI<4e(«,0 S ftHf(A)
»0>log2j>
~ '* "'
»=[log2<>]
< Ac(a, Op-Clog p)'" 1 .
(5.2)
Let m ^? o. If one of the relations |m,| < 2'« (1 <J i ^ s) is not satisfying, then it is known that C t (m) = 0. If |m,| < 2'' for all i, where 1 ^ i ^ s, and #0 < log2 p> then p can not divide all the m'iS. Hence by Lemma 1.1, we have
5L< sup 2 " S'|C,Cm)| — S e^"'^"*-^' f*Hf(A)
t0
P
2<>
i=\
S ' l0,Cm)|.
|m.|<2',-
Consequently, by Schwarz inequality and
ll9.lk<-llq»«IK^Ca,02-'»,
where
l
S2 < C* - 1>-1/2 sup S " ( S l)* ( S f^HfCAU^log^
< (« - 1>-1/2 sup S "
im.l<2'i
2 2
V
'
\C,CmWf
| m .| <2 '<-
y
llVtlk
f(fj(A1 t0
< 4c(«, S)p-]/2 2 "
2-(a4>0
r^logjp
= Ac(a, s)p-l/2 ^
2"1" ^ ' C* + D'" 1
1=0
Ac(a, s)p-1'2, for 1 ^ a > —, <•
(5-3)
1
s 1+
Ac(a, s)p~°(logp) " *i. = , for
y ^ « > 0 .
By substituting (5.2) and (5.3) into (5.1), the theorem is immediate.
391
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Remark. For the case a > 1, we propose to use the uniformly distributed sequence of sets (1.6) and the following result holds true: Let p be a prime and n = p 2 . Then we obtain
sup
|/(/)-iii:/^f,.--,f)j
Proof. Evidently, we have for / € E?(.O that
I v v f f l *i1 ... o£iT\ n^i^'lp'p ' P ) p2
o=l / = 1
l
= /(/)+ P~ S' C(m) n^H
S
1.
f-ro^^EO (mod p)
If at least one of m's is not dividing by p, then the number of solutions of the congruence ml + • • • + m,i'"' = 0(mod p) (1 < j < p) is at most s — 1.
Therefore
II
The assertion is proved.
"
f»,+—+mJ;f~%l) (mod p) i
< C • 2SP"1!; —^—- < C • c(a, s)n-i. llmlr VI.
INTERPOLATIONS
We use the notations A = sup
/ ( * ) - ±-
2J
/ (-^-J
2 J
e
Theorem 6.1. / / ^ e congruence (4.1) te mo solution in the domain (4.2), then it yields A < B • si 1 / 2 c(«, e)'i!f"l<'I>a+e,
where «, = [M" Ca) ],
in which i>(a) is defined by (1.8).
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339
Lemma 6.1. Let 1= (llt •••,£,) be a vector with integral components such that \\l\\>
3'. Further let N satisfy 1 < N < ^j~ . Then
1|B^N
<
\\l+m\\'
. V_Z!_ *! c(«) || ; ,| a , for « > 1.
Proof. Suppose first that 1 S* a > 0.
I
y
£& (ir+^D-
Since N < — for s = 1, we have
<(*-)'™-. v
2 / it
Thus the lemma is true for s = 1. We suppose that k is a positive integer and that the lemma holds for s = 1, • • •, k. Now we proceed to prove the validity of the lemma for s = k + 1. Clearly, it follows from m,- • -my+i < N < ' ' 'fc'+f+1 that there exists at least an - I m; such that m,- < —L, where 1 ^ i ^ & + 1.
z
2
— = —
1
3
Hence
— < s , + ••• + s*+i,
where a.-ffl^J.
1)
Suppose that JV <
s
2
" fc k+1 .
Then by our inductive hypothesis, we have
. < w t l jg2..«fc+1<^. ( ( ^ + m 2 )- • -0k+1 + m fc+1 )) a
(^i • • •
2)
~~~ ^
Suppose that iV >
?2
h+i)"
' ' ' fc ^ +1 .
Then
£2
0-2,
where ff =
i
2" 77
h _
-v1 ZJ
3fcy
\-> ZJ
a
—=rzzz=
1
H f c + I < ^ . (S.h + m,) • • • (ifc+i + m f c + 1 )) a
,
393
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SCIENTIA SINICA
Vol. XVH
and °2 = 17 h
ZJ
=
2-I
7,
3icN
_g_ ( ( j 2 + m 2 ) . . . (j f c + 1 +
B
•
ms;+1))
a
•
It is evidently known that <J
<2!
V
(w, ' • • wfc+1)'-°+6
V
^ c(a, e)kN1-"*
y.
Ui
(m,)°
h+i)
By our inductive hypothesis, we have
^ a • • • k«)"
'
3) From 1) and 2) we obtain
Since the 2J' S satisfy the same relation, hence the lemma follows by mathematical induction immediately. We may treat the case a > 1 similarly. The lemma is thus proved. Remark. For the case 1 Js a > 0, the conclusion of the above Lemma may be replaced by
,£iN
m+my-
<s!c(o
°
inn- •
Lemma 6.2. Suppose that Q ^ 1 awd 1 > a > 0. / / Wie congruence (4.1) /ias no solution in the domain (4.2), then («, m)=0 (mod »)
|| Will
Proof. By Theorem 8.1 of [1], we have («, m)S0 (mod „)
|| tn ||
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Consequently we note ]>]'
1
(«, m)=0 (mod „) !im :'(<£>
\\ m ||°
^ Ql-a+s
V'
!
(a, m)=0 (mod „)
II " * ll
Take
for « > 1,
[[log2 MwrI+T] + 1, for 1 > a > 0. By Minkowski inequality, we have A < A 1 + A2 + A3,
(6.1)
where sup
At= A2 = sup
/ ( * ) — 2 "
2LJ 1 "?'(*)
Z J «?«I
) ZJ
e
»
and
A3= sup ^ S / f ^ ) S . 1 " ( - " J f )
- 2" | ± *.(•£) S .-'<—^->|| W
» 0
1)
N n
/=1
L
' llmlKs,
:
By Minkowski inequality and Lemma 2.2, we have
A, < sup 2 " H^IL, < B 2 " 2"<"».-. /ee°W
» 0 >r
» 0 >r
(
< Bc(a, e)' 2- "2)
£)T
< Bc(«, e)' ilf-»w°+«.
(6.2)
A2<^cr1 + (T2, where
ff!= sup
2 J
ZJ
( ^*( m )
ZJ
)e
j
6
and
/eO°(B) r0
S l|mll>«,
Ct(m)e2'^
. ^
Since
C(«) - J- S Jj±)e-2<- ^ = and
n
j =1
\ n /
2 u,/)=o (mod»)
|| 1|| < 2 ' | | m||- | l ' + m i l , therefore we have
0.(1 + «)
395
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SCIENTIA SINICA
^< sup
2"
f(Q°(B)
2
2'
llmlKfl!
to
= sup S <^ c («y
s
CtU+m)e>""-'>\f "LJ
U , l)=0 (mod „)
fe"
1(0°(.B) \\m\\
Vol. XVH
2'
10,(1+m) | Y
(«,7)=0 (mods)
(
2'
'
in',,.)'
IIZI!<2'+rBl
2'
II m !!<•>! (a,/)=0(modo) \IIII
Evidently, we may suppose that nt < — . o Lemma 6.1 and Lemma 6.2 that 1
... • 1 M ||, || 1 "t" HI ||
2'
-ji4p-
(«,/')=0 (mod») ]ll'l\<2t+T»,
|| « II
Hence it follows by Theorem 8.1 of [ 1 ] ,
(B2s\ c(a, e)'
for « > 1, for 1 > a > 0.
By Schwarz inequality, we have
a\= sup
£
< sup
S
( 2 " |0,(m)|Y 2" 2 ~^2"2^|C,(m)|>
U0f(B1 llml^a, »0
2
«0
ro
llatl^j,,
2^|C,(m)|2
S " 2 ^ || ^Hi, _
i!°
log2 »,
Therefore A2 < B • s\in c(a, e)' if-"»'« +t . ON
3)
»
II ^ T ' 1 K~<
/ ?O!\
A3= sup 2 J - S ? ' - ^ /«r
w
j=l
X~»
2
(6.3)
e l*i(m,x-—)
\ « / || m ||< O l
tj
<2" / X ll ^' ll ( S i)*<*S"2-.( 2 fo>T
'«",(»
^llmlKn,
< B c(«, e)' nj+? 2~aT+^
/
»0>T
X
lln.ll< B ,
< B c(a, e)' M-" (a ' o+£ .
The theorem follows from (6.1), (6.2), (6.3) and (6.4). Remark. The conclusion of the above theorem may be replaced by A < B c(a, s) M-V(a)a (log 3M)"-(a),
4
^r) II "» II
'
(6.4)
396 No. 3
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where
I
s — 1, (1 - 5a + a?Xs — 1) + 2«2 dl a •-, 1 + 4 a _ ^
343
for a > 1, forl^«>0.
Now we apply the sequences of sets (1.2) of [1] to the problem of interpolation. We use the notations h = (1, \ , • • •, hg) where (h1; • • •, hq; n) denotes the set of integers defined by (1.2) of [1]. Then we have Theorem 6.2. We have
A = sup f ( , ) - i j / (•&) 2
e-1"^ e*«»->
-»(a)a(j+l) j_r
M
2
].
,
where M,
= [n
VII.
«
CONTINUATION
Theorem 7.1. Let p be a prime and n^ = [p"
sup
Then there exists a vector
2 ,"•"<-^~<--»||
Kx}-±±f(J±)
< B • S!^(a,e);p-"('1'o+'. Proof. By Theorem 6.1, it is sufficient to prove that there exists an integer a such that the congruence (a, m) = mx + m2a + • • • + msas~l = 0(mod p)
(7.1)
has no solution in the domain
II m ||
(7.2)
where
Evidently, we may suppose that c(s, e)p'~s > 1. Take a vector belonging to the domain (7.2). Then the number of solutions of the congruence (7.1) is at most s — 1 in the interval 1 < a < p. Consequently, the total number of solutions of the congruence (7.1) in domain (7.2) and 1 ^ a =SJ p is at most
2J
2J
llmll
K(«-I) ^
v
'
2-*
l|m|1
|lm!!
<( s -lM,, e ) ! ,(_S s L-)-<|.
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Hence, there exists an integer a satisfying 1 < a < p such that the congruence (7.1) has no solution in (7.2). The theorem follows. Theorem 7.2. Let n be an integer > 3. Then for any integer satisfying n > Mi > 1 and any vector with integral components a = (au • • •, as~), we have
A- sup
/Or) - j-± f (if)
£
e-M^e^\\
>-±—n--<>.
Proof. Evidently, we may suppose that ^ = — 1 and (a 2 , n) = 1. Let -2s denote the i-th convergence of •£* = —. 1 = g0 <
Let
Suppose that «; satisfies
' • - <
" " # < In = «•
-^ = (p*8* — 3iPfc) •
It then follows
\K\ < - ^ - < — . Take a function of E^A): / g2«iCKar,+)rj)
A
/(:
+1
4(23r)» \
|Z|«
„— 2»
+
\KY
gJnifBj+Dr,
+
(«i + 1)'
g— 2»i(n 1 +l)i J \
+
(«. + ! ) " /
Then we have A2
>
S
S'
(CCZ + m,, g-fc + m2) + C ( - Z + m,, - gfc + m2))2
> _S + 2(
Wi + m1; 12 + m2) 2 + 2 (—-^— r ) J (Wl + 1)-"
j-
)2 («, +1)- 2 - > CCK, iy + C(-K,-
iy
The theorem follows. Remark. It follows from Theorem 7.2 that Theorem 7.1 does not permit further substantial improvements, apart from some possible improvements about the order p* for the case a > I . VIII.
THE APPROXIMATE SOLUTION OF FREDHOLM INTEGRAL EQUATION OF THE SECOND TYPE
For simplicity, we use capital Latin letter to denote a vector of s-dimensional space. Now we study the approximate solution of Fredholm integral equation of the second type
398 No. 3
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= i [ JT(P, 0M0)d0 + / ( P ) ,
345
(8.1)
where /(P)€B,'U) and K(P, Q) € fliU). Let
denote the Fredholm kernel, where Z
( n " ' "' n") =
det( X ( P ;
'
Q
^'
(1<
*' J' <
v)
•
Further let A( A) = det (»„ - A- K(Mi, M^),
(1 < », j < n).
We assume, moreover, that~D(X)^? 0. Theorem 8.1. If the quadrature formula sup
If F(P*)dP - — 2
K ^ j ) < Ac(a, s)
(8.2)
^oMs w^A,
(1 < i < ») •
(8-3)
Tfee« #ie solution of the equation (8.1) cam 6e written as V (P)
= /(P) + 2 Z(P, MjWMi) + O((p(n)), i=l
wTiere tte constant implied by the symbol " 0 " depends o?% ow A, Z' awd /. Lemma 8.1. Let a,-,- 6e real numbers and let l,(ft) = det (a;,),
(1 < i, j < n),
where a'H = 1 + o,-,-(l < * < fc) « ^ otherwise a'a = aH. Further let AJJt) least upper hound of the absolute value of An(Ji) under the condition
denote the
where r is a constant. Then there exist two constants Yi ^nd y2 such that for any positive integer n, we have Ajjn - 1) < — and An(n) < r2n
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Lemma 8.2. / / the integral formula (8.2) holds, then there exists constant n0 = no(X, A, a, s), such that for n > n0, we have
\AW\>±-\DW\. See Kopo6oB[7].
Lemma 8.3. The solution of Eq. (8.1) belongs to H?(A'^), A' being a constant which depends on X, K and f only. Proof. From
we have
| ( v ( P ) - / ( P ) ) ( - 9 - - - - 8 ' ) | < sup \K(P,QyS"--°^\
\i\ \
Hence
where 9U • • •, 9S = 0 or 1.
\
(1 < j < »).
From (8.3), we have the system of linear equations n
Zj = 2
7c = 1
where
afkzk 4- 6, ( 1 < i < n),
a, =
{- j^KCM,,
M^), • • -,1 - j-K(Mk,
Mh-), •••,
with Then we have 2 =
'
A,(A)
A(IT' ^ < i < w ) -
-j-K{Mn!Mk-)J
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As n sufficiently large, we have
|AO0| >-i-|DU)| > 0 by Lemma 8^2. Further since n
n therefore by Lemma 8.1, we have
|A,| < |6,B,| + 2
Kfc<»
n
I&AI
where Bfc is the cofactor of bk in Aj.
fc=l
Hence
«, = 0(
( K j < n ) .
The theorem follows. Let
*<-({*}•{*}'-•{¥}) «<'<•' be the sequence of sets defined by (1.2) of [1], Then from Theorem 4 of [1] and Theorem 8.1, we have Theorem 8.2. Suppose that s = r. Then the solution of the equation (8.1) can be written as 9(P)
= / ( P ) + - ^ 2 K(P, M,)
where
T H E PROOF OF THEOREM 4
The solution of the equation
y)
(9.1)
(9.2)
is given by the Neumann series
where
(
(xv— i
x Cx1
o Jo
•••
Jo
^ ( x ; a;,, • • •, x^)dxx • • • dx»,
flva+1(2("+1)("+1)4'-1-1).
xJ)f(%J).
401
348
SCIENTIA SINICA
Hence o
•••
Jo
Vol. XVU
<&,•••<&„< -
v\
— .
Consequently there exists
± 9,GO < ± r^^<
ciA^^eUae)p_,a)a+,
(9 3)
Let P fr[ X
y: j *
2.]
X^
\
P)
\P
PJ
e-inUm1+mJa+—+mva"~')Ut
\ P
P I
€2*i(.m1xl+—+mvx
Then it follows by Theorem 7.1 that there exists an integer a such that q>X%) ~ \ • • • 1 Jo Jo
Gj&Xy • • • dx» ^ \ • • • I Jo Jo
| G, — G» | dxt • • • dx»
< ([" • ••["'"' dxr • -dx^WG. - ff-Ht, < Av+1c(a, eyp-»w°+' (Kv
(9.4)
The theorem follows immediately from (9.2), (9.3) and (9.4). Bemarks. 1. The present method can be used to treat more general integral equation of the type
'• • • I Jo
K(xu
• ••, xs+i, yu
••, y s + d d y l • • • d y s + l + f ( x l t
••-,
• • •,
y!+l)
»,+,),
where s > 0, I > 1, f 6 H? +/ (A) and K 6 HL+«(A). 2. The present method can also be used to treat the Fredholm integral equation of the second type when X is sufficiently small. 3. The sequences of sets defined by (1.2) of [1] cannot be used here, since we don't know the explicit relation between r and c(9t r , e), where c(JRr, e) is the constant appeared in Theorem 2 of [1]. REFERENCES [ 1] [2] [3] [4]
[5] [6] [7] [8] [9]
Him, L . K . & Wang, Y . : On uniform distribution and numerical analysis, ( I ) , Sci. Sin., 16 (1973), 483—505. Weil, A . : On some exponential sums, PNAS, U S A , 34 (1948), 204—207. BaxBajioB, H . C : TeopeMU BJioHcemiH flJia KJISCCOB (JiyHKrmft c HecreuibKHMH orpaHiweHHbiMH npoH3BOflHHMH, Bee. Moc. yn-ra, 3 (1963), 7—16. BaxBajioB, H . C : 0 6 onrHMaJibHUX oueHKax CXOAHMOCTH K Ba^paTypHbix npoueccoB H MeTOflOB HHTerpnpoBaHHH THna MoHTe-Kapjio Ha K^accax (JjyHKUHS, Ron K omyp. euH. Mar. u Mar. tpu.3; U3d, «HayKa». Moc. (1964), 5—63. KopofioB, H . M . : ripH6jiHM<eHHoe BbWHMeHHe KpaTHbix HHTerpajlOB c noMOmbro MeTOAOB TeopHH <ween, UAH CCCP, 115 (1957), 1062—1065. Kopo6oB, H . M . : O npH6jm>KeHHOM BbiqacjiemiH KpaTHbix HHTerpajlOB, UAH CCCP, 124 (1959), 1207—1210. Kopo6oB, H . M . : TeopeTHKO-iHCJioBbie MeTOAbi B npnfijiHJKeHHOM aHajiH3e,
402
S C I E N T I A
Vol. XVHI No. 2
S I N I C A
Mar. - Apr. 1975
ON UNIFORM DISTRIBUTION AND NUMERICAL ANALYSIS (III) (NUMBER-THEORETIC METHOD) HUA LOO-KENG (ffSJ^gg) AND WANG YUAN (3£ 7 6 ) Received Jan. 3, 1974.
ABSTRACT
We present in this paper, a study of the uniformly distributed sequences of sets defined by elementary symmetric functions of roots of /GO = 0, where f(x) is the minimal polynomial of a Pisot-Vijayaraghavan number (cf. Jaeobi-Perron algorithm). We obtain the estimations of their discrepancies and then apply them to the problem of numerical integration. For practical purposes, two suggestions concerning the polynomials /(a;) are given. I.
INTRODUCTION
The Pisot-Vijayaraghavan numbers (PV number) are those algebraic integers co > 1, whose conjugates other than co itself lie in the open unit circle (Cf. [2] Chap. VIE). Let to be a PV number of degree s ( ^ 2) with conjugates co(= wm = eoi"), to™, • • • , a>w so that «>1,
|co«| < | i » « | < • • • < |co w | < 1 .
(1.1)
Further let u> satisfy the irreducible equation / ( w ) = 0, where a0, au
•••,
fix) = x! — «,_!£'"' — • • • — axx — a0,
(1.2)
af_t are integers.
The conventions and notations introduced in [3, 4] are also used here. Theorem 1. Let b = (&<,, h, •••, &*-i) be an integral vector ^ o. Let ( § „ ) • ( = ( ( $ / ' ) ) be a sequence of integers defined by the following recurrent formula (0
Qi-h
Qn = a,_, gfl_, + as-2Qn-2 + • • • + a,Q n - s+l +
(1-3)
Then for any given large number N, there exists an % = m0(co, b, iV). When n> n0, it holds true that \QJ>N (1.4) and
^ - c o
(1.5)
where P
=
-
logco
•
(1
-6)
403
No. 2
HUA & WANG: UNIFORM DISTRIBUTION & NUMERICAL ANALYSIS
185
It follows from Theorem 1 immediately that -&±>L - to* < c (a.,b)|10 B |- 1 -' >
(l
(1.7)
Since 1, w, • • •, u'~l is linearly independent over rational field R, hence by the Theorems 6.1, 6.2 (with a slight modification) and 8.1 of [3] and Theorem 4.1 of [4] we have Theorem 2.
The sequence of sets
({4-}' {-%f4 • •'' {j21f1}) a < * < KM,«>«.)
(i.8)
is uniformly distributed and has discrepancy
(1.9)
where e is ant/ pre-assigned positive number. Theorem 3.
Suppose •••
Sup
that a > 1.
27ie«
/ ( # ! , • • •, xs')dx1 ••• dxs
lo.l £iT\Q.'
'
Q. '
< C • c(w, b, a, e) 10. | "^"T- +E , Theorem 4.
Suppose that 1 > a > 0.
(n > % ) .
(1.10)
T/iew
• • • f /(a;,, • • •, icjcfo;, • • • dx,
Sup ftH°(.A)
Q. /I
I •>»
JO
10.1 i ^ T\Q.'
Q.
< A • c(», b, a, e) |
' + E
' ,
Qn ) \ (»>«„).
(1.11)
For practical purposes, we give two suggestions. tion
1) Take a0 = at = • • • = as-x = 1. We denote the largest real root of the equaF{x) = x' - x'-1 - • • • - x - 1 = 0,
(1.12)
by TJ(= JJO' = ij«>) and its other roots by rf2\ ••-, rf'K Let ( F n ) ( = (*£>)) be the sequence of integers defined by the recurrent formula Fo = F, =
F _ 2 - 0,
F _ , = 1,
Fn+S = Fn+S-X + F n+J _ 2 + • • • + F B+1 + F o ,
(« ^ 0).
(1.13)
As usual, (P n ) is called the generalized Fibonacci sequence of dimension s. From the
404 186
SCIENTIA SINTCA
Vol. XVTII
theory of Jacobi-Perron algorithm, we have U m Z»±i. = »•»»
(1.14)
v
Fn
(Cf. [1] Chap. 7). In this paper, we shall prove that rj is a PV number and -Z2+1. — ^ < c(jt)FZl"vt^~~^,
(1.15>
(n>s).
Consequently, if we take ((?„) == (F B ) in Theorems 2, 3, 4, then the right hand sides of (1.9), (1.10) and (1.11) become c(rf)F»2
*+1i°*>, C • c(n, a~)Fn2 *I+ii°<* and
a
_a
A • c(.v, a)Fn^ 2'+11°*2 respectively. 2) Take a0 = 1, ax = • • • = a,_2 = 0 and a,-! = L, where L is an integer ^ 2. "We denote the largest real root of the equation G(x) = xs — Lxs~l — 1 = 0
(1-16)
by r ( = rCl) = T' 1 ') and its other roots by r(2), •••, r w . Let ((?„)(= (G1!")) be the sequence of integers defined by the recurrent formula Go =
GU = 0,
(?„+, = L
(?,_, = 1,
(» > 0 ) .
(1.17)
In this paper, we shall prove that r is a PV number and -^±i_ Gn
t
< c ( r ) GC1"^"1" u-i)lio,t. ~u-i]L<+3, (jn>$).
(1.18)
Consequently, if we take (Q») = {Gn} in Theorems 2, 3, 4, then the right hand sides of (1.9), (1.10), (1.11) become c ( r ) ^ 7 ~ i ^ + 1 7 = I ^ F r , C • c(r,
a)G~J~1^5+1J=^
and A • e(r, a ) ^ 1 " 1 5 ^ + a " o t logL respectively. The results given here are slightly rougher than the corresponding results obtained by the real cyelotomie field 9i, in the previous papers [3,4]. However, compared with (.h, • • •, hs; n) in [3, 4], the magnitude of elementary operations required for finding ((?„) is decreased. Finally, we shall give some numerical results obtained by Wang 520 calculator as an example Sup
/eE2(c)U0
•••
JO
/ ( z j , • • • , x^dxi
•••
dx4
_ J _ V / (J-, Z-fc E^L Iz£) < 0.0054C, F19£i
\ Fl9'
F19 F19 Fo 1
where Fl9 = 10671, F20 = 20569, F21 = 39648 and F22 = 76424.
(1.19)
405
No. 2
HI7A & WANG:
UNIFORM DISTRIBUTION & NUMERICAL ANALYSIS
II.
T H E PROOF OF THEOREM
187
1
1) Elementary Symmetric Functions. Let 8, = co™ + aP» + • • • +
M
w/
,
1 = 1, 2, • • •.
(2.1)
It is known that Si can be evaluated by the following recurrent formula of Newton 51 = as-u
52 = 0 , - A + 2a,_2,
(2.2)
#, = a,-i#,_i + as^28s-2 + • • • + a ^ ! + sa0 and £„ = <»,_,£„_! + aJ_2-Sffl_2 + • • • + o^,,-^! + ao8,-s
(2.3)
for « > s. It is evident that
| s . - » - | < S l» ( 'M"<«-i1=2
Hence we have \Sa\
= |fif. + 1 -a.flf.||S.|-'=
S »«•(«'» - co) |flf.|-i ,-=2
< (co + l)(s - r)\aM\*\Bn\^ < (» + 1)(« - I V I S . | -'2) Suppose That the Initial non-negative integers and
p
;
= (co + l)(s - Dm-'-ISj-' (« > «„).
(2.4)
Vector b = ( 0 , • • •, 0, 1 ) . L e t llt • • •, I, be a set of co(»'.,
A(J 1; • • • , O =
u>V>\
••-,
toM't
.
(2.5)
co(1)S coC2>S •••, cow'^
Let
(2.6) Then P ; (s — 1) is an algebraic integer since AQl, s — 2, • • •, 1, 0) may be divided by A(s — 1, s — 2, • • •, 1, 0). Consequently Pt(s — 1) is a rational integer since P,(s — 1) is a symmetric function of co(", • • •, w^. It is obvious that P 0 (s - 1) - P,(j - ! ) = • • • = P ^ / s - 1) = 0,
?_,(« - 1) = 1.
(2.7)
406
SCIBNTIA SINTCA
188
Vol. X V m
Since a(.On =
a«)«-/u(/).
=
Bw»-((flj_iU(n<-i +
as_2ClJ<.D>-2
+ ... +
aiiau)
+ fflQ-)
= a^^ 0 '"- 1 4- a,_2co°')B-2 + • • • + a,^""'-1"1 4- aou>M«-', (0 < t < s — 1), for w ^ s, we have P B 0 - 1) = ^-iP-.-tCs - 1) + as..2Pn-2(s - 1) + • • • + o ^ - ^ C s - 1) + a0Pa-sCs - 1), (« > s).
(2.8)
It follows from (2.6) that
(2.9) Consequently, we have the Diophantine approximation P + (S
" '
~ V - co < c ( c o ) | P n ( s - l ) | - 1 ^ , ( « > « 0 ) .
P»(s — 1) 3) The Proof of Theorem 1. Let
P,(O , A(« - 1, • • •, * + 1, ?, » - 1, • • •, 0) A ( s - 1, •••,0)
(0
(2.10)
(2.11)
Then we have evidently that P;(O = */.;,
(0<»lK«-l),
(2.12)
where 5,-, ( denotes Kroneeker symbol and
P.CO - o^xP-.CO + aJ_aP._,(O + • • • + OjP.r-H.lCO + OoP.-XO, (0 < * < » - 1),
(2.13)
for ti > s. Let 6; = &»P/(0) + 6^,(1) + • • • + b^P,(s - 1),
G = 0,1, • • • ) .
(2.14)
Then Qt(l = 0,1, • • •) satisfy CO < * < » — 1),
Qi = h and
Qn = a^Q^ + as.2Qa-3 + • • • + a ^ ^ ^ ! + aaQn-s, for w > s. Let Wj'' denote the eofactor of w«" in A(s — 1, • • •, 0). Then PiCO = —;
7
r S TTj^wW',
A ( s — 1, • • • , 0 )
/=
(0 < » < s - 1).
(2.15) (2.16) (2.17)
i
Henee we have from (2.14) that
Ql = —
^
r S co«w § Wi'',
A(S — 1, • • •, 0 ) y = i
,=0
G - 0,1, • • •).
(2.18)
407 No. 2
HTJA & WANG:
UNIFOBM DISTBIBUTION & NUMERICAL ANALYSIS
189
Since M ca<-i }
...
m (2)i+l
.. .
;
ww<-i
fflMi+1
1, •••, 1 -= ( - l y - ' + ' ^ w C c o ® , • • -, « w ) T r ^ , where <7;(co(2), • • •, cow) =
^
wC/l>
2<(,<—«;
(o < * < s - 1 ) ,
(2.i9)
""' '°C'') denotes the elementary symmetric
function of &3(2), • • •, coCrf, therefore we can prove easily that *,(«<» • • •, coCrf) = gr,(co),
(0 < I < s - 1),
(2.20)
where £f;(co) is a polynomial of degree I and has rational coefficients. Its leading coefficient is equal to ± 1. Hence Wo, Wlt • • •, Ws-t is linearly independent over rational field R, since 1, co, ••-, ca1'1 is a set of basis of the algebraic number field .E(co). Especially, we have h0W0 + • • • + 1,-iW,-! *f 0.
(2.21)
It follows from (2.18) that Ql = (&oW0 + • • • + b^W^W A(s — 1, • • •, 0)
+ 0 ( | aM
| /)
where the constant implied by the symbol "O" depends on co and b only. theorem follows from (2.21) and (2.22).
(2.22)
Hence the
Bemarh 1. For practical purposes, we suggest to take the initial values Qo = Qi= ••• = Qs-2 = 0 and Qs-t = 1 as usual (Cf. [1]) 'Remark 2. For the case s = 2, a0 = ax = 1 and b = (0,1), we have from (2.18) the well-known formula
»'-j;R-¥L)'-(i=JL)')-
«-M.-->-
<"»
Hence the formula (2.18) may be recognized as the generalization of (2.23). III.
THE ESTIMATION OF r\
2
—< 2I-I
Lemma 3.1. We have
and
hw|
TJ
< 2- — 2s
(3.1)
(2<*<s).
(3.2)
408 190
SCIENTIA SINICA
Lemma 3.2.
Vol. XVHI
If the coefficients of the polynomial g(x) = asxs + fflj-jx'"1 + • • • + atx + a0
satisfy as ^ a,_t > • • • ^ ^ ^ a0 > 0, ffeew no root of the equation g(x~) = 0 has modulus greater than 1 (Cf [7]). Proof. Since | ( 1 - x)g(x) | > a, | a; | '+1 - ((a, - o_,) | as |' + (a,-, - a , _ 2 ) | a ; | ' - 1 + ••• + C « i - a o ) | x | + a0) > a f |a;|'(|a;| - 1) > 0 for | a; | > 1. The lemma follows. The proof of Lemma 3.1.
1 ) Denote
Q(s;) = (x - l)F(a;) = xs+1 - 2xs + 1. Then we have
4-i)=(2-ir-<2-i)'+i - 1 -(-i)'i- 1 -( 1 -lFr)'>« >
and
=i-(2--i-Y^-=i-2(i-^y V
2s~l 1 2S~X
\
2s)
Let »(«) = 2' - 1 - 2 S ~' . Then gr'CO = 2' log 2 - 2 J ~^ ( l + ^ \ log 2 - 2 ' ( l - 2 - J ( l + -^))log2. Since
^ ( 1 + 7)'that is gi'(s) > 0 for s ^ 2, therefore 0O) is increasing for s ^z 2. 2s — 1 — 2'~ 7 > 0, 2^--^I->l. 2i-'
Consequently
409 No. 2
HXIA & WANG:
Hence Q \2
UNIFORM DISTBIBTJTION & NUMERICAL ANALYSIS
^ - j < 0.
191
(3.1) is thus proved.
2) Let F(x) = {x — r{)B{x) and B{.x) = x'-1 + /9,_2x'~2 + • • • + ftz + /?0. Then • - J?/?0 = •
A
. " ^
t
" "
Ps-2~V=
1, 1
'
(3-3)
"I-
Hence we have PC"—,
ft-
^
,
,
ft-3-
^_ 2
Let ^
Since
=
jft/ft+n for 0 < j < s - 3 ; r?'+1 +
x
T?'
+ • • • + r?
is an increasing function of x for x ^ 0 and r < = —H r-* '— +1 x+ 1 v> + rf + • • • + v + 1 (0 < j < s — 2), we have r , _ 2 > r , - 3 > ••• > ro. (3.4) Let x = ft-jj/. Then B(y) = B(ft_2i/) = iSpJir 1 + ftz\y'-2 + PsS-ly'-'
+ • • • + £ & _ # + ft.
From (3.4) we have /SJIi > ft-3/9^^ > • • • > ftft-2 >
ft.
(.3.5)
Hence it follows by Lemma 3.2 that the moduli of roots of E(y) = 0 are all < 1. That is, the moduli of roots of i?(a;) = 0 are all <1 ft_2 = JJ — 1. The lemma is proved. Lemma 3.3. s = 3.
We have h ( 2 ) | = i?"1 for s = 2
Proof. Obviously, for s = 2 we have | j?t2) | = IJ"1.
and |ij (M | = |?jC3)| = if1 For s = 3, since
z3 — a;2 — x — 1 = (a; — ^)fa;2 + (?? — 1) x + —J and ( , _ 1)2 _ ± = ^3 - V + v - 4
=
- y2 + 2V - 3
<
Qj
for
410
192
SCIENTIA SINTCA
Vol. XVIH
hence if®, ?;(3) is a pair of conjugate complex numbers. Therefore | i?C2) | = | if® | = ?; 2 . The lemma is proved. IV.
T H E ESTIMATION OF t
Lemma 4.1. We have
L
(4.1)
and (L + L~^T^i < I ra) I < CL - (L - l)"^)"^1,
(2 < i < 0-
(4-2)
Proof. 1) (4.1) follows immediately from G(L~) = - 1 < 0 and
G(L + L- ( '-°) = L-'-'-'KL + L-wy-1
- 1 > 0.
2) Let X be the real root of the equation g(x) = xs + Lx'~l — 1 = 0 in the interval (0, 1). Since gr'(o;) ^ 0 in (0, 1), therefore X is the only root of g(x) = 0 in (0,1). It follows from ^(O) = - 1 < 0 and g(l) = L > 0 that g(X — 5) < 0 for any d satisfying 0 < 8 < X. Hence on the circle | x \ = X — 8, we have 1 > \xs + Lxs~l\. It follows by Eouche's theorem that 1 and (?(#) have the same number of zero in the domain \x\ < X — 8. Therefore (?(») has no zero in the domain | x \ < X — 8. Let 8 -> 0. Then the moduli of roots of (?(a;) = 0 are all > X. Since g((L + i" 7 ^)"^) = ((£ + L~~l)~~l + LXL + L~~lTl - 1 < (L~~l + L) (L'731 + i)" 1 - 1 = 0, we have
i_
i)
\t<- \>X>{L
s 1
i_
s
+ L ~ } ~\
(2
3) Suppose that L > 2. Let Q be the only real root of the equation Kx) = xs — La;'"1 + 1 = 0 in the interval (0,1).
Since K0) = 1 > 0
and h{T) = - L •+• 2 < 0,
411 No. 2
HUA & WANG: UNIFORM DISTRIBUTION & NUMERICAL ANALYSIS
193
we have h{Q + S~) < 0 for any 8 satisfying 0 < 8 < 1 — Q. Hence on the circle | a; | = Q + 8, we have |liar-11 > |z' + l | . It follows by Rouche's theorem that xs~* and G{x} have the same number of zeros in the domain \x\ < Q + 8. Therefore G{x) has s — 1 zeros in the domain \x\ < Q + 8. Let 5 -»• 0. Then G(x) = 0 has s — 1 roots in the region \x\ < Q. Since
= ((L - (L - 1 ) " ^ ) ~ - L)(L - (£ - l ) " ^ ) - 1 + 1 = (L - (L - l)" 7 ^)- 1 ((L - (L - l)"^ 1 )" 7 ^ - (L - l)" 7 ^) < 0, we have lr(»] < a < (L - (L - l-)~1')~l,
C2 < i < s).
4) Suppose that L = 2. We proceed to prove that the equation
=0
roots with moduli > 1. Let 8 > 0. Then it follows from Rouche's theorem = ys + (2 + 8}y — 1 and y have the same number of zero in the domain Therefore JT S G/) has s — 1 roots with moduli > 1. Let 8^-0. Then also has s — 1 roots with moduli ^ 1. Since SQy) = 0 has no root satisfy1 , -HXJ/) = 0 has s — 1 roots with moduli > 1. The lemma is proved.
Lemma 4.2. s = 3.
We
have
\ r (2) | = r" 1
fors
= 2and\
Clearly, for s = 2 we have | rC2) | = r" 1 .
Proof.
r0) | = | r0) | = r~7
for
For s = 3, since
a;3 - ia; 2 — 1 = C* - T)fa;2 + ( r — L~) x + — \
and /•
T\I
^r — ivj
4 _ T3 — 2Lr2 + L2r - 4 _ — i r 2 + L2r - 3 . „ <^ u,
r r hence -rC2), rC3) is a pair of conjugate complex numbers. Therefore | rC2) | = | rC3) | = r The lemma is proved. T
V.
2
.
IRREDUCIBILITY OF POLYNOMIALS
I n this section, we prove two theorems concerning the irredueibility of polynomials.
412 194
SCIENTIA SINICA
Vol. XVffl
Let w{x) = xs + a^x'*1 + • • • + atx + a0,
(5.1)
"where a0, alt • • •, as-t are integers. Theorem 5.1. (Xie Ting-fan and Pei Ding-yi [8]) If
k l > I oS"11 + \a*-i<~2\
+ ••• + \a2a0\ + 1 , a0 * 0,
(5.2)
then w(z) is irreducible over rational field R. Proof. From (5.2), we have
|a,a o | > \a'0\ + k - i < C M + ••• + \a2a20\ + k l • It follows by Rouche's theorem that w(x) and x have the same number of zero in the domain \x\ < \ao\. Therefore wQa;) has only one zero & in the domain \x\ < \ao\. It follows easily from (5.2) that the equation w(x) = 0 has no root with modulus | Oo I - H w(x) —u{,x)v{,x), where u(_x~) and v(x~) are polynomials with integral coefficients and with degrees > 1 and if M(9) = 0, then the moduli of roots of v(a-) = 0 are all > \ao\. Hence
kl = k(o)| = |«(oMo)| > Ko)| > kl, which leads to a contradiction.
Thus we have the theorem.
Theorem 5.2. If | a01 = 1 and w(x) = 0 has only one root 9 with modulus ^ 1, then w(x) is irreducible over rational field R. Proof. If w(x) = u(x)v(x~), where u(x) and v(x) are polynomials with integral coefficients and with degrees ^ 1 and if u{9~) = 0, then the moduli of roots of v(x) — 0 are all < 1. Hence | u ( 0 ) | < 1, which leads to a contradiction. The theorem follows. It follows from Lemmas 3.1 and 4.1 immediately that F(a;) and 6f(x) are irreducible over rational field R. Hence we have Corollary 5.1. -q and r are PV numbers. Remark 1. Theorem 5.1 improves a result of Perron [5] and a result of Bernstein (Cf. [1] Theorem 12) too. Remark 2. The polynomials z'-z'-'-l
(* = 2, 3, •••)
are not all irreducible over rational field R. For example, x% - x* - 1 = (a;2 - x + l)(a; 3 - x - 1) (Cf. Theorem 11 of [1]).
413 No. 2
HUA & WANG: UNIFORM DISTRIBUTION & NUMERICAL ANALYSIS
VI.
195
RATIONAL APPROXIMATIONS OF r\ AND r
Lemma 6.1. If « > s, then -f^-rj
< c(n)F~1'^^~^ri.
(6.1)
Proof. From Lemma 3.1, we have p
log \vM\ >
_
log??
iog(n - 1) ^ ~ log i1 ~ fr)
""
>
log 17 1
2' log 2
"^
log 2
+-JL.. 22<+1
Hence the lemma follows from Theorem 1 and Corollary 5.1. Lemma 6.2. If « ^ s, then ~±
- t < c ( r ) G~' ~ ^ + C s -^ l o e L ~ ^ 7 I ^ _
(6_2)
Proo/. Since !_
i_
log (.L - (L - 1) s~l) = log L + log (1 - L~l (L - 1) ' - ' ) > logL - i~ l (£ - l ) " 7 ^ - £-' (L - l ) " 7 ^ > log L - L"1 - i" 2 and log ( i + £-<'-«) = log i + log (I + L-0 < log L + L~', we have _ log \rO)\ ^ log£ - X-1 - L~2 logr " (s - l)Qog L + L~0
>^_(l_^ s-l\
LlogL
^_i_(i__i
L_Vi
L 2 logL/\
1
s —IV
LlogL
L 2 logL
s—1V
Llogi
LS+3J'
L_^ L'logLJ
L_+
L'lagL
1 \
£ J + 1 log 2 i/
Hence the lemma follows from Theorem 1 and Corollary 5.1. Remark. If we use Lemmas 3.3 and 4.2 to replace Lemmas 3.1 and 4.1 in the proofs of Lemmas 6.1 and 6.2 respectively, then the corresponding right hand sides of (6.1) and (6.2) may be improved to C(T ? )^< 2) - 2 and c ( r ) ^ 2 ' - 2 for s = 2 and c(7j)Fi3)-3/2 and c(r)6?^3)-3/2 for s = 3.
414 196
Vol. XVIH
SCIENTIA SINTCA
VII.
EXAMPLES
Let n be an integer ^ 1 and let a1} • • •, as be integers.
Denote
O.Cn-[Jo1 ---f 1Jo /C*., •••,xs)dx1---dx1-^±f(^.,.--,^). n £r\ \ n
nJ
(7.1)
Theorem 7.1. We have Sup [
(7.2)
where
^«.,..,«,)=L ((l+fy+(l _D- (l+f) when 2 | «, i« which fi denotes the number of odd integers in av(l ^ v ^ s ) .
(Cf. [ 9 ] )
Lemma 7.1. We have
±.^-i-f + fu-«.»-. Proo/. Since f1 ( i _ i d + Jd QI _ 2x")2N) e-2"''"11^ = mr\ Jo\ 6 2 / the lemma follows. The Proof of Theorem 7.1. From Lemma 7.1, we have Q .JL, ^ - v g2ni(tiiml+—+asms)k/n HEhc)
"
"~
n
*= 1
(TOj • • • W,V
-(ign^-l-K 1 -^})')- 1 ) =
C(H(n;
a
u
• • • , a s)
—
1).
(7.3)
415 No. 2
HUA & WANG: UNIFORM DISTRIBUTION & NUMERICAL ANALYSIS
197
The theorem is proved. Take n = F^,
w
.ff(m; ai, oi, a3)
5(w; oi, 02, as,at)
149
1.17442
401
1.36254
927
1.01102
2872
1.02416
1705
1.00480
10671
1.00540
Table 2 « = 5, 6 n
H(n; a,, •••, aO
n
H(n; 01, • • • , a*)
13624
1.07428
29970
1.14458
'Remark. In addition to the two generalizations of Fibonacci sequence in our papers, there is another generalization given by G. N. Eaney [6]. Let Q = Qn = (a,-;0, where a,-j = 1 for i + j ^ n + 1 and a,-,- = 0 for otherwise condition. Let <£>B,0 = (1,0, • • •, 0)' and 4>n,i+1 = Q4>n,d(d = 0,1, 2, • • •). The sequence of column vectors 4>n, d is called the generalized Fibonacci sequence. Let Dn(A) = det (Q — 17) be the characteristic equation of Q. The D»(A) is irreducible over rational field B for n —
, where p is a prime ^ 5, but not
always irreducible for other n. When Dn(X) is irreducible, we may obtain by the given method of this paper, n sequences of integers 4>n, Ar^(r = 1, 2, • • •, n; d = 0, 1, 2, • • •) satisfying the recurrent formula
'"
[T'1 /
where the initial values are the row vectors of the matrix (<£n,o> •••» 4>n,n-i) respectively. Then we have
Since Dn^/l) is not the minimum polynomial of a Pisot-Vijayaraghavan number in general, i t seems that we can not expect the rapid convergence of 4>n,d+Ar^>/4>n,dt
416
198
SCIENTIA SINTCA
Vol. X V m
Corrigendum. In §3, 1) of [3], we may write the co's as w, = 2eos-—2^ P where g is a primitive root mod p . defined as (.ffO
( K K c ) ,
The transformations <J'\1 ^ j ^ g) should be co; - • col+j
(1 < I < r ) .
REFERENCES
[1] [ 2] [ 3] [ 4] [ 5] [6] [ 7] [ 8] [ 9]
Bernstein, L.: The Jacobi-Perron Algorithm, Its Theory and Application, Lee. Not. in Math; Spr. Ver. Pre; (1971), 207. Cassels, J . W. S.: An Introduction to Diophantine Approximation, Camb. Univ. Pre., (1957). Hua, Loo-keng & Wang Yuan: On uniform distribution and numerical analysis (I) (Number-theoretic method), Sci. Sin., *, (1973), 483—505. Hua Loo-keng & Wang Yuan: On uniform distribution and numerical analysis CH) (Number-theoretic method), Sci. Sin., 17 (1974), 331—348. Perron, O.: Grundlagen fur eine Theorie des Jaeobisehen Kettenbruchalgorithmus, Math. Ann, 64, (1906), 1—76. Eaney, G. M.: Generalization of the Fibonacci sequence to n dimensions, Can. J. of Math., 18 (1966), 332—349. Uspensky, J . V.: Theory of equations, MH book Com. Inc; (1948). Xie Ting-fan & Pei Ding-yi: On irreducibility of polynomials, Kexue Tongbao (to appear). See "Applications of Number Theory to Numerical Analysis", edited by S. K. Zaremba, Aead. Pre., (1972).
417 Vol. 26NO.6
KEXUE
TONGBAO
June 1981
A NOTE ON UNIFORM DISTRIBUTION AND EXPERIMENTAL DESIGN WANG YUAN (3£
7c) AND FANG KAITAI
(^Jf^)
(Institute of Mathematics, Academia Sinica) Reseivei December 29, 1979.
I. INTRODUCTION
If there are s factors of which each factor has 2 levels in an experiment, where q > 1, then the orthogonal array is often used. The number of experiments is 0(g 2 ) which is too large when
1. Suppose that the integers a{ |(l < | i s£l s) satisfy a^ = 1, 1 < a, < q(2 ^ i ^ s),at *? a>(i ^ q) and g.c.d. (a,, q) — 1(1 < i < s), then we use the set of points PqW
= O«i,
fc«j,[*--,
kas)
( m o d q ) , k=>l,
2, ••-, Q
8(2.1)
to arrange the experiments by the usual way. Obviously, s needs to be ^
where pu
p2, — p m
are different
418 486
KEXUE TONGBAO
Vol. 26
large when q>(q) is comparatively large. 2. Suppose that q = p. We suggest to use the set of points Q f O) = O ; kb, •••,&Z><-1)
(mod p), k = l, 2, ~-,P
(2.2)
to arrange the experiments, where b is an integer satisfying 1 < b < p and b' ^ 6'(mod p)(i ±? j). We use the notation b = ( l , b, • • • , b'" 1 ). It is noted that b is often chosen among the primitive roots modulo p. 3. Suppose that q = p — 1. Then the set of points may be obtained by omitting the last coordinate of the points of (2.2). III.
UNIFORM DISTRIBUTION
Let Ot be the s-dimensional unit cube. Let 1 < % < w^ < • • • be a sequence of integers and Rnl(k) => (x^Ck), •••, a^"'3(fc)) ( K fc < %) be a set of points Gs. For any given r = ( r ! ; r2, • • •, rs) 6 Gs, let Nn,(r) denote the number of points Rnl(k) ( K k < «;) satisfying 0 < x)"$(k)
K.Ks.
If sup ^ - - \ r \
=0(1),
S
as w ; ->oo, where \r\ — J J r,-, then the sequence of sets {Rn,(k)}(l
< ih < w2 <
1 = i
— ) is called uniformly distributed on Gs. For simplicity, we omit the index I. Let sup
Z«(rl _ | r |
=D(W).
Then D(n) is called the discrepancy of fin(fc)(K fc < n). such that the sets of points
We will choose a and b
( K K f )
({T"}'-"'{^-})
(31)
and
(3.2) have lower discrepancies. We use D(q, a) and D(p, b) to denote the discrepancies of (3.1) and (3.2) respectively. The set (3.1) with q = p was first introduced by H. M. KopoSoB121 and B . HLawkaC3] independently in their studies on the theory of numerical integration over Os and the set (3.2) was proposed by H. M. Kopo6oB[4]. We use the notations x = m a x ( l , \x\),
\\x\\ = x\, •••,
! =1
xs and
419 No. 6
KBXTJB TONGBAO
487
the scalar product of two vectors x and y. Lemma 3.1 (Erdos-Turen-Koksma). Suppose that n is a positive number and h is an integer ^ 2, then
#O) = 2
T ^ - 2 e2"(m>'R*Cfc> + O(v) + OCn-'hrKinhY-1),
where 2 ' denotes a sum with an exception m^ = • • • = ms = 0 (cf. [ 5 ] ) . Lemma 3.2. Suppose that g. c. d. (a,, q) = 1(1 < i < s), Wiew /or any integer
r>\,
—?/2
(cf. [ 5 ] ) . It derives from Lemmas 3.1 and 3.2 that Lemma 3.3.
D(q, a) =
2'
l O f + O(q-K^q)') •
(a,m)so(mod«> »™"»
Lemma
3.4. Suppose that x€ ( 0 , 1 ) , tfTiew ^—. plnimx ±
S
^
=
1
n
a
- — ln(2sin«a!) + —2— ,
where \&\ < 1 cwwZ (x) = min({a;}, 1 — {a;}) in which {%} denotes the fractional part of*(c£. [ 5 ] ) . Theorem 3.1.
D(9
'^
=
I § H C1 - ~ln (2sin» {^})) -
X
+ O(q-Klnqyy (3.3)
The theorem follows from Lemmas 3.1, 3.3 and 3.4. Since the second term and third term of the right-hand side of (3.3) are — 1 and 0(q-l(\nq)') respectively, hence the comparison of the quantity D(q, a) may be reduced to the comparison of the quantity
««••>-*§ n(i-i*(»—ft})). Sometimes Dx(q, a)'s are difficult to make a difference, hence we introduce
where avK = ovft(mod q), 1 < avK < q for 1 < k < q, and «„, = g. Theorem 3.2.
#*(«» «) = Di(q> a) +
O(q-\lnqy).
420 488
KEXTJE TONGBAO
Vol. 26
Proof. Clearly D2(q,a)=J+ O(q-Xlnq)s)' where 1 "ST* TT (i
7
q fri ^ V
2 , (n •
*
V
avk g
\\
+ 1/7
and
- 0 (V 1 S (lng)-Z1-1 ) = O(q-Xlnq)0. The theorem follows. The integral vector a such that D2(q, a ) attains its minimum with respect to a may be used to define the set of points (2.1) which is called a uniform design. Similarly, the set (2.2) with minimal D2(p, 6) is also called a uniform design. IV. TABLES
Since the congruences (ar,m) = 0 (mod q) and (b, ni) ^= 0 (mod p) may be written by a^\a, m) = 0 (mod q) (2 < ft < s) and Zr' +1 ( 6 , m ) == 0 (mod p) respec tively, hence the calculations of Dt^a^a, q) ( 2 < ^ ^ s ) and D2(b~c'~w6, p) may be neglected. 1. For q =• p, we use the set of points (2.2). If q =• p — 1, then we have the set of points by omitting the least coordinate of corresponding set of points of the level p. By the calculation on computer DJS-2, we have the tables of the uniform design for q = 4, 5, 6, 1, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 28, 29, 30, 31 (cf. Table 4.1.). Table 4.1 The Uniform Design for Prime p and p — 1 ^ ~ ~ \ - ^ * 5 7 11 13 17 19 23 29 31
2
3 4 5 6 7 8 9 10111213141516171819 202122 23 24252627 2829 30
2 3 7 5 10 8 7 12 12
2 2 3 3 3 3 7 7 7 7 7 7 7 7 4 6 6 6 6 6 6 6 6 6 1010101010101010101010101010 8 141414141414141414141414141414 17171717171515151515 7 7 1717171717 7 7 7 9 16161616 8 8 8 8 8 141414 8 8 8 8 8 8 8 8 8 18181818 22 221212121212121212 2222 22 2222 22 221212121212121212121212
2. The uniform designs of q = 8, 14, 20, 24, 26 may be obtained by the uniform designs of q = 9, 15, 21, 25, 27 by omitting the corresponding last coordinates. For the case q = 9, 15 and 21, we may use (2.1). For the case q = 25 or 27, we may
421 No. 6
KEXUE TONGBAO
489
use (2.2), since there exists primitive roots mod 0. although 0. is not a prime (ef. Table 4.2). Table 4.2 The Uniform Design for 1 and q — 1 g \
2
3
9
4
4, 7
4, 7, 2 4,7,2,5
15
11
4, 7
4.7.13 4,7,13,2 ^X?1' (x ;a 1„1' ? ! '1i , H,14» ' '^> » i 3 [ 8
21
13
4. 10
4. in lfi 4. in i
25
11
11
11
11
4
9
8
27
8
8
20
20
20
16
W
14
15
4
5
4
7
8
9
10
11
2,4,5, 8.10,11, 16,17,19
2,4,5,8, 10,11,13, 16,17,19
8
8
8
16
16
20
5
16
17
>7£2'5'
9 A. 7
12
g \
6
O A T
«!*»'»
18
19
20
9
15 2,4,5,8,10,11,13,16,17, 19,20
Z,
£1
25
8
27
5
8 20
8 20
8 20
8
8
5
5
8
8
8
5
Table 4.3 *7>(54) ^ ^ - J ^ 1 2 3 4 5
0
"
1 1
2 2
2 3 4 5
4 1 3 5
3
4
4 3
3 1 4 2 5
2 1 5
3. Example. For s = 4 and q == 5, we have uniform designs by Table 4.1 (fc, 2k, 22k, 23&) (mod 5 ) , 1< k< 5 or the form of Table 4.3 and we use the notation Z7s(54) similar to L^b6) in the orthogonal array. Data in the uniform design may be treated by regression or step wise regression. REFERENCES
[ i ] *Bf*^KS¥mftift&tm,^#*i.ft^ttiJKtt, (1977). [ 2 ] KopofioB H. M., UAH CCCP, 6(1959), 124, 1207—1210. [ 3 ] EOawka, B., Mm. Math., 2(1962), 66, 140—151. [ 4 ] KopoBoB H. M., Bee. MW, 4(1959), 19—25.
[5] #S^S,lS,im4ffi
422 f£25 5j£H?2$ 1982 ff 3 M
i S t # ^ J | x .
Vol.25, No. 2
ACTA MATHEMATICA SINICA
Mar., 1982
On Diophantine Approximation and Approximate Analysis (I)* Wang Yuan ( £ x ) (Institute of Mathematics, Academia Sinks')
§1. INTRODUCTION
We use y = ( T U , • • • , 7n, • • • , fi,, ' • • , Tw) to denote a point in ^-dimensional Euclidean space Rst. We use the notations y, = (y,,-, • • • , y « , ) ( l < i < 0 and y,; = ( r / 1 5 •••, r , v ) ( l < / < * ) . <7 = (?i> ' " " > ?<)' * = ( ^ i ' ' " " ' Ks) and m = ( m l ; • • • , raj will be x
vectors with integral components and (x, y) — 2
iyi
t le sca ar
'
'
P r °d u c t °f * ^ d y . We
< = i
use also 6 to denote any pre-assigned positive n u m b e r ,
c(f, g, • • • )
a
positive constant de-
pending on / , g, • • • only, but not always with the same value, and o , c, a, b, • • • a b s o lute constants. Let Gs be the s—dimensional unit cube, i.e. the set of x = (* 1 5 • • • , x^), where 0 ^
*,- < 1, 1 < i < s.
Let Pn(O = (^»>(^)> • • • > 4 ' } ( 0 ) C1 < ^ < »)
b e a s e t of
P oints
a
in G^. For any a— ( « , , • • • , » , ) 6 G 5 , let Nn( ) be the number of points P B ( ^ ) ( l ^ ^ ^ » ) satisfying
0<*i"(O<«.-» K » < * . Then N
D ( n ) = sup a6Gf
is called the discrepancy of Pn(l0
° ^
— \a\ ,
n
|a| =a1---oJ
(1 ^ ^ ^ » ) .
For a real number * , we use [ * ] , {*} and {*) to denote respectively the integer part of x, the fractional part of * and the distance from x to the nearest integer, i. e. s
(x) = minO — {x}, 1 + {x} — x).
We also denote M(m) — J J wj(Inm,-)*, where
a ^ 1 ,£ ^ 0 and ? = max(l, | * i ) . Theorem 1. Suppose that "7 is an integer ^ 2 and that
f [ <(r,, m))M(m)>c(r,
a,b)
(1)
< = i
holds for a n y m ^ O .
Then the set ( ( y 1 5
« ^ t) has discrepancy
* Received January 23, 1980.
qr),
• • • , ( y , , g ) ) ( m o d l ) ( l < 1,<, 1,
K
423
2 Jf^
Wang Yuan- On Diophantine Approximation and Approximate Anafysis(l)
249
a, b, s, t)i->+2l"-°-iXfo
D(i')
where 5,, denotes the Kronecker symbol. T h e o r e m 2. Suppose that 1 and r satisfy the conditions of Theorem 1. Suppose further r +1 +1
that n is an integer ^ 1p
and that
r,,-bi
Then the set (^
1J
^a
•. .,
(modl)(l <
D ( ? 0 < c ( r , a, b, s, O?- (+2 " ( '- 1) (ln?)'' J+ ' +iS '-'' where hi = (hLi, • • •, hti), 1 < ;' < s. Similar results were obtained by Hua and Wang [1>2] and Niederreiter ^ for the case t = 1. It may benefitcial in practical use to make suitable chcises of r and t. § 2. THE PROOF OF THEOREM 1.
Suppose always that r and c ( r , a, £ ) satisfy ( l ) . We use Pt,M to denote a ^-dimensional parallelepiped defined by at < where J J (*-• ~ «<•) < M .
Xi
< bt, 1 < « < / ,
Denote also Q = [ 2 ( a + * ^ M ( / n ) c ( r , a, b)~1].
> = i
Lemma 1. There is at most 1 point ( ( r l 3 At), • • • , ( r , , fe)) in any parallelepiped of -the type P,,Q, where | \i\ ^ | mt \ , 1 ^ (' ^ s. Proof. If there are two points ( ( r 1 5 k'), • • •, (rt, A')) and ( ( r 1 5 A") 5 • • • ,(rs, k")) in a P,,B, where At' ^= A" and | £• | < | m,-1 , | £•' | < |ra,| (1 < »' < s). Then
]1 <(r,-, *' - k")) < p. i = l
On the other hand, it follows from ( 1 ) that
I[
a, b)M(K-k")-1
< = i
> c (r, a, b)2-°'\\m\\-« (flh^y ^ < = i
>c(r,
t +4)
«, Z > ) 2 - "
'
1
W(/7i)- >£),
which leads to a contradiction. Hence the lemma follows. Lemma 2. Suppose that / is an integer ^ 1 and that M satisfies 0 < M ^ — 2 the ^-dimensional domain
Then
424 250
£fc
3*
M < * , - • • * , < Mf,
^
?R
M <*,<—,
25 S 1 < I" < y
(2)
can be covered by at most 2-t~1/(log2 M" 1 )*" 1 parallelepipeds of the type Pt,M. Proof. Since the domain ( 2 ) is the interval M =SS *j ^ min I / M , — ) for the case s=\, \ 2/ it can be covered by at most / intervals of the type [ c , c + M], where c is a real number. Hence the lemma holds for s = 1. Suppose that ^ is a positive integer and that the lemma holds for s = ^, Now we proceed to prove the validity of the lemma for s — \ + 1. Divide the domain M < xr • -xk+1 < IM,
M < *,• < — , 1 < ,• < \ + 1
(3)
log2 — subdomains L Mi
into
2-'- 1 < xk+1 < 2-', 1 < / < [log, ^ ] , where the left hand side of ( 4 ) should be replaced by M
(4)
for i =
log2 — . L Mi
Consider a
subdomain 2-'- 1 < ^^ +1 < 2 - , where * <
(5)
log2 — . Since L Ml 2iM
<
_M_ < Xn+i
Xi...
XK
< M_ < iV^M. Xk+1
Hence it follows from the assumption of the induction that the above ^-dimensional can be covered by at most 2Kl (log 2 M~ 1 )*~ 1 = L (say) parallelepipeds of the type Using these P^MI' as bases and 2~' as height, we obtain a set of Pk+i,M and so ( 5 ) covered by at most L parallelepipeds of the type P$+i,M. Now consider the subdomain with / =
domain P^,M2'. can be of ( 4 )
log2 — .Since L Ml
i<M2f'^ 2
<Xl...H
*
M
=
2l-±, 2
it can be covered by at most L parallelepipeds of the type P^,i/2. Using these P^,i/2 as bases and 2M as height, we obtain a set of PK+»M and so the subdomain of ( 4 ) with i =
[log
— can be covered by at most L parallelepipeds of the type P*+I,M. Consequently domain Ml ( 3 ) can be covered by at most 2
L log2
[
M] < ^O*^" 1 )*
paralldopipeds of the type P^+^M. The lemma follows by induction.
425
2 $f]
Wang Yuan; On Diophantine Approximation and Approximate Analysis(l)
Lemma 3. Suppose that a ^ 1.
251
Then
S'
where 2 7 denotes a sum with an exception k ^f 0 = ( 0 , • • •, 0 ) . Proof. Let ThQ be the number of points « O i , £ ) ) , - • • , < ( r ( , £ ) ) ) ( ! & I < I *«* I » 1 ^ < ==S f) in the domain
*r • •*« < '£>> p < * , < — , l < » < /. 2
Since
n
>•=i
i=i
by ( 1 ) , we have <(r,, k)) > 0(1 < i < /) and TUQ = 0.
Hence by Lemmas 1 and 2
and thus y^'
i
^
y ^ TI+UQ — ThQ
1 = 1
< C(«, O^Clnp- 1 )'- 1 S - + KO^-Clnp-1)'-1 i=i <
O£»-°(ln£»)'-1+J>'«.
Lemma 4. Suppose that A is an integer ^ 1 and a ^ 1 and that g(ni) is a nonnegative function of m. Then
Y; Km) '5J1* llm|| < y-1 «n! y> -^
x S ••• 2 where ^ j ^ "
0.
i
Proof.
1 0 1 6 8
a s u m m
^
S
I
- S'K*),
w h i c h l = (*',, • • • , » , ) r u n s over all t h e p e r m u t a t i o n s of ( I , - - - ,
426 252
m.
^
¥
ffi
25 ^
Similarly
Hence
lt»;l<« \ml'
lm,l<* l|nt||
" 'ms-lJ
S
m,= l
,
+
a
l*,l«m,
h
S ~r=-—-—^{ S
1
h
h
x 2 ... S
S
s-l
m7
"~l 2 ' K ^ n •••>»»,_!, ^)
1
••• S'K*).
The lemma is proved. Lemma 5. Suppose that a ^ 1 and A is an integer ^ 2.
Then
'^^(llmll n <(r,-, m)>)° ; = i
Proo/. From Lemmas 3 and 4 , we have
2'___J '» ! * (l|m|in<(r,-,in)>)" < = i
« 2J
h
2-i
2-i
2.1 jz—..
„
v+i
427 2 J5
Wang Yuan; On Diophantine Approximation and Approximate Analysis(l)
v ... v
x
... v '
v
'*•;<* l*.-/+i"s--m
i*.-,!**
253
1
' ^ ' ^ H <(r,,m)> > = i
« V AA-'- VhA'- V . . . V (ln^)^° +< - 1+g « 2_i 2j Z =J1 Z J 7Z wa-i>a+i m m ;=0
i
™',+
(o 1)
<SC /^ - "(ln/0*"
c+< 1+s
-
tni=-\\ il+i
i,)
H
"'."" '.«
where the constants implied by the symbol <*C depend only on r , a, a, b, s, t. is proved.
The lemma
Lemma 6. Suppose that x is a real number. Then
Lemma 7. (Erdos-Turan-Koksma). Suppose that r and h are positive integers such that where TJ satisfies 0 < T\ < 1/6. Then the discrepancy of the set P o ( ^ ) r ^ 1 and h>rli\, (1 < /^ sg; „ ) of Gs satisfies
*>(») < S ' ^ ^ " - E «J"("'p-<*)> + (5* + 6), lm/l<*
I»
m
B
(t=l
(Cf. Hua and Wang [ 2 ] ) . The proof of Theorem 1. By Lemmas 5 and 6 , s
'K© ll*m|| Vfa'*"' <
tit Z.i TTZTnll "
g
Z Je
\mi\
'
i*«. \\nm\\ JJ
r
m
)>
a, 6 , f, O ? " ^ ( " " ' 1 ) ( l n ^ ) * i + ' + J S ' " ' .
'
= 1,7} = ( 4 ? ) " ' and A = (4?) 2 1 in Lemma 7 . The Theorem follows. § 3. THE CONSEQUENCE OF THEOREM 1.
Lemma 8. For almost all r € Rs,, we have t
I]
where a = 1 and Z> = / -1
s+ t
.
(6)
428 254
fC
*
^
ffi
25 $
(Cf. Wang and Yu [ 5 ] ) . From Theorem 1 and Lemma 8, we have Theorem 3. Suppose that 1 is an integer S& 2. Then for almost all r £ Rst, ((Vi> <7)> • • • > (y,> <jO)(modl)(l < ? , • < * » 1 < i < s) has discrepancy
the set
D ( < ? 0 < r O , s , s, /)?-'(In ?)"+' + < + e
(7)
Now we shall give another proof of Theorem 3 based on an idea of Littlewood Schmidt [ 4 ] ) which leads to an improvement of the logarithmic order of (7)
(Cf.
Lemma 9. Suppose that m ^ 0 and 3 > 0. Then
'On) = ( (IT
Then nil ^? 0 and
K « 0 - ( ( I I <m»i) I In
/(«.) = » r (""••• ("' ( n
/(m) - (
^ • • • <Jy,_ (
Let rfiBi! +
( f [ (0,- + r,-^,) I In (0, + r,,m,) \l+s )"'
- ( ( IT <'»«'> I ln
e,
s,
i)9~'
Proof. We know from Lemma 9 that the series
S ' ( l [ ^,(ln^)1+s V1/(m) ^1 = 1
'
is convergent and the series
S' ( n «.- (ln».-) is convergent for almost all r € Rsl. inequality ( 6 ) is satisfied. Then
II <(ry,m))|ln<(r;, m)>|^)
(8)
Take r such that the series ( 8 ) is convergent and the
429
2 ^
Wang Yuan; On Diophantine Approximation and Approximate Analysis(l)
255
|ln<(r,, m)>| < |ln C (r, e)| + In J[ « , ( L ^ ) '+~^ i=i
< c(r, s, s, /)ln||m||. Put 8 = —-—•. s+ t
Then
V'
1
i<,« \\m\\\\ <(r,, m)> >= I
<
max f[(M*ty+» I [ Iln<(r,, m)> |»+»
X S ' ( l [ S.-Cin^;)^ n <(r,,m)>|ln<(r,,m)>|^)~ 1
e, s, 0(lnA) o + < K 1 + s )
Take r = 1, ij = ( 4 ? ) - ' and A = ( 4 ? ) ' in Lemma 7.
t)([ahy+t+\
The theorem follows.
§ 4. THE PROOF OF THEOREM 2 .
Lemma 10. Suppose that the sets Pn{\) = (rf'OO, ' * * > ^ ' ' ( O ) (1 < k <= ») and iP/^) = ( y ? ^ ) . • ' •» /» } ( O ) ( K ^ < ») have discrepancies D(») and E(n) respectively and that W'XO - ySn)(OI < s, i < -t < n, i < i < s.
(9)
Then | D ( » ) - E ( » ) | <*« Proof. For a€ Gs, let a' = (os'15 • • •, c£) and a" = (aj', • • •, a7)j where ra,- — 6, a, = < 10,
if a, — 5 ^ 0, otherwise.
f«i + 5 , 11,
if a, + 5 < 1, otherwise.
and
Let Afn(a) and M n (a) denote respectively the numbers of points Pn(\) and Q„(_!() 0 < ^ ^ TZ) belonging to the domain 0 < Xi < «15 • • • , 0 < *, < a,. Then we have M,(tf')<2V,(«)<W.(«") by (9).
Denote
„,-
i£^^-|«|
and
* - *^Q-|.| .
430 256
$j
j£
^
jg
25
£
Then W |
^
-
| a | <max(
Since
o < |«"| - |«| < 8 + « , ( ] ] c « ; ' - n « , ) < • • • < ^ s , \ ' =2
1=2
'
we have ff2<M£(O_|a"|
+ | a "| _ | a | <£(„)+, 5 .
n 1
o , also satisfies the inequality. Hence D ( « ) < E ( « ) + *<5. Similarly E ( « ) < D ( « ) + *5. The lemma follows. The proof of Theorem 2 .
Since
it follows by Theorem 1 and Lemma 10 that the discrepancy of the set ( ^ " q \ ••-, i—^—1} \ n n I (modl)(l < 1i< 4> K i < 0 satisfies D ( ? ( ) « q~t+2stl*-v>(\aqys+t+ssiia + qn-i-P where the constants implied by the symbol ^C depend only o n r , a, b, s, t. is proved.
The theorem
REFERENCES
[ 1 ] Hua Loo Keng and Wang Yuan, On uniform distribution and numerical analysis (Numbertheoretic method) (I) Set. Sin, 4: 16(1973), 483—505; (II) Sot. Sin, 3: 17(1974), 331—348; (III) Sd. Sin, 2: 18(1975), 184—198. [ 2 ] Hua Loo Keng and Wang Yuan, Applications of number theory to numerical analysis, Science^ Press, Beijing, 1978. [ 3 ] H. Niederreiter, Methods for estimating discrepancy, Applications of Number Theory to numerical Analysis (8. K. Zaremba, ed.), Academic Press, New York (1972), 203—236. [ 4 ] W. M. Schmidt, Metrical theorems on fractional parts of sequences, Trans. Amer. Math. Soc, 110(1964), 493—518. [ 5 ] Wang Yuan and Yu Kun Rui, A note on some metrical theorems in diophantine approximation, IHES/M/79/297.
431
$
%25%%3M 1982 f£ 5 %
t
#
#
&
Vol.25, No. 3
ACTA MATHEMATICA SINICA
May, i9sz
ON DIOPHANTINE APPROXIMATION AND APPROXIMATE ANALYSIS (II)* Wang Yuan ( £ x ) (Institute of Mathematics, Academia Sinica)
§ 1. INTRODUCTION
Let a and C be two positive constants. Let E J ( C ) be the class of functions of Gt>
Where the Fourier coefficients satisfy
The notations introduced in WangU11 are also used in this paper. Let I be the least integer ^ a and pn, u t be the integer defined by
Theorem 1. inequality
Suppose that n is an integer ^ 2 and a > 1. Suppose further that the t
JJ <(r,, m»M(m) 5s c(y, a, b),
(1)
> = i
holds for any m ^? 8-
Then
l«c
< f7f(y, a,a,b,s,
t)n""V~*^''(in w
„)»""<-'->+'*.... [51
Wanguo:!, Hua and Wangt8*9', Similar results were obtained by Bahvalov , Haselgrovc 031 15 1 Korobov and Niederreiter' - for the case / = 1. It may beneficial in practical use to make suitable choices of y and / . § 2. T H E PROOF OF THEOREM 1
Suppose that / € E ? ( C ) . Since a> 1, / h a s the absolutely convergent Fourier expansion
/O) = £C(mV"«—»,
|C(m)| < - ^ _ .
• Received January 23, 1981. Correction: The first part of this paper, titled «On Diophantine Approximation and Approximate Analysis (I)» was also receiveu on January 23, 1981.
432
324
&
3*
^
JR
25 ^
Since C(0)-( /(*)<** and «» >
1
{./« T 1./
qK--ni i = l t
- C(0) + S'C(m) I I (—i— S *"^.->«*Y, we have by Lemma 6 of [ 2 1 ] ,
sup \\ K*)dx-
*
2
n^.».,y/((yi»v)»--»(v*.«))
^ Hmll- M 2« + 1 , ^ » ^ Hmll- M 2« 4- 1
^
< c(S, + SO, where
(In
-t" 1 ^
ImJtKA
M
, , - T-T / /
>...
y= i
ana
in which h is an integer ^ 2 that will be determined later. By Lemma 5 of [ 2 1 ] , we have S i < c(y, a, a, b, s, />-«A' ( "- 1 '"(ln A)*"" + '-' + ' { >«, Evidently £2 < c(a, s)h-+1. at
Take h — [ f l " " 1 ] .
The theorem follows.
From Theorem 1 and Lemma 8 of [ 2 1 ] , we have
433
3 $3
Wang Yuan; On Diophantine Approximation and Approximate Analysis (II)
T h e o r e m 2.
325
Suppose that n is an integer ^S 2 and a > 1. Then the estimation
sup I ( f(x)dx -
*
2
I I *»'• «,-/((r»' * ) » • • • . ( y » « / ) )
< C*r(y, a, e, , , » V ° ' ( l n ») a " + ' + '~ 1 + S
(2)
holds for almost all y€ Rsl. Remark.
By the use of the inequality
r —^
<( r —r^—Y c->o
m
^<*<' I'1"" IT < ( r » ' m » /
«*
and the argument of the proof of Theorem 3' of [ 2 1 ] , it may be proved that the right hand side of ( 2 ) may be replaced by Cc(y,
/ > - " ' ( I n »)» ( '+' ) + \
a,e,s,
§ 3. T H E CASE a ™» 2.
For the case a = 2, the weight fin,i,^
in Theorem 1 may be simplified;
Lemma 1. Suppose that x is a real number. Then
2 o - k i y'*• < mm («», --/-). T h e o r e m 3. Under the assumption of Theorem 1, we have
sup
I ( / o v * - .1 2 n f1 - ^ /((v- «)»•••» (V" * » < Cc(y, a, b, s, t)n-it+w"-l\\a
n )"'+'-'+'
5
..».
Proo/.
J_ 2
n(i--[^)/((yl,^"-,(yJ^))
= «"2' S I K " " lftl)SC(m> - 1 = C(0) + Z'C(m)n-* f[ ( 2 O - l*l>ta"r''">'')» Heace by Lemma 1, we have
sup If }(x)dx-±
2
I[(»-lftl)/((y.»«').---.Cy,»v))
434
326
$
IWI
<%
m
•%
25 $
I n «ry, m)y I y= i
The theorem may be proved easily by the argument of the proof of Theorem 1. § 4. THE NUMERICAL INTEGRATION OVER
Q?(C).
Let
[(eo.(—log,|*|))\ i f - ! < | * | < 2 , otherwise,
[0, »(x)
- f<2»-'*)
and /toGO = 1 — 2
»= i
A«/(*).
suppose that / ( * ) is a periodic function of s variables of which each variable has period 1 and that / ( x ) has the Fourier expansion
If the series
SCCrnXm)^"- 10 is convergent almost everywhere, then its sum is denoted by / ( j c ) © i ( / n ) .
Let
Let Qf(C)(a
> 0, C > 0) denote the set of continuous functions of G, satisfying sup |qt>i| < C2~"'»,
Lemma 2.
h —h
+'••+*,.
£, a (C) c E?(C2').
Lemma 3. If / ( * ) € £,°(C), then where 2 " denotes a sum in which f runs over all integral vectors with non-negative integer components. (Cf. Bahvalov"'31, Hua and Wang" M ). T h e o r e m 4. have
Suppose that 0 < a ^ 1. Then under the assumption of Theorem 1, we
435 3 553
Wang Yuan; On Diophantinc Approximation and Approximate Analysis (II)
sup I [ f(x)dx - i j /((y15 g), • • •, (YJ, «r)) < Cf(y, a, a, *, s, / V ' ^ ^ C l n »)*'+'+'*"«. /Voo/. Suppose that /€ C??(C)- Let
sCO - [ /(*V* - -7 T, /((y» « ) . • • • , (y»«/)). Then from Lemma 3, we have SCO -
2"5(
where 5(
c
Hence sup | 5 ( / ) | < S i + S i , where Si—
sup
2 J " ls(
and 2 J — sup
t*O"(C)
2 J
tt<\oean'
|5(q»f)|.
Obviously
«0>log2 n'
/=[log2 »']
Since C,(m) — 0 for ||m|| > 2'°, we have
«;=1 llmll<2'<>
- »'c(o) + 2 ' c«(m> n ( s ^/
j = l \fl
'
and so
|5(«P,)I
<»-• s ' ic'Cm)i - i — l|jnll<2'e
TT
//•
\\
•
327
436 m
328
¥
JR
&
25 •&
Hence by Lemma 2 and Lemma 5 of [ 2 1 ] , we have
Sa<»-' £ ' -7
'"-'<"' I I <(',. "•)>
S"|C,(m)|
i= l
<«-• 2' i c <»i-— ll«nll<«'
TT
//•
\\
11 <(r ; , m)>
y= i
-c-
'
_J
S'
^
S
M<
"" ' llmll yI=Ii <(»•/' m »
< C
sup I ( K*)dx - -L j ] /((y» « ) , • • • , ( r » »)) j < C
0 » ~ o ' O »)" + ' + ' +B .
holds for almost all y € Rst. Remark.
The right hand side of ( 3 ) may be improved to Cc(y, a, e,s, t)n-'KIn
n)'+'+'
by the method used in the proof of Theorem 3 ' of [ 2 1 ] . § 5. T H E ERROR TERM OF QUADRATURE FORMULA
Let
s(»,y,/)=(
/(*>*--7 2
( i - - ^ ) / ( ( y i . «),-••>(*»*)).
Theorem 6. sup |5(», y, / ) | < CW(n, f), where
W{n, /) - »"' 2 (l - ^ ) I I (1 + 2*JB2({(y,, g)») - 1.
(3)
437 3 J5j
Wang Yuan; On Diophantine Approximation and Approximate Analysis (II)
329
in which B2£x) — x' — x -i
denotes the Bernoulli polynomial. 6 Proof. It follows from the argument of the proof of Theorem 3 that the W ( » , / ) may be given by
FK(«,/)=»- ! 'S' T vsn(''-i'?*i)'
v=i
llmll «,=-» *=i
Since
Sincehave We have
2_£^!=1
+ 2 ,^ 2 ( M ),
WC, /) - »-" 2 S f[ (1 + 2*»B,({(y», 9 )})) ~ 1 _ „-> i ] (l - -M) f[ (l + 2«»B2({(yv «)})) - l.
The theorem is proved. Remark.
In practical use, it may be suggested to take r,-,(l =^ »'<S/, 1 ^ /' < s)
irom an integral basis of a cyclotomic field Q\ cos—), where
Concerning the assumption (1), we propose two conjectures. 1°. Suppose that s~$st and r,-,- ( l <1 i <1 t, 1 ^ ;' ^ s) are « real algebraic numbers (ru, '",ru 1_ 0 \
such that the determinants of all * X t minor matrices of l \r,i, arly independent over Q. Then
•••» ru
0
' •.
I arc linc1 /
t
I I <(r,, m)>||m||»+« > c(y, e),
< = i
m ^ 0.
2°. Suppose that f ^ / and rti — e r '; where r,,- ( 1
\ r « , • • • , rlt
0
1 /
438
330
»
^.
*£
ft
25 ^
r
IJ <(r,, m)>l|/n||1+e > *(y, «), « * 0. i = i
1 ° and 2 ° arc Schmidt'1'1 theorem and Baker141 theorem respectively for the case / =~ 1. Let n be an integer ^ 2. Consider the systems of linear congruences s
2 ] aijXi + *,+,- - 0 (mod «), /=i
1< i < *
(4)
i
(5)
and i
2
a
ay + yi+« • ° Cmod »),
1=1
and their solutions satisfying —
^ *,-, yt ^
— ( 1 <J /> ; < a ) . Evidently *! = • • -
= *,+, = 0 and y t = • • • — y,+, = 0 are the solutions of ( 4 ) and ( 5 ) respectively and they are called the trivial solutions. The other solutions are called the non-trivial solutions. Let q = q(an, • • •, a,,) denote the minimum of the product * i - • 'xs±t, where *u • • • > xs+t runs over all non-trivial solutions of ( 4 ) . Further let q* — ? * ( » ) — max q(an, • • • , a,,). l
For equation ( 5 ) , we may define Q = Q(au, ••', a,s) and 0 * = £>*(n) similarly. Concerning q, Q and q*, we have the following two conjectures. 3 ° . Let n be a positive integer. Then q(n) and £)(«) satisfy the inequality
£ ^ < , ( , , /)»"+"<'-".
(6)
**(«) ^ K'» 'V.
(7)
4°. Let />; denote the Z-th prime number. Gelfond proved the inequality ( 6 ) for the case a = pi, s^ 1 and / — I (Cf. Korobov[14]) and Wang, Wang and R e n M generalized the result to the case n = pi and s,t i~5 1. Concerning the inequality ( 7 ) , it is well-known that
q*(Fl+2) > cFl+2 for the case s = t=\,
1 /71+A/TV+2
where F, + J = -J=[[
*
)
(\— \/~5\'+2\
\
2
)
)
( i > 1) denotes the Fibonacci sequence. Bahvalov [1] proved q*(pi) ^ c(f)/>;/( In/»;X for the case y > l , « = 1 and Wang, Wang and R e n M established q*(p,)>c(s, OpJ/(ln?;)'+'"x for s > 1, / > 1. 5°. There exist a set of integers ( « ; ) ( l < / z 1 < « 2 < • • • ) a — o ( » , ) — ( a , , • • • , «,)(/ — 1 , 2 , • • • ) such that
and integral
vectors
W{ni) - sup ( f(x)dx - -L 2 / ( i £ , • • •, ±£) tZE"(C)
jG
>
n
l *=1
X
"/
< Cc(a, s)(In »,>-'/»/(« > 1). 171
Hua and Wang
M
and Bahvalov
"I '
(8)
proved ( 8 ) independently for s — 2, »/ — F;+j and
439 3 JJ!
Wang Yuan; On Diophantine Approximation and Approximate Analysis (II)
a = ( 1 , Fi+i~) (/ — 1, 2 , • • • ) • that W(pi)^Cc(a, by BahvalovM.
f)
***' pt
331
Korobov'121 established that there exists an a = a ( p j ) such
and it was improved later to W(pi)^Cc(a,
*)
nPl
' Pi
Sarygin'171 proved that W(n) > Cc(a, s) L i 2 £ i — holds for any integer
n ^ 2 and integral vector a . Conjecture 5° may be derived from conjecture 4° for the case t = \ 6 ° . There exist a set of integers ( » ; ) and integral vectors a -= a ( n ) such that the set
(\—}'
" ' {—IV1 < * < «i) has discrepancy
Zaremba"31 proved D(Fn-2) < c
ln F|+2
F i+i
for * = 2 and a - ( 1 , F !+1 ).Korobov 1131 and
Hlawka1'1 established independently that there exists an a=a(^pt) such that D ( p ; ) < f O ) for s^2.
Niederreitertlfl proved that there exists an a — a(n)
D(n) < c ( » (
lp n
n
n
?'^-
such that the relation
^' holds for x > 2 and all integers n > 2.
It seems that the set ( « ; ) in, 5°, 6° may be even proposed to be the set of all positive integers ^ 2. 7°. For almost all y — ( r u , - - - . r , , ) , ( l ^ ^ , - ^ ^ , l ^ / ' ^ y ) has discrepancy D(q')
the set ( ( y 1 5 «y), • • - , ( ? „ « ) ) (mod 1)
6, s, Oq-'Clnqy^-l+\
(9)
111
By the use of continued fraction, Khintchine' proved the conjecture for s = / = 1. Schmidt'1" proved that D(q) < c ( y , e, /)^~'(ln^)' + 1 + £ holds for almost all y for the case s > 1, / — 1 and later WangUI] generalized his result to D(q') < c(y, e, s, t)q~'(laq)s+t+* for ; > 1, / > 1. REFERENCES
[ 1 ] Bahvalov N. S., Approximate computation of multiple integals, Vettnik Moscow Univ. Ser. Mat. Meh. Astr. Fig. Him., 4 (1959), 3—18. [ 2 ] Bahvaloy .N. S., On embedding theorems for class of functions with bounded derivatives, Vestnih Moscow Univ. Ser. Mat. Meh. Astr. Fie. Sim., 3 (1963), 7—16. [ 3 ] Bahvalov N. S., Optimal convergence bounds for quadrature processes and integration methods of Monte Carlo type for classes of functions, Z. Vycisl. Mat. i Mat. Fie., 4 Suppl (1964), 5—63. [ 4 ] Baker A., On some diophantine inequalities involving the exponential function, Canarf. J. Math., 17 (1965), 616—626. [ 5 ] Haselgrove C. B. A method for numerical integration, Math. Comp., 15(1961), 323—337. [ 6 ] HJawka E., Uniform distribution modulo laud numerical integation, Comp. Math., 16 (1964), 95—105. [ 7 ] Hua Loo Keng and Wang Yuan, Remarks concerning numerical integration, Sci Bee. New Ser^ 4 (1960), &—11. [ 8 ] Hua Loo Keng and Wang Yuan, Numerical integration and its applications, Science Press, Beijing,
440
in
m
m
&
JR
25 ^
1963. [ 9 ] Hua Loo Keng and Wang Yuan, On uniform distribution ami numerical analysis (Number theoretic method) (I); (II); (HI), Sci. Sin., I (1973), 483—505; 3 (1974), 331—348; 2 (1975), 184— 198. [10] Hua Loo Keng and Wang Yuan, Applications of number theory to numerical analyses, Science Press, Beijing, 1978. [11] Khintchine A., Ein Satz fiber Kettenbriiche, mit arithmetischon Anwendungen, Math., Z. (1923), 289—306. [12] Koroboy N. M., The approximate computation of multiple integrals, Dokl. Akad. Hauk SSSB, 124 (1969), 1207—1210. [13] Korobov N. M., Number-theoretic methods in approximate analysis Fizmatigiz, Moscow, 1963. [14] Korobov N. M-, Several problems in. the theory of diophantine approximation, Uspehi Mat. Na.uk SSSB, 3 (1967), 83—118. [15] Niederreiter H., Application of diophantine approximations to numerical integration, Diophantine approximation and its applications (C. F. Osgood, ed.), Academic Press, New York, 1973, 129—199. [16] Niederreiter H., Existence of good lattice points in the sence of Hlawka, Monatsh. Math., 86 (1978), 203—219. [17] Sarygin I. F., A lower estimation for the error of quadrature formulas for certain classes of functions, Z. Vycisl. Mat. i Mat, Fie., 3(1963), 370—376. [18] Schmidt W. M., Metrical theorems on fractional Parts of seguences, TAM8., 110(1964), 493—518. [19] Schmidt W. M., Simultaneous approximation to algebraic numbers by rationals, Ada Math., 125 (1970), 189—201. [20] Wang Yuan, On numerical integration and its applications (Number-theoretic method), Shnxue Jinzhan, 5 (1962), 1—44. [21] Wang Yuaa, On diophantine approximation and approximate analysis (I), Ada Mathemulica Sinica, 25: 2(1982), 248—256. [22] Wang Yuan, Wang Ling Xiang and Ben Jian Hua, A note on a transference theorem of the systems of congruences (to appear). [23] Zaremba S. K., Good lattice points, discrepancy and numerical integration, Ann. Mat. Pure Appl., 78 (1966), 293—317.
___
441
Chin. Ann. of Math. 11B:1(199O),51—65.
NUMBER THEORETIC METHOD IN APPLIED STATISTICS*** WANG YUAN ( i
TL) *
FANG KAITAI (^r ff $.) **
(Dedicated to the Tenth Anmuersary of OAM)
Abstract This paper gives 3ome applications of number-theoretic method (or quasi Monte Carlo method) for numerical evaluation of probabilities and moments of a continuous multivariate distribution over a special domain such as cube, ball, sphere, simplex:, etc., where the uniformly distributed sets of points'm such domains, which are useful in experimental design, simulation, geometry probability, etc., are suggested. Some applications of numbertheoretic method in optimization are discussed also.
§1. Introduction The problem of numerical evaluation of probabilities and moments is really a problem of numerical integration. The number-theoretic method (or quasi Monte Carlo method) for numerical evaluation of multiple integrals and for optimization is based on the theory of uniform distribution (u. d.). Let -£T= [%, 5J X ••• X [a8, 5S] be a rectangle of i2*, 6 = (6j, •••, bs)', a? = (a;1, •••, %,)', andi^(ar) be a continuous monotone distribution function on K, whioh satisfies F(b) = 1 and F(x) = 0 whenever at least one of the a;, is «(. Note that a't a and Z>is may be defined to be — oa and oo respectively. We use x<:b to denote that xt
gup
N( P
- ' ^
- F(x) i =D (n, P ) .
F a n I D,(n, P) is called the I'-disorepanoy of P with respect to F(x). If Pn=(a;$n), •••, OJ^') is a sequence in K such that hn-+oo as «-»oo. and if BP(Jcn, P) =o(l) as n->oo, then Pn is called an i^-uniformly distributed sequence. If K = Js, where J = [0, 1] and F(x) is uniform distribution on I', i. e., F(x) =x% ••-x3, then we omit the F in the above notations (of. Weyl [12], Hlawka and Muck [5], and Niderreiter [9] ) .
Manuscript received March 15, 1989. * Institute of Mathematics, Academia Sinica, Beijing, Ohina. ** Institute of Applied Mathematics, Academia Sinica, Beijing, China. •** This Work is Supported by the Chinese National Science Foundation and the Science Foundation of the Chinese Academy of Sciences.
442
52
CHIN. ANN. OF MATH.
Vol. 11 Ser. B
If P ={X]c, Jc = l, ••; n} is a get I* with deorepanoy D(n, P) or B(n) foralmpllfy and /(a?) is a function of bounded variation in the sense of Hardy and Krause with total variation F ( / ) , then it is known that
f f(x)dx-±±f(xll)
[<7(/)D(n)
(1.1)
(See Koksma [7], Hlawka [4], Hua and Wang [6]). Let B be a domain (for example, ball, sphere, simplex, etc.) in iJ*. In this paper we shall pay more attention to numerical evaluation of
I-f f(x)dv,
(1.2)
JD
where dv is the volume element of B and D has a parameter representation. First by using a transformation a quadrature formula over D can be transferred to a quadrature formula over J*, where t is the dimension of D. Another approach to this problem is to use a u. d. sequence in D. We shall start from a u. d. sequence in I*, and then derive the u. d. sequences with respect to oertain distribution functions, in particular, to some uniform distribution functions in some special domains: ball, sphere, simplex, etc., which are often useful in simulation, geometry probability, experimental design and many problems in statistics. More details are given in our next paper with the same title. Another application of the u. d. sequences in D is in optimization. Let f(x) be a continuous function on D, we want to find its global maximum M in D. There are many gradient methods for this kind of optimization problems (of. Avriel [2]). Unfortunately, there appear only few oases that the global maximum oan be reached, and we can obtain in usual a local maximum if the function / is not unimodal, and the dimension of D is large, for example, dimension of D>5, because the solution, in general, depends on the ohoioe of initial point. Therefore, we use the following algorithm to find an approximate value of M. _ fTO,,,
if / (a^+i) < mj;,
Wjt+1
~t/(« f c + 1 ), if/(«* + i)>m*, where (xit xa, •••) is a u. d. sequene in D, i. e., Pn={Xi, •••, #„} is a u. d. sequenoe. After a large number n of steps, we may reasonably expeofc that m* is close to M, if f(x) satisfies some regularity conditions. We often use the following quantity to measure the uniformity of distribution of these points d(n, D)=max mincZ(#, x^), m&D l
(1-3)
where d(x, OJ») denotes the Euolidan distance of x and ar*. d(n, D) is called the dispersion of the set {xk, #=1, ••• , n}. However one can show that if D=I', then y/Tn-1/a
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(of. Zielinski [13] and Niederreiter [10]). This means that it is true that m* is oloaed to the globle maximum M if n is large. In Section 4, we shall generalize the above result to some kind of D's and give some applications in statistics.
§2. Numerical Integration Let D be a bounded domain in 22s. We are required to calculate the integral 8
(1.2).Assume that tha dimension of D is s, dv = JJ_dx,=dx and Del'.
Then it may
be simply suggested to use the following formula
I-jpf(a>)ID(.O!)dx, where Isipe) is the index funotisn of D(of. Hua and "Wang [ 6 ] ). This will lead to a big error sometimes, since/(*)Jz>(#) may be discontinuous on the boundary of D. However, the domain D is often very special in statistics, so it is possible to reduce the integral over D to an integral over J*(*<s). More precisely, suppose bhat D has a representation Xj=Xj{q)l7 •-,
and let jr(9))=det(rr)1/a. When i = s, J(q>) is just the Jaoobian of transfomation from x to
I=jj(x)dv=jj
(a?(»)) J(q>)dv,
(2.2)
t
wheredq>^Yldq>t. Therefore a quadrature formula over I* induces a quadrature formula over D. Denote by v(D) the volume of D. Then «(2))-f J(
(2.3)
Suppose further that
«(2>)-vG»)-g/i(po, where/4(?>,) is the density functions of p(, »=1, •••, i, and the corresponding distribution functions Jo
satisfying ^",(0) =0 and i^(l) =1, *=1, •••, *. Let Ft(jDt) =y, and let FTl{yd denote the inverse funotion of FiQc,), i = l, •••, i. Then
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CHIN. ANN. OF MATH.
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j ^ / (?) J (?) dq> = v (D) J/( / (* (F-i (y) ) dy),
(2.4)
where J F- 1 (y) = (JPr1 (2/1). -> Fj^Vt))', and % = gdy t . For a given get {6s,•= (Z»si, •••, &M)'> A = l, •••, w} of I* with discrepancy D(n), we have a Bet {Cj; = F~y(b^), Jc = l, •••, w} which has .F-dasorepanoy DF(n, {oJt})=D(«) too, where F(x) =JJFi(xi).
Hence by (1.1), (2.2) and (2.4) we have
I J/(*)
Jj/(«)(tos-i.ig/(*(6»))J(6») and
(2.6)
f /(«)
Both formulas have the same order of aoouraoy. Define a set of D by P={xlc-=x(c]e), * = 1, »., n}. The volume v(q>
(2.8)
so that
v(.9
sup ^(p
E0z<Ml-F{y)
= D(n). (2.9) We thus suggest an algorithm for obtaining a set P of B that is scattered uniformly in D from a known set with lower discrepancy in T. Now we give some examples. Example 1. The domain D is a simplex J. s ={a;: 0
445 No. 1
Wang, T. f Fang, K. T. NUMBBB THEOBETIO METHODS IN STATISTICS
55
where q> <E I'. We have (=1
and Therefore
are density functions over Is with corresponding distribution functions
For a given set {6fc, k = l, ••-, n} in I s with discrepancy D(n), we have a set in J* with .F-disorepanoy
JD(*I)
too, and a set P of As:
**=(«** —, <%»)'» * = 1» "•» »»>
(2.10)
^ = t e C 8 - i + 1 ) , * - l , - , n, j = l, - , s.
(2.11)
where The set P satisfies (2.9). Example 2. Let D be the s-dimensional unit ball Bs={oc: xl + —+x^
j=l, ••; g - 1 ,
x, =
where S* = sin ( w ^ ) , OS; = oos(5r^j6), * = 2, •••, s - 1 , iS, = sin(25r?>,) and 0,=00B(2trcp,) in whioh p € J". Then we have s l «/(0>)=2£7rs-vr< =ln£ ~ 2
J
and
«(£.)-£/(*)<2* 2
rr R (1 B
"TU \T
s-i+l\
—2—>
because for any integer m>0,
[ o (dn(«))-d,.ii.^ J2+L) Therefore
C
-D'r1,
if*=i,
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OHIN. ANN. OF MATH.
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are density functions Over 2* with, corresponding distribution functions
For a given set {6S, A«=l, •», n} i n P with discrepancy D(n), we have a set {cR, & — 1, •••• »i}, where -^<(OfcO=&fc,5 * = 2, —, s, k = l, •••, TO, in I* with J'-disorepanoy D(n) too, and finally a set P of B,: **-(flto •", »*»)', * = 1, "•» *»,
(2.12)
where
7
(2.13)
in which S w = sia((7row), O r w =cos(src w ), » = 1, •••, * — 1 , 5rii, = sin(25rofc,), (7*,= cos (2ovcu), A = l, •••, re. Example 3 . Let Z) be the s— 1 dimensional unit sphere whioh has a representation
•where 8i = sin(t!vg}l), Ot = cos (ovq>t), a = l, •••, s—2, /S,_1 = sin(25rp,_ a ) and 0,-1 = 1 COS(2OT0>,_I) in whioh q»6 J*" . Then i < 1 OT
/(,»)=2 «- nV -
and
Therefore / . C ^ ) - ^ - " 7 - 5 ( 1 / 2 , («-*)/2), » = 1, .», s - 1 are density functions over I with corresponding distribution functions
***- B(U%l-i)/*)fc*mt>"**> °
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Wang, Y. #• Vang, K. T,, NUMBER THEOEBTIC METHODS IN STATISTICS
*»=(ato —, %»)', * = 1, •••, n,
57
(2.14)
where «ta- U SMOW, i = 1, • • •• s - 1 ,
(2.15)
8-1 *=1
in wihoh /Sw = sin(5row), C^ = COS((7F,(C<), & = 1, —, s - 2 ,
/S t , s _ 1 = sin(25ro*,,_1) and
O»,,_i = cos(2i7rote,_0, * = 1, •-, n. Example 4. The domain D is a part of the boundary of s-dimensional unit simplex 2 7 «-i={«: «iH •which has a representation
haj, = l, sc^O, * = 1 , •», «}
det (TT')=L>-t
nS2i<'-i>-1Ol)adet(SS'),
<=i
\
/
where -1
S
0
01 -1
8101 ... SI ... &2_2 0?_i 81 0§
. . . SI
- 1 : i .-
». ^ 2 _ a
O^
£§ .»
3
SU
5fs2_i\
fl^
JS*_J I
:
f
0 0 0 »• -1 1 / Note that det(S5i') is invarint if S is replaced by AS, where A ia an (s—1) x (s —1) matrix with det A— ± 1 . We now prove that there exists an (s —1) x (s—1) matrix A with det A= ±1 such that 1-1 1 0 ••• 0\ 0 - 1 1 ... 0 AS= I I i :•• • - F _ j , 0 0 0 — 1 \ 0 0 0 ». - 1 / say. In faot, if s = 2, then S = ( —1, 1), and the assertion as true. Suppose now that s>2 and the assertion holds for s—1. Then / I -81 — 0\ 0 1 — 0 I
/-I 1 0 — 0 I 0 - 1 Of ... 81 ••• SLi
\o
\
0
— 1/
0
0
0 — -1 1
By induotion hypothesis, there is an [(*—2) X (s—2) matrix A3. with det .4.1= ± 1 and
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CHIN. ANN. OF MATH.
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O% -
Vol. 11 Ser. B
81 - 8U\
41 7::: ; U\
0
0- -1 1
/
Therefore
and the assertion follows. We have / 2-1— -1 2 •••
0 0\ 0 0
0 0— 2- 1 \ 0 0 — - 1 2/ say. Since ^ = 2 and 4 = 24_i — ^ { _ 2 (i>2), we have -ds_i = s. Henoe anct
«(T._i) =J/s J(?.)^=s 1/a /(s-l)!. Therefore /*(?><) - (•-•)/S?t-l)-1cr1, *=1, - , . - 1 are density functions over I with corresponding distribution functions For a given set {6to A = l, •••, n} in I s " 1 with discrepancy D(m), we have a set {ct, i«=l, •••, n}, where c^^AOarcsinC^ 2 8 - 2 0 ), i=l, —, s-1, 4 = 1, •••, «. Finally, we have a set P of r s _i: (2.16) *»=(»*i. —, **s)'- * = 1, •••> w, where
f^=II^ <s - o (l-^f^). i = l, - , s-1, Z s
U,s=n6^ -«, *-i,...,«.
(2-17)
§3. Some Applications In this section we shall pay our attention to applications of number theoretic method in numerical evaluation of probabilities and moments of a continuous
449 No. 1
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59
multivariate distributions. The bagio quadrature formulas are given by (2.6) and (2.7). There are a number of methods to produoe sets of points {ba, h = l, •••, n} in P (see Hua and Wang [6]). In view, of our experiences, we will recommend using the following algorithm: Let (hit •••, hs; n) be an integral vector, where hi = l, 0
ji(xy=a+Bx, where a and B are f x l and txs matrices of regression coefficients and X belongs to a rectangle K = [oi, bi] x • • • x [os, 6J . Suppose that for eaoh x£K, we have /j. (x) ~ Nt(a+Bx, S), the multivariate normal distribution, where a, B, and 1? oan be used by their least square estimators. Let Tit i=l, •••, t, be the constants such that the Steel is said to be qualified i f / i ^ T j , i=l, •••, t. Thus the probability that the alloy steel corresponding to a; is qualified is equal to
K * ) = j ~ - £ % (V, /*(*), 1?) dp
(3.1)
where y = (yi, •••, yt), and %(y, p. (x), 2) is the density of Nt (/», 2). The integral (3.1) oan be evaluated by applying (2.6) if we choose suitable numbers Alt a = l, •••, *, such that
p(as)ezj^~j\(.V, (fi*), s) dy -"EK^-Z1,) —£n,(z», ft (x), S),
(3.2)
whore a»-(B«, - , 2w)=(ri+Ui-2 7 i)6 S : i, - , Tt+(At -Tt) ikt), & = 1, •••, n and {6^} is a uniformly distributed set of points in / ' . To illustrate the computational aoouraoy, set S = I5, the 5 x 5 identity matrix and /* (x) =0, T{ = — 1, ^ ( = 1 , « = 1, •••, 5 in (3.2). We have rx-
/-I
• P = Ji "*J-i n^V' °' ^
d|/
'
Now we have by (3.2) the following: Table 1 shows that 5-digit accuracy for numerical evaluating a 5-fold integral is
450 6O
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Table 1. n
approximate values of p
1069 2129 5003 8191
0.148299406 0.148295351 0.148291410 0.148291358
m
0.148291347
obtained by the use of 1069 points only. Example 6. The moments of order statistics. Let Xlt •••, X , be a sample from the population with distribution function F(cc) and density/(a;). Let Y, = X(i )
/i(m., -,«*)-«![ Uyf'fWv,
(3.3)
where D* — { — oo<2/s<2/,_1<...<2/1
E[[(a+(6-a)8i)" y /(«+(6-o)«i)]*>
Jo j-i
where J5 is defined in Example 1. By Example 1, (2.6) and (2.7) we suggest the following two formulas for the calculation of fi(ms, •••, mi):
K^, -,«*)&«! (&-«)» i ± nk?*"- 1 /^^)],
where <^=ai+(& — a)&jy and {6K, 4=1, •••, n} is a uniformly distributed set of points in I', and fi(m.< —' % ) = s ! ( 6 - a ) s — S n C a + ( 6 - a ) c w ] m ^ / ( a + ( 6 - a ) o w )
=s! (6-a)s ii!n[ai+(6-a)6jf-^1)]^/(a+(6-a)Jj//-'+1)) where {c*} and {6ft} are given in Example 1. Since the mixed moments of order statiftios of uniform distribution U(0, 1) on J = [0, 1] can be formulated. We give an example in Table 2 which shows the (see Table 2)aoo uracies. Example 7. In his study of compositional data, Aitohison [1] introduoed in 1986 a so-oalled additive [logistic normal distribution. Let Tn be a domain denned in Example 4 with n=N — l. Any X in Tn is called a composition. For a given x£Tn, we denote by *_y the n-dimensional vector formed by the first n components of * .
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Table 2. Mixed moments of order statistics of U(Q, l),s=7 n
B(XmXmXm)
E(Xa)X%Xfa)
418 597 828 1010 1220
0.03887518 0.03888518 0.03888511 0.03887286 0.03889159
0.00326433 0.00339034 0.00332116 0.00332084 0.00325168
0.03888889
0.00326340
co
|
Let f/ = log(ar_y/XJf) = (log(X1/XJf), - , log(X B /X*))'. (3.4) The equation (3.4) yields an onetoone mapping from TntoiJ". A random veotor x£Tn is said to have an additive logistic normal distribution ANn (/*, S) if its corresponding V~Nn 0 , 2 ) . Aitohison gave the formulas for E(log(Xi/X])), E(Xi/Xj), Cov(log(X,/X,), log(Xfc/Xi)), OovCXj/X^X^/X,). It seems difficulty for him to oaloulate E(Xt) and Cov(Xi,X/), which are required in many praotiotical problems. The density function of JJV,(fi,lf) is given by
(2 5 r)-/ 2 (detl?)-V^nXr 1 )e X p{-^log-|^— M ), ^ ( l o g ^ g L - , * )} (3.5) and the mixed moment of a; is
E(X?">X& = (2w)-'/2(det^)-1^ f ft x^ ezP{- (1/2) [(log^-A* ), l f ' ( l o g ^ i - | . )]}dy, where dv is the volume element of Tn. By Example 4, we have
where q>ei", dqt^fidy,, teal
0=(2/ov)-"/s>(d.eiS)~i^N1/annd
Q(v)-«xp{-(l/2)(flr(i»)-#»)'5-1(flr(9»)-M)} in which 9(9) = Q-°g(Ol)-log(Sl-Sl), ..., log(8l>~8l.10l)-log(8l-8*)y, -2(log01-i31ogfiffc logOa-±logSt,-,
log^-Slog/S,)'.
By (2.6) or (2.7) we may obtain approximate values of any mixed moments of
AN.fa 2).
Example 8. We meet the problem of direotional data in which some statistical distributions are defined over S"'1 (see Example 3). The so-called Langevin distribution which is an extension of Von Mises and Fisher's distribution has a density function
452 62
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Oex.p{kfi.'a;},ac£8s-1, where /JL £ S''1, &>0 and O is the normaling constant (of. Mardia [8]). Another distribution called Soheidegges-Watson distribution has a sensity functor Oexp-OO, *)2}> xZS*-1, where O, To, and /*. have the similar meaning as before (of. "Watson [11] ). We can apply Example 3, (2.6) and (2.7) to calculate probabilities and mixed moments for these two kinds of distributions.
§4. Optimization In this section we shall generalize inequlity (1.4) to some domains which have been discussed in the past sections. Then we give an example to show that the algorithm mentioned in Section 1 is powerful. We suppose that the get of singularities of the transformation X=x(q>), i. e., the set of solutions of J\
i < / < (.
1
where 9»£Z*\(£. Then dFf fa)/dq>t, i = l, ••; tare positive and bounded over J*\© too. First we take a set b]l = (bkl, •••, 6M)', i = l, •••, n in I*\& with lower discrepancy D (n). Then we have shown that
c,=F-1(6fc) = (j?'r1(M- - , Fr\bM)Y, t-i, -, « is a set in I* with ^"-discrepancy D(n) too, and finally we have a set in D: X% = x(Ck), h = l, •••, n. Leix = x(
••-, da>,)', d y -
Then
dx'dx=dip' TTdtp=d^'STT'Sdijt. Since the elements of S and T are bounded in I*\@, we have
d (x,
x^^r^idx'dxy/3 JxW
= r (d*'fir2T/sa^)i/»
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Wang, Y. # Fang, K. T. NUMBEB THEOKETIC METHODS IN STATISTICS
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Thus if a?K, J = l, •••, n, are scattered uniformly in D, then the maximum value of a function on these points may be taken as an approximate value of the global maximum of the function on D. Example 9. Additive logistic elliptical distributions. The so-called additive logistic elliptical distributions defined on T,-i (see Example 4) are extensions of the additive logistic normal distributions mentioned in Example 7 and have density functions of the form /(a?) - (det Sy^flx^gix),
(4.2)
where ^)=^((log-^-M)'^-1(log^-/*))
(4.3)
and *_s is an (s—1)-dimensional vector formed by the first s —1 components of x. The mode of/(*) can not be analytically formulated so far. However we may use uniformly distributed sets on T,-% to oaloulate the approximate values of the mode. When the function g in (4.3) has the form g(u)=O (1+u/m)-', p>s/2, m>0, (4.4) where O- («m)-"r(p)/r(p~s/2), the corresponding distribution is called additive logistic elliptical Pearson Type VII distribution. In this case to find the mode of / ( » ) is equivalent to obtain the maximum of AO*)=]W
ri+(iog-^-M) W i o g ^ - M W F
over Ts_i. The results in Table 3 show that the approximate values of the mode in the case of s = 3, p = 9, m = 5.5 and
are olosed to those of the mode (1/3, 1/3, 1/3)'. Table 3 Approximate values of the mode n
Mn
x1
%i
xz
233
26.55203
0.3271035
0.3306723
0.3422242
377
26.82553
0.3061467
0.3598099
0.3340434
610
26.48600
0.3604561
0.3150540
0.3244899
4181
26.89095
0.3256147
0.3368701
0.3375152
10946
26.99783
0.3334971
0.3337690
0.3827339
17711
26.98296
0.3292827
0.3388427
0.3318746
We note in Table 3 that when n is increasing, the corresponding M„, in principle, is increasing also. But sometimes Mn<Mn>, where n>n', because the sets {cfc} for
454
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OHIN. ANN. OF MATH.
Vol. 11 Ser. B
different n may be completely distinct. Henoe we suggest using a sequential method to improve the above result. The following program is designed for our presented problem, Step 1. Ohoose a uniformly distributed set of points {xk, 4 = 1, ••. , r^} j n Do = T,_i with suitable ng. Find the maximum M0 of the function among these points, and assume that it is attained at x%= (ofa, •••, a*),)'Step 2. Find a small domain D± of Do such that DiCD0 and arJcrDi. For instance, D± is a domain with *J located near the gravity of Da. More precisely, w« odoose a,, i=l, •••, s, suoh that O^a^Xot, i=l, ••; s. Set o = o i + " ' + o, and bi=ai+l — a, i<=l, •••, s. Then 8
«
a
i = l,-" ,s. Denote Di = {x=(x1, ••; xs)':att
8 = 1, •••, s, 4 = 1, •••, %,
which is uniformly distributed over D%. Denote by Mi the maximum of the function on X'K 9 which is attained at the point x\. Step 3. Suppose that in the yth step we have found the maximum M} of the function and the correrponding point ae). By a similar method we can reduce the domain Bj to A+i» a n d make a set of points on Di+i, by which we can find another maximum Mj+i of the function and the corresponding point x*+1. Eepeat Step 3 until the searoh domain is smaller. The last maximum Mj+i is expected to be closed to the global maximum M of the function. Applying the above program to our problem, we get Mo=wi=--- = 223 for each step, and the results are given in Table 4, which improve those in Table 3. Table 4. The seqnencial method for optimization No
««
1
0.0000
2
0.3000
3
0.3300
4 the
6f
Mi
x\
x\
%%
1.0000
26.55203
0.3271035
0.3306723
0.3422242
0.4000
26.97836
0.3304331
0.3343395
0.3352274
0.3400
26.99543
0.3327104
0.3330673
0.3342224
0.3330
0.3340
26.99994
0.3332711
0.3333067
0.3334223
global
maximum
27.00000
0.3333333
0.3333333
0.33333333
We have done many examples which all show that the present sequential method is advantageous.
455 No. 1
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References [I] [ 2] [ 3] [ 4] [ 5] [6] [7] [ 8] [ 9[ [10] [II] [12] [13]
Aitohison, J., The statistical Analysis of Compositional Data, Chaman and Hall, London/New York, (1986). Avriel, M., Nonlinear Programming, Analysis and Methods Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1976. Fang, K. T. & Wu, C. Y., The extrime value problem of some probability function, Ada Math. Appl. Sinica, 2 (1979), 132—148. Hlawka, B., Funktionen von besehrankter Variation in der Theorie Gleichverteilnug, Ann. Mat. purrs Appl., 54, (1961), 325—333. Hlawka, E. & Muck, R., A transformations of equidistributed sequences, in "Applications of Number Theory to Numerical Analyssi" (Zaremba.S. K. ed). Acad. Press, New York, 1972, 371—388. Hua, L. K. & Wang. Y., Applications of Number Theory to Numerical Analysis, Springer-Verlag (Heidelberg) and Science Press (Beijing), 1981. Koksma, I, F., Een algemeene stelling uit de theorie der gelijkmatige Verdeeling modulo 1, Math.. B (Zutphen), 11, (1942—1943), 7—11. Madia, K. V., Statistcs of Directional Data, Acad. Press, New York, (1972). Niederreiter, H., Metric theorems on the distribution of sequenes, Proo. Symp. Vv/re Math., 24, AMS Pro. B. I., (1973), 195—212. Niederreiter, H., A quasi-Monto Carlo method for the approximate computation of the extreme values of a function, Studies in Pure Math, Btrkhauser, Basel, (1983), 523-529. Watson, Q. S., Statistics on Sphere, Wiley, New York, (1983). Weyl, H., Uber die Qleiehverteilurg der Zahlen mod Eins, Math. Ann., 77. (1916), 313—352. Zielinski, B., On the Monte Carlo evaluaton of the extreme values of a function, Algorytncy, 2 (1965), 7—03.
456 Chin. Ann. of Math. 11B:3(199O), 384—394.
NUMBER THEORETIC METHODS IN APPLIED STATISTICS (II) WANO YUAN
(i
?£,)*
FANG KAITAI
(^r^$.)**
Abstract I n this paper, the authors give some applications of F-uniformly distributed sequences, which are suggested in their previous paper under the same title, in experimental design, experiments with mixtures, geometric probability and simulation.
§ 1 . Introduction In our previous paper0133 we proposed a method to produce sets of points which are uniformly distributed over a domain B of R' and gave some applications in numerical evalution of probabilties and moments of a continuous multivariate distribution and optimization. In this paper we shall give the applications of this kind of uniformly distributed sets of points in experim ental designs for both independent factors and experiments with mixtures in Section 2 and Section 3, and in geometric probability in Section 4. We also give examples to show the comparison between the number-theoretic method and some other methods. The definitions and notations given in our paper [13] are retained hereafter.
§ 2. Uniform Design If there are s factors and each factor has n levels, then the number of all possible ex-periments is n'. The orthogonal array is to choose 0(n2) experiments among them by the theory of orthogonal Latin squares and group theory. However the number of experiments in orthogonal array is still large if n is comparatively large. The number of experiments may be decreased by BIB (balanced incomplete blocks) method for the case of s=2 only. Hence it requires to find a method for decreasing the number of experiments. Wang and Fang (1981) proposed a kind of experimental designs, the uniform Manuscript received March 15, 1989. • Institute of Mathematics, Academia Sinica, Beijing. China. >* Institute of Applied Mathematics, Academia Sinica, Beijing, China. •*« This work was supported by The Chinesse National Science Fund and Academia Siniw.
457
No. 3 -Wang, Y. # Fang. K. T. NUMBEB THEOBETIC METHODS IN STATISTICS (II)
385
designs, by the number theoretic method whioh has been applied satisfactorily in designs of new products in textile industry, metallurgical industry, engineering industry and agriculture in China. "We offer a set of tables Un(h") of uniform designs, where n denotes the number •-*> of experiments, 6 the number of levels and cthe maximum number of factors. For example, if there are 3 factors A, B, O and eaoh has 11 levels J+, Bt, O» in a design, then a possible ohoioe is to use the table Un (II 6 ) listed in Table 1. UnQl9)
Table 1, numbers\eolumns
1
1 2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10 11
2 2 4 6 8 10 1 3 5 7 9 11
3
4
5
6
3 6 9 1 4
5 10 4
7 3 10 6 2 9 5
10 9 8 7 6 5 4 3 2 1 11
9 3 8 3 7 1 6 11
7 10 2 5 8 11
1 8 4 11
There is a table attaohed to eaoh Un(b") which indicates how to select columns for the s factors. For UV^ll*), the attaohed table is Table 2. Table 2. Table attaohed toU^ (II6) number of factors 2 3 4 5 6
recommended
columns 1
1
1
1 2
1 2
4 2
5 5 4
3 3
5
g
4 4
5
6
For our problem, the oolumns 1, 4, 5 are recommended. Finally we list the design of exrerimenta in Table 3. Therefore only 11 experiments are deigned for 3 factors and eaoh has 11 levels. Tables of uniform design are obtained by an integer vector (hi, •••, h,; n) in whioh hx =1, ht
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CHIN. ANN. OF MATH.
Table 3. The design of experiments No.
A
1 2 3 4 5 6 7 8 9 10 11
B B5 Bio B4, Be B3 Bg Ba B, B± B6 Bu
Ax At Aa Ai As A6 A, A» A, Ala Ait
O Gi Ca Cio Ct C2 d d Ci Ct Gi Ca
Since Kh(
w-«n(i41
(2-2)
•where p runs over the prime factors of n. Since —k=n—h (modm), the rank of matrix (^w) is at most l+$(n)/2 if n>2, i. e., the number of factors must be< /
' j possible ohoioes of «=(Ai, •••, h,)' since
hx*=l. We want to obtain the "best" h among h's. Wang and Pang (1981) noted that a best h is the minimum of the function
*<*> 4 1 ^ - M H ^ T } ) )
(2 3)
-
with respect to h, where {*} denote the fractional part of x, and its corresponding set of points is (2.1) which is called a uniform design. When n is largo it will cost much time for finding the best h. We suggest to use a set of points of the type & ( * ) - ( * , *6, - , W- 1 ) (modrc), 1
459 No. 3 Wang, Y. f Fang. K. T. NUMBER THEORETIC METHODS IN STATISTICS (II) 387
numbers of factors and levels. But we oan treat the data by regression or stepwise regression. Excmnple 1. To design a Vinylon product we shall consider the following factors: A: Temperature (0), B: Time (m), 0: Concentration of methanol (g/1), D: Concentration of sulphuric acid (g/1), E: Concentration of mirabilite (g/1). Each factor has 7 levels listed in Table 4. Table 4.
Factors and levels
factors\levela
1
2
3
A
64
66
68
B
14
16
18
4
5
6
7
70
72
74
76
20
22
24
26
C
IS
20
22
24
26
28
30
D
206
212
218
224
230
236
242
E
70
70
85
S5
S5
100
100
If we use Latin square design, then we require 49 experiments which lead to the following linear regression equation y=-42.37+0.56*i+0.38*2+0.26* 8 +0.10* 4 -0.04*5 (2.5) with a multiple correlation coefficient 5 = 0.97 and a standard deviation o-=0.83. Here y stands for the quality of Vinylon. Now we use uniform design and choose Uj.4(145), whioh designs for 14 levels. We repeat the original levels twice by the quasi-level method, and obtain the regression equation as follows (2.6) y= -57.97+0.37*1+0.46CP 2 +0.38* s +0.17*4+0.04*5 with .8 = 0.96 and cr=1.13. Equation (2.6) is olose to (2.5). The result is not too bad because only 14 experiments are arranged.
§3. Experiments with Mixtures lit factors Xlf •••, X, are non-negative and satisfy Xt-\—X, = l, then the experiments are called the experiments with mixtures, whioh offen appear in designing the chemical and metallurgical produots. In the last two decades, a lot of works appeared in the statistical literature have proposed some kinds of designs. Soheffe (1958) introduced the simplex-lattice designs and the polynomial models. He (Soheffe (1963)) suggested an alternative design, the simple-eentroid design, to the general simplex-lattice. Draper and Lawrenoe (1965) proposed to use the designs
460 CHIN. AHN. OP MATH.
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whioh minimized the bias in. the fitted model as well as the variance through minimizing the mean square error of the estimate of response over the experimental region. Thompson and Myers (1968) considered an elliptical region inside the simplex factor space by rotatable design. Snee (1973) suggested techniques for the analysis of mixture data. Cornell (1975) gave a suggestion of axial design and ho ((1973), (1981)) gave a thorough review of this subject. In this section we give a different approach to the experiments with mixtures by a uniformly distributed set of points over the simplex. This kind of designs has the same ad-vantages as the uniform design mentioned in Section 2. We pay attention to the best formulation of ingredients, and call this design the uniform design fo r experiments with mixtures (UDEM). Let T,-i = {(xi, —, as,): xt>0, » = 1, —, s, Xi+ — +a>t-l} be a part of the Surface of the unit simplex in 22s. The idea of uniform design for experiments with mixtures is to design n experiments whioh are uniformly distributed on T,_i. Let {, » = 1, •••, s—1, * = 1, •••, «} be a uniform design denned by (2.1). Then {but} with
6
*~ ? z §r L ' *=1''" n'i=1''"' s~1
(3 i:>
-
1
is a set of uniformly distributed points in I'" . In Section 2 of [13] we suggest a method to produce a uniformly distributed set of points {P*, i = l, •••, n} on Ts_i with /»„= (asm, —, aha)' and
f %*= rh^a-^'/-"), j-i, -, t-i «
(3.2)
Figure 1 shows the uniformity of the set (3.2) when s = 3 and n = 31. Example 2. Consider a regression model
•where e stands for a random error. Since Xi_-\
hX, = l, it can be reduced to the
form Y=e+ 2 e^«+ 2 euXiXj+e.
(3.3)
Consider the following special model Y=X1+X3-3Xl-3Xl+X1Xa+e, 2
(3.4)
where e~N(0, a ). When a is small (for example, cr = 0,005), we get the following da,ta for n-=17 and s = 3 by the use of UDEM (3.2). The corresponding regreooion model is
461. No. 3 Wang,-T.
Table 5. DATA No
Xt
Xa
1 2 3 4 5 6 7 S 9 10 11 13 13 14 15 16 17
.829 .703 .617 .546 .486 .431 .382 .336 .293 .252 .214 .178 .143 .109 .076 .045 .015
076 .253 .102 .307 .045 .284 .664 .215 .520 .110 .439 .798 .328 .708 .190 .590 .029
Y -1.100 - .541 - .391 - .157 - .160 .038 — .230 .146 - .103 .163 .031 - .889 .134 - .644 .155 - .388 .000
f = -0.0376 -1-1.1162X1+1.1197Xa-3.0842Xf
-s.ossoxf+.ssseXiX.,
(3.5)
whioh is olofle to the model (3.4). The multiple correlation coefficient of the equation (3.5) is 5 = 0.9999 and the estimate of standard deviation is 8 = 0.0054 whioh is close to
Xi
1 a 3 4 5 6 7 8 9 10 11 12 13 14 15
0.817 0.684 0.592 0.517 0.452 0.394 0.342 0.393 0.247 0.204 0.163 0.124 0.087 0.051 0.017
Xt 0.055 0.179 0.340 0.048 0.201 0.384 0.59a 0.118 0.326 0.557 0.809 0.204 0.456 0.727 0.033
Y 8.508 9.464 9.935 9.400 10.680 9.748 9.698 10.238 9.809 9.732 8.933 9.971 9.881 8.892 10.139
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Y^lO+Xi-aXlSXl+XiXz+e
(3.6)
•where e~iV(O, a3) wither = 0.30, and 15 experiments by (3.2). We obtain the data (Table 6) by simulation. The oorrespondind regression equation now becomes Y = 10.0908+0.7972XX - 3.4542X1 - 2.6733X1+0.8884XiX a
(3.7)
with JJ = 0.9003 and &=0.2891. Note that this regression equation deviates from the model (3.6), because there are high correlations between Xi and X\. In the original model the response Y reaches its maximum 10.0857 at Xx=0.171
and X 2 =
0.0286. From the regression equation (3.7) it is easy to show that Y reaches its maximum at X1== 0.105 and X a = 0.0196 which Y in (3.6) is 10.0728 that is close to 10.0857. Hence, from the point of view of the best formulation of ingredients, the result of this example seems nice.
§4. Geometric Probability and Simulation In this seotion, we use two "case study" methods to illustrate the applications of uniformly ditsributed sets of points over B to the problems in geometric probability and simulation. The readers can understand the general principle from these two examples without essential difficulties. The questions come from practical problems, and have no satisfactory solution for a comparatively long time. Now we propose algorithms for finding their feasible solutions by the use of number theoretic method. A. The area of the intersection between a fixed circle and the union of a set of random circles. Given a unit circle K with oentre at the origin. There are m random circles Ox,—, Om with centres Pi, •••, P m and radii Blt —, Rm respectively. Assume that P,~tf s (0, cr?Ja), where
flf-jrn(QiU-uo.). It is required to know the distribution of 8. It is easy to find the distribution of 8 for the case of m=l, since the overlap area of two oiroles can be expressed explicitly in terms of the distanoe between their centres. When m > l , it seems difficult to find a feasible method for finding the distribution of 8. This is a problem of geometrical probability. A. natural way is to use simulation. The classical method is the sooalled lattice points method. Let ABOD be the circumscribed square of the unit circle K as shown by Figure 3. Divide the square ABOD into r? equal subsquares of side 2/(»—1). We have n* lattice points
463 No. 3 Wang, Y. $ Fang. K. T. NUMBER THEORETIC METHODS IN STATISTICS (II) 391
in ABOD. Suppose that there are N points lying in K. "We now use oomupter to produce m reandom oiroles with oentres P4~2V2(0, of/ 2 ) and radii .8,(6 = 1, •••, m). Suppose that M points among the N lattice points are oovered by these m random circles. Then we get an obvservation TCM/N for the distribution of 8. We then generate other m random oiroler and obtain another observation. Continuing this process, we have an empirical distribution of 8. This method is called the method I. Its convergent rate is slow. The more serious thing ia that its accuracy is low even if we take N large, because there are O(«jN) points among the N lattice points located nearly the boundary of K. However we may use a set of points (3.1) of s = 2 with a linear transformation instead of the above na lattio e points and do Simulation as before (of. Figure 3). We call this method the method II which gives faster convergent rate and higher accuracy than the method I. For example, we take w = l and compare the results of 8 given by these two methods with the exact value of 8. It takes more than 180 minutes by the computer IBMPO/XT and the method I to get a sample of size 1, 000 with an error 0.15, but it needs only 4 minutes by the same computer and the method II to obtain a sample of size 1, 500 with an error 0.02. In general, the method II is faster than the method I about 100~1000 times. Note that the set of points (3.1) of s=2 is denned in ABOD, but not in K. Inspiring by the numerical interation over K Stated in [13], it is possible to define a set of points that is uniformly distributed over K by means of the set (3.1). Let r» = rcos(2
.
0<6<2TC,
0
(4.1)
and lot (#j, r) (l
2% S ru to approximate the areas oovered by these m random circles. We oall this method the method III. In order to compare the methods II and III, we also takeTO= 1 and consider the intersection area of two unit oiroles K and O with distance d between their centers (of. Figure 6). We use a set of 1069 points of the type (3.1) in which
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844 points are lying in K for the method II, and a set of 828 points of (3.1) is used for the method III. The result is given by Table 7. Table 7 errors\d
0.1
method II method III
-0.55% 0.00%
0.75 0.07% 0.04%
0.8 0.19% 0.12%
1.3 0.013% 0.08%
Prom Example 2 of [13] we may obtain a uniformly distributed set of points denoted by {#*, ife = l, •••,?i} over K, by which we may have another method method IV. Now the simulation ia based on {a?*, & = 1, •••, n} with the same manner as method II. Our simulation results show that the methods III and IV almost have the same aoouraoy. B. The problem of coveringthe sphere by random belts with a fixed width. This problem oomes from the steel rolling [1]. People wish to increase the life of the roller by using a random rotary ball roller instead of a fixed roller. Its mathematical model may be stated as follows: Let 8 be the unit sphere xl+xl+%l = ± and 8 a constant satisfying 0 < 8 < 0 . 3 . Let R be a great oirole which is uniformly distributed on 8 and Ga(B) be the belt on 8 with width 8 and with R as the equidistant partition curve. Let G^, ••••, (?«„, ••• be a sequential sample of the population G>(R). For any x£S, we denote by Dx(x) the number of belts which cover x in the first N random belts. If there is a point in 8 that is covered by m telts whereTOis a given positive integer, we say that the roller is useless. For a given integer m, let Tm be the minimum of N such that DK(x)>m for some x£8, i. e., Tm = min{iV: Z), (x) >m, for some x£ 8}. (4.3) The Tn stands for the life of the roller. We wish to obtain the distribution of Tm and to find some way to inorease the life of the roller. It seems difficult to give a formula for the distribution function of Tn, which leads us to do the problem by simulation. In our simulation the following facts are used: From Exmple 3 of [13] we oan obtain a set {**-= (0*1, $** x^)', Jc = l, •••, n} which is uniformly distributed on 8. More precisely, a;kl=cos(5rcw), (4.4) wM = Sin(oFOni)cos(2i7roM), A = l, •", n,
{
Xxa = Sin (won) sin (2i7rcM), where i?<(cw)=5w, * - l , 2, jfc=l, - , n, {&»=(&*!, &»)', * = 1 , - , n} ia a uniformly distributed set on P, and
465 No. 3 Wang, X. # Fang. K. T. NUMBEE THEORETIC METHODS IN STATISTICS (II) 393
*\T'
~2~)
i. e.,
F% (x) = A (1 - cos (txx) )
and
Faix) =x.
Therefoi e
5 M =-i- (1 - cos (orom) )
and J a3»2 = 2N/6J(1 -6?i cos(2OT6M),
(4.5)
[ asw=2N/&w-&|iSiii(2w&M), * - l , 2, —, n. Let T* = min{2V: .D* (#*) >m, for some A, l < i < » } , When n is sufficiently large, T*m is close to Tm and the distribution of T*m is close to that of Tm. Our simulation is based on Tm. Given a point v on #, it corresponds a great circle B such that the normal direotion of the plane including B is the direction of OV. If we identify the points V and -v, then it has a one-one 'correspondence between the points on 8 and the great circles and consequently the belts with width 8 on 8. Thus, to generate a random belt GS(B) which is uniformly distributed on 8 is equival ent to generate a point v£8 which is uniformly distributed on 8. We also use Qt(v) to denote the belt corresponding to V. Our problem of simulation includes the following steps: Step 1. Give m and 8, for instance, m = 20 and 8 = 0.2. Step 2. Choose a suitable n (for example, M = 1069) and produce a set of n points {**, i = l, •••, n} which are uniformly distributed on 8. Step 3. By a standard technique of the simulation, generate sequentially the points »!, v2, ••-, which are independent and uniformly distributed on S and consequently we have the corresponding belts (?a(t>i), Gt(va), •••. Step 4. If there are JV(iV = l, 2, •••) random belts to be generated in the current step, account the number of belts covered x* and denote by !)»(*»). If •#»(**) =«i for some k, go to Step 5, otherwise go back to Step 3 and generate the (N+l)th random belt. Step 5. Account the number of random belts generated already. This number is an oboervation of T*m. Repeat the above process «<, times and obtain a sample of size «<> of T*m. We take no=6000, the respective Sample mean and the sample standard deviation are 5^=99.7 and o-(T*m) =9.8. Furthermore, tiie corresponding empirical distribution is close to the normal distribution.
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By the same way, we obtain 20 samples of size no = 5000 (total 100,000 observations) and find that the results are very close to each other. Since the Tm (or T*m) stands for the life of roller, we note that sometimes T*m can be reached at 125 in 100.000 obcervations by the above simulation. We denote the corresponding normal directions by v\, vl, •••, l?J2s- This means that if v^v*, « = 1 , 2, •••, are fixed, we always have 27J, = 125 in the case of 8 = 0.2 and m=20, which is better than the above random ohoioes of {vt}. Is it possible to find another set of ***, vl*, •••to beat the above {v*, i = l, •••, 125}? We may use the uniformly distributed sets of points on I 2 to produce the sets of {#£*, &=1, ••-, n} on 8. First we ohoose ??.=126 and find the corresponding T^ = 126. Then we increase n one by one untill the Tjj can not be increased any more. Finally we find a set {vt* = (via, %2> vxa) '» * - l , 2, - , 155} by which 2 ^ = 155! The vf is given by \ vw = 2 N/ iw - bl-L cos (2w&M), •• «»3 = 2's/6jii — &I2 sin (25T6S2), where the set {6^= (5fcl, 6 l2 )', k = l,—, 155} is produced by (hx, h2; n) = (1, 20; 155) (See [13], Section 3 for details). This example indicates that the number theoretic method wins the champion of 100, 000 experiments by Monte carlo method in our simulation. Acknowledgement: We wish to thank Mr. Wei Gang for his excellent computer works.
References [ 1 ] Cheng, P., An open problem in steel rolling, Mathematics in Praotice and Theory, 2 (1983), 79—79. [ 2 ] Cornell, J . A., Experiments with mixtures: A review. Teohnometrics, IS (1973), 437—455. [ 3 ] Cornell, J . A., Some comments on designs for Cox's mixture polynomial, Teconometrics, 17 (1975), 25—35. [ 4 ] Cornell, J . A., Experiments with Mixtures, designs, models, and the analysis of mixture data, 1981, Wiley, New York. [ 5 ] Cox, D. B., A note on polynomial response functions for mixtures, Biometrika, 58 (1971), 155—159. [ 6 ] Draper, N. B., & Lawrence, W. E., Mixture designs for three factors, / . Boyal Statist. 80c. B, 27 (1965). 450—465. C T ] Hua, L. G., & Wang, Y., Applications of Number Theory to Numerial Analysis, Springer-Verlag and Science Press, 1981, Berlin/Beijing. [ 8 ] Scheffe, H., Experiments with mixtures, J. Sfvyal Statist, 80c. B, 20 (1958), 344—360. [ 9 ] Seheffe, H., The simplex-centroid design for experiments with mixtures, J. Boyal Statist. Soc. B, 25 (1963), 235—263. [10] Snee, E. D., Techniques for the analysis of mixture data, Technometrics, 15 (1973), 517—528. [11] Thompson, W. O. & Myers, B. H., Besponse surface design for experimonts with mix-tures, Technometrics, 10 (196S), 739—756. [12] Wang.Y. & Fang, K. T., A note on uniform distribution and experimental design, Kexue Tongba, 26 (1981), 485—489. [13] Wang, Y. & Fang, K. T., Number theoretic methods in applied statistics, Chin. Ann of Math., 11B: 1(1990), 41—55.
^
467 No. 3 Wang, J. $ Fang. K. T. NUNBEK THEOBETIC METHODS IN STATISTICS (II)
^mrp-or-Hi i n c L.
-\
7__
A
f__II_I
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 6
L
IJ
395
468
Vol. 39 No. 3
SCIENCE IN CHINA (Series A )
March 1996
Uniform design of experiments with mixtures* WANG Yuan ( £
%)
(Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, China)
and FANG Kaitai (-jfJV B ) (Hong Kong Baptist University; Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China) Received July 3, 1995 Abstract
Consider a design of experiments with mixtures: 0
where a,, &,, l^>'<s are given constants.
1-JC,= 1,
A method is proposed to treat this model by the theory of uniform
distribution in number theory. Keywords: experimental design, uniform design, experimental design with mixtures, number-theoretic net (NT-net), discrepancy.
As an application of the theory of uniform distribution in number theory to the experimental design, we propose a method, the so-called uniform design'11. Suppose there are s factors x,, •••, xs. We may assume without loss of generality that the experimental domain is the unit cube C s =[0, l] s and each point in Cs corresponds to an experiment. The idea of uniform design is to find out a set of n points
3* = {ck=(ckV-,cks), \
We call the set
^
There are a number of methods which can produce the NT-nets with different s and n to be compiled in tables'3' fl. Then we may arrange n experiments on 3" and obtain a point c which has the best experimental result among these n results on &. However, in practice, especially in chemical experiments and chemical engineering, some constraints should be added to the factors, for instance, O^a^x^b^l,
K i ^ s , Exf=l,
(1)
where a,, bt are given constants. The experimental design with constraints is called the experimental design with mixtures. Cornell'51 gave a comprehensive exposition on the experimental design with mixtures, for example, he changed domain (1) by an ellipsoid and then transferred the original problem by an (s-1)-dimensional problem. The aim of the * Project partially supported by the National Natural Science Foundation of China and Institute of Mathematics, Taipei.
469 No. 3
UNIFORM DESIGN OF EXPERIMENTS WITH MIXTURES
265
present paper is to apply the theory of uniform distribution to the experimental design with constraints (1). The simplest model in (1) is that a, = 0 and b,= l, \
f " 1 S=j J = (XI,-,XX):X(>0, Ki
~^-, Ki<£, 1<7<S, K j ^ s are continuous on C. Denote Oq>.
and J(q>)=det(TT')112. J(q>) is the volume element of D with respect to x. When t=s, J(v>) is the usual Jacobian. Then it follows by the independence of (p\s that
where V(D) denotes the volume of D and pt((p) the probability density function (p.d.f.) of
wL/ ( ? ) d ? = a w i
where dfl> = nci
and F^
(2)
denotes the cumulative
i-l
distribution function (c.d.f.) of
be a set of points on D and Xi=x(
470 SCIENCE IN CHINA (Series A)
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Vol. 39
where I(A) denotes the indicator function of A, i.e. UA\ = \ 1> if ^ o c c u r s > ^ | 0 , otherwise. Then the empirical function of *„•", xn may be defined by n
i-i
Then s
(3)
W\Fn(r)-F(r)\
is the Kolmogorov-Smiraov distance for the goodness-of-fit test of F(r). If F(r) is the uniform distribution of C, then by (2),
where V(t
Now we denote (3) by V(t^r)
n (g»\ =sup f /•-•» _
This is a measure for uniformity of the set J'on D and it is called the F-discrepancy of J1. When s = t, D=C and x, = 9i, K i < s , D f ( ^ ) is just the discrepancy of^* in the sense of Weyl[2! and is denoted by IX,^"). Let
^
={ck=(ckl,-,ck,),l
be a sample from the uniform distribution on C with discrepancy D(^). F,~l(r) the inverse function of F/lr), KKt, and
Denote by
xk=x(F-\Ck)), where
*- 1 (O=(^r l (c«).-,J 7 r l (cj), ^ f c ^ w Now we proceed to find the F-discrepancy of the set J'* = {xk=x(F'\ck)),
*•„«=- I/(*-!(q)
i-i
n
1=1
Kk^n}.
Since
471 No. 3
UNIFORM DESIGN OF EXPERIMENTS WITH MIXTURES
267
we have
Df(^*)=sup - I/fo^FMJ-lWi) =SUP - Z/(ct<«)-riu, =D(^). This means that the F-discrepancy of 3s* is just the D(^). Therefore, starting from a set J1 on C with discrepancy d, we can induce a set ^ * on D with F-discrepancy d. This method is called the inverse transform method and we often call the uniformly scattered set of n points with F-discrepancy o(n~v*) the NT-net. We can construct the NT-nets with F-discrepancy O(w~1(logn)'~I)[3> M2 The domain S(a, b)
Suppose a=(a 1 ,-", a) and b = (bl,m", b^ are two given vectors satisfying and a = i > i < l and & = £ > , > 1. ;«]
(4)
i-l
Denote by S(a, b) the domain % » ) = x=(x l ,-,xJ:a / <x l
'-' J
I
Note that (4) is a necessary and sufficient condition that S(a, b) is not empty. In fact if (4) holds, then S(a, b) is not empty clearly. On the other hand, if a ^ l or b^l, then s
s
£xi>l or £x,
i=l
(5) Then S
S
S
i-l
i-l
i-l
Z (b,-a^y,='Z x,-Z at= 1 - a .
Write
We have Z4y,= l, d,>0, Ki<s. i-l
Therefore S(a, b) and the condition (4) become S(d) =
ly=(yl,-,yyyeC\tdiyi=l,di>O,l^s\
472 SCIENCE IN CHINA (Series A)
268
Vol. 39
and
£4>i,
(6)
i=l
respectively, where
S*W =
Let
\y=(y»--%y):yl>O,l
First we give an algorithm for finding an NT-net on S*(
(7)
Then 4= C (
(8)
where
The proof will be given in sec. 4. Hence the volume of S*(d) equals K(S*(rf))=|
Adz=c(d)\
z r 2 - z ^ 2 d z 1 - d z , . 1 = c(rf)/(s-l)!.
We conclude that Z,,---, Zs_, are mutually independent and the p.d.f. and c.d.f. of Z; are p,= (s-j)z-'-1 and
Gj= pj(t)dt = z°-J, Jo
z€C\Kj<s-l
respectively. We can use the inverse transform method to get an NT-net on S*(d) as follows: start from an NT-net on C*~l M. Since the inverse function of G,(t) is i
G7\z)=zs-i ,
1<;<J-1,
we have an NT-net on S*(d), {yt=(y*,-,yJ,i*kZn},
(9)
473
No. 3
UNIFORM DESIGN OF EXPERIMENTS WITH MIXTURES
269
where
Substituting them into (5), we have an NT-net {xk=(xkl,•••, x^, Kfe
£ x f = 1 >,
where
{xk!=(\-a)c^ -c^
(\-cf)+as
(xta=(l-a)cij^-c^_2ct,_1 + ar Kj<s-l,KKn. 5
Since the Jacobian of transformation (5) is a constant n(fy~ai)>
tne
F-discrepancies of
i-l
{xk: 1 ^k^n} and {j>t:l ==;&<«} are of the same order. So we study the S(d) only in the latter. 3 Volumes of S*(d) and S(d) Our purpose is to get an NT-net of n points on S(d). The algorithm is that we choose (9) an NT-net J" of m points on S*(d) of which there are nearly n points falling on S(d). Since ^* is uniformly scattered on S*(d), the ratio — should be approximately n equal to the ratio of the volumes of S*(d) and S(d), i.e.
m_ „ V(S*(d)) V{S{d)) ' ~ n Hence the estimation of the number m is reduced to the estimations V(S *(
F(S*(rf))=c(
(10)
Now we proceed to evaluate V(S(d)), V(S(d)) = c(d)N(d), where
N(d)={,z\-2-zs_2dzl-dzs_l.
(11)
474 SCIENCE IN CHINA (Series A)
270
Vol. 39
First we find out a recurrent formula which reduces the (s— l)-dimensional integral N(d) to an (s — 2)-dimensional integral. Secondly, we give the expression of N(d) for the case s = 2 and finally we get the expression of N(d) for s = 3, 4,••• by successive iterations. 3.1 Reduction We may suppose without loss of generality that
(12)
d&d^-d, Otherwise, we may change the order of variables such that (12) holds. (i) If ds>\, then S(d)=S*(d), and
(13)
*»-W-
(ii) If dx>\ and ds
Zl
^ where (zj,-,
Z^^GC1"2
' f ,.=z,-v1,2
and (yj,--, ^ e C 8 " 1 . When z,e(0, dX we have -^- >l and (13) z
i
with s replaced by s - 1 . JV(rf)=| z\-2dzA zr 3 -z s - 2 dz 2 -dz s _, J s .(i.,..,i) Jo
+ zrl
[
I,(i,,i)' r '"' z - dzr ''' z -'
It yields by (4) or (6) that z, must satisfy 1-2
in the latter integral; otherwise, this integral is equal to zero. Therefore we obtain a recurrent formula
(14)
475 No. 3
UNIFORM DESIGN OF EXPERIMENTS WITH MIXTURES
271
(iii) If rf,< 1, z, must satisfy 1 -z,
zr2N
tf(rf) =
V zi
JI-J,
z
i /
dz,
(15)
3.2 s = 2 JV(l,i 2
(16) (d,
(17)
3.3 s = 3 If dj^l, d 3
/•min(l,i,+<(,)
N « - 4 - +f 2
Ji,
/
J
J
\
J2
C1
J
J2
i
2
. V A . A U . A +f ZlAdZi=,3_A. \ zi
z
i /
2
JJ,
z
(18)
Similarly we have by (13)—(17) the following
JV(rf)-(4,+4)mm(l,4,+4)- ***lf+&
- ^ L
(
N{d)=dx- i - (^+^+^+l)+(^+^ 3 )min(l,rf 2 + f i 3 )- l m n ( 1 ' 2 d 2 + d 3 ) 2
(19)
( d 1 < l , l - i 1 < ^ > (20)
^ ( ^ = 4 , ( ^ + 4 , - 1 + ^ - ^ (^ 1 <1,^ 3 ^1-^ 1 <^ 2 ),
(21)
JVW= y (4+<* 2 +4-l) 2 W < l , i l < l - r f 1 ) .
(22)
We can find complicated analytic expressions for the case s^4, but we prefer to suggest the use of statistical simulation for an estimation of the ratio r. take an NT-net on Cs~\ where m is sufficiently large. If S1 has discrepancy d, then the induced set •&r*=={yk= (JW> JO* Kfc^m} on S*(d) has F-discrepancy i (see (9)). It yields from the uniformity of S1* on S*(d) that the number of ^** falling on S(d) is asymptotically equal to
476 272
SCIENCE IN CHINA (Series A)
f
Vol. 39
V(S(d)) 1
where [x] denotes the integral part of [JC], i.e. — may be regarded as an approximate value n of the ratio V(S*(d)) V(S(d)) 4 Hie evaluation of A RecaU (7) and (8): <4 = (detffiT)7, where
Z,
Z
Z,
2
I
Z2
Z
z,-i
-'
First we shall prove the following lemma on determinant. Lemma. Let at>0, l^i^s,
and
al + ap ^ _
a2
~a2'
0
-a2, +a
3'
•. .
0
• . . '
«s-2+as-i
~a,-\
-a,.,
a s -, + as
Then
Proo/. The conclusion is obvious if s = 2. Now by the mathematical induction we assume that s > 3 and that the conclusion is true for 4> where 2 < t < s - l . Then a2+a3, 4=(«i+«2)
-a3, . .
'• • •
0
0 .. .
«5-2 + a s -i
~fl»-i
-a,_,
«,-, + «,
_ _ ^ _
477
No. 3
UNIFORM DESIGN OF EXPERIMENTS WITH MIXTURES
<J3 + a4, ~a4>
_a2
273
0
-at, a4+a5, " • - . . .
a s -2 + a s _,
s
-as_,
s
= (a1 + a2)a2---as^a~1-a^3---asga,r1 S
5
5
i-2
i-2
i=3
S
(-1
The lemma is proved. The evaluation of A: let A be an (s—1) x(s—1) matrix with det/<=±l. Then 4 is invariant if H is changed by <4//. Let Atj=(auv), K M , y^s —1, where auu = \, l ^ « ^ s - l , a,7=g and aul) = 0 for the remaining u, u. Then the difference between //and AHkes only on the fact that the i-th row of Afl is equal to the i-th row of H plusing g multiple of the y-th row the //. Using these operations, we can find out a matrix A such that
2 1
AH=
3
1
Hence 4 2 =det (flH") = det (AHH'A') = (zf~2- • • z.^detG, where
/^r 2 +^ 2 \
o
-^ .
.
o •
:
:
•
•
:
-c2,
•
•
-
^
\
dZ+d;1/
'
By the lemma it follows that detG=(d,-4,)- 2 ti? = c(rf)2, i-l
and therefore 4 = c(rf)zr2-z,-* 5 Examples Example 1. Suppose a = (0.6, 0.15, 0.05) and A=(0.8, 0.25, 0.15). Then a=0.6+0.15
478 274
SCIENCE IN CHINA (Series A)
Vol. 39
+ 0.05 = 0.8, b = \.2 and rf=(l, 0.5, 0.5). By (10), (11) and (19), we have N(d) = l - j
- - ^
=0.25
and V(S*(d)) V(S(d))
_ 0.5 0.25
=2
Hence we may get an NT-net of nearly [n/2] points on S(d) (or S(a, b)) which is induced by an NT-net of n points on C2. For example, if an NT-net of 17 points on C2 is generated by the good lattice method, we can obtain 9 of them in S(a, b) as follows: x, = (0.765 7, 0.175 2, 0.059 1), x2=(0.740 6, 0.176 2, 0.083 2), *3 = (0.723 3, 0.161 3, 0.115 4), x4=(0.709 3, 0.227 4, 0.063 3), x5=(0.697 1, 0.207 5, 0.095 4), x6=(0.686 2, 0.180 1, 0.133 6), *7=(0.667 2, 0.239 9, 0.093 0), x8 = (0.658 6, 0.204 1, 0.137 3), *9=(0.635 5, 0.232 2, 0.132 2). Example 2. Suppose «=(0.3, 0.2, 0.1, 0.05) and ft=(0.6, 0.5, 0.4, 0.2). Then a = 0.65 and d=l —, —,
—, — I. By (14) we have
+ f T Z\N(-%- ,-$- , ^ - )dz,+ [' Z\N(^J1
\ 7z,
For /„ we have -^->h 7z,
7z,
lzx )
)§_
\ lzl
and so Nl^-,
-%-,
•=-)=
\ /zl
I
l
lzx
(T
2A
lzx )
,J-
lzx
\
I
,-f- )dz1 = / 1+ / 2+ 7 3 , say. 7z, )
by (13). Therefore
26
For 72, we have —— >1 and —— <1, and so by (18), 7z,
/Z[
For /,, we have ~ ^ 1 , 1 - - ^ - < - ^ - , - ^ - + — ^ 1 . 3 7z, 7z, 7z, 7z, 7z,
Therefore we deduce by (19),
479 No. 3
UNIFORM DESIGN OF EXPERIMENTS WITH MIXTURES
i
f
1,15
(
81
\ iA
275
88
Hence JV(
ns*(d)) K(S(rf))
=
£i_ 276/6 xV
=
j43_ 276
=12427536623... L242/
^W)2i
•
This means that we may obtain an NT-net of nearly n points on S(d) which is induced by 3 an NT-net of -r=— 276 n points on C .
L
J
References 1 Wang, Y., Fang, K. T., A note on uniform distribution and experimental design, Kexue Tongbao, 1981, 26:485. 2 Weyl, H., Uber die Gleichwrteilung der Zahlen mod Eins, Math. Ann., 1916, 77:313. 3 Hua, L. K., Wang, Y., Application of Number Theory to Numerical Analysis, Heidelberg and Beijing: Springer-Verlag and Science Press, 1981. 4
Fang, K. T., Wang, Y., Number-theoretic Methods in Statistics, London: Chapman and Hall, 1993.
5 Cornell, J. A., Experiments with Mixtures, Design, Models, and the Analysis of Mixture Data, New York: Wiley, 1990. 6 Wang, Y., Fang, K. T., Number-theoretic methods in applied statistics, Chinese Ann. Math. (Series B), 1990, 11: 51. 7 Wang, Y., Fang, K. T., Number-theoretic methods in applied statistics (II), Chinese Ann. Math. Ser. B, 1990, 11:384.
481_
LIST OF PUBLICATIONS BY WANG YUAN
I. Articles 1. On the representation of large even integer as a sum of a product of at most 3 primes and a product of at most 4 primes, Ada Math. Sin. 6:3 (1956) 500-513, in Chinese. 2. On the representation of large even integer as a sum of a prime and a product of at most 4 primes (Conditional result), Ada Math. Sin. 6:4 (1956) 565-582, in Chinese. 3. (with A. Schinzet) A note on some properties of the functions
482
16. On the representation of large integer as a sum of a prime and an almost prime, Sci. Sin. 11:8 (1962) 1033-1054; see also: Ada Math. Sin. 2 (1960) 168-181. 17. A note on interpolation of a certain class of functions, Sci. Sin. 10:6 (1960) 632-636. 18. (with Hua Loo Keng) On the calculation of mineral reserves and hillside areas on contour maps, Ada Math. Sin. 11:1 (1961) 29-40. 19. On numerical integration and its applications (Number-theoretic methods), Shu Xue Jiang Zhan 1 (1962) 1-44, in Chinese. 20. (with Hua Loo Keng) Finiteness and infinity, discrete and continuity, Kexue Tongbao (1963) 4-21, in Chinese. 21. On the estimation of character sum and its applications, Shu Xue Jiang Zhan 1 (1964) 78-83, in Chinese. 22. A note on the maximal number of pairwise orthogonal Latin squares of a given order, Sci. Sin. 13:5 (1964) 841-843. 23. (with Hua Loo Keng) On diophantine approximations and numerical integrations I, Sci. Sin. 13:6 (1964) 1007-1008. 24. (with Hua Loo Keng) On diophantine approximations and numerical integrations II, Sci. Sin. 13:6 (1964) 1009-1010. 25. Remarks on the interpolations of a certain class of functions, Sci. Sin. 14:4 (1965) 429-431. 26. (with Shieh Shen Kang and Yu Kun Rui) Remarks on the difference of consecutive primes, Sci. Sin. 14:5 (1965) 786-788. 27. (with Shieh Shen Kang and Yu Kun Rui) Two results concerning the distribution of primes, Proc. Univ. Sci. Tech. China 1 (1965) 32-38, in Chinese. 28. (with Zhu Yao Cheng and Jian Yun Cui) Remarks on the number theoretic methods in numerical analysis, Proc. Univ. Sci. Tech. China 2 (1965) 213-218, in Chinese. 29. (with Hua Loo Keng) On numerical integration of periodic functions of several variables, Sci. Sin. 14:7 (1965) 964-978; see also: Proc. Univ. Sci. Tech. China (1966) 1-12. 30. On interpolation of a certain class of a functions, Kexue Tongbao 9 (1966) 387-389, in Chinese. 31. On the maximal number of pairwise orthogonal Latin squares of order s (application of sieve methods), Ada Math. Sin. 16:3 (1966) 400-410, in Chinese. 32. (with Hua Loo Keng) On uniform distribution and numerical analysis (I) (number theoretic methods), Kexue Tongbao 3 (1973) 112-114, in Chinese. 33. (with Hua Loo Keng) On uniform distribution and numerical analysis (II) (number theoretic methods), Kexue Tongbao 4 (1973) 165-166, in Chinese. 34. (with Hua Loo Keng) On uniform distribution and numerical analysis (I) (number theoretic methods), Sci. Sin. 16:4 (1973) 483-505. 35. (with Hua Loo Keng) On uniform distribution and numerical analysis (II) (number theoretic methods), Sci. Sin. 17:3 (1974) 331-348.
483 36. (with Hua Loo Keng) On uniform distribution and its applications for multi-dimensional functions of bounded variation, Proc. Univ. Sci. Tech. China 1 (1974) 39-67, in Chinese. 37. (with Hua Loo Keng) On uniform distribution and numerical analysis (III)— Number theoretic methods, Kexue Tongbao 12 (1974) 559-560, in Chinese. 38. (with Hua Loo Keng) On uniform distribution and numerical analysis (III) (number theoretic methods), Sci. Sin. 18:2 (1975) 184-198. 39. (with Pan Cheng Dong and Ding Xia Xi) On representation of large even integer as a sum of a prime and an almost prime, Kexue Tongbao 8 (1975) 358-360, in Chinese. 40. (with Pan Cheng Dong and Ding Xia Xi) On the representation of every large even integer as a sum of a prime and an almost prime, Sci. Sin. 18:5 (1975) 599-610; see also: Proc. Shan Dong Univ. 2 (1975) 15-26. 41. Remarks on a theorem of Davenport, Ada Math. Sin. 18:4 (1975) 286-289, in Chinese. 42. (with Hua Loo Keng) A note on simultaneous diophantine approximations to algebraic integers, Sci. Sin. 20:5 (1977) 563-567. 43. On Linnik's method concerning the Goldbach number, Sci. Sin. 20:1 (1977) 16-30. 44. (with Xu Guang Shan and Zhang Rong Xiao) On number theoretic method in numerical integration of multi-dimensional space I, Ada Appl. Math. Sin. 1:2 (1978) 106-114, in Chinese. 45. (with Hua Loo Keng and Pei Ding Yi) On a set of independent units of cyclofomic field, Ziran Zazhi 5 (1978) 6, in Chinese. 46. (with Yu Kun Rui and Zhu Yao Cheng) Remarks concerning a transference theorem of linear forms, Ada Math. Sin. Ad. 22:2 (1979) 237-240, in Chinese. 47. (with W. M. Schmidt) A note on a transference theorem of linear forms, Sci. Sin. 22:3 (1979) 276-280. 48. (with Wang Lian Xiang and Ren Jian Hua) A note on a transference theorem of the systems of linear congruences, Ada Math. Sin. 24:2 (1981) 303-307; see also: Proc. Northwestern Univ. 2 (1979) 12-23. 49. (with Yu Kun Rui) A note on some metrical theorems in diophantine approximations, IHES/M 297 (1979); see also: Chin. Ann. Math. 2 (1981) 1-12. 50. (with Hua Loo Keng) Applications of number theory to numerical analysis, in Recent Progress in Analytic Number Theory, eds. H. Halberstam and C. Hooley, Vol. 2 (Acad. Press, 1981), pp. 111-118. 51. (with Fang Kai Tai) A note on uniform distribution and experimental design, Kexue Tongbao 6 (1981) 485-489. 52. On diopantine approximation and approximate analysis, Kexue Tongbao 5 (1982) 468-472.
484
53. (with Xu Guang Shan and Zhang Rong Xiao) On number-theoretic method in numerical integration of multi-dimensional space II, Ada Appl. Math. Sin. 4 (1982) 414-417. 54. On diophantine approximation and approximate analysis I, Ada Math. Sin. 25:2 (1982) 248-256. 55. On diophantine approximation and approximate analysis II, Ada Math. Sin. 25:3 (1982) 323-332. 56. A note on the approximate solution of the Cauchy problem by number theoretic nets, Chin. Ann. Math. 3 (1982) 451-456. 57. On additive equations in an algebraic number field, Kexue Tongbao 5 (1985) 583-587. 58. (with Shan Zun) A conditional result on Goldbach problem, Ada Math. Sin. (New Series) 1 (1985) 72-78. 59. On a system of diophantine inequalities, Kexue Tongbao 17 (1987) 1162-1165. 60. Bounds for solutions of additive equations in an algebraic number field I, Ada Ari. 48 (1987) 117-144. 61. Bounds for solutions of additive equations in an algebraic number field II, Ada Ari. 48 (1987) 307-323. 62. Number theoretic method in numerical analysis, Cont. Math. AMS 77 (1988) 63-82. 63. Diophantine equations and dipohantine inequalities in algebraic number fields, Cont. Math. AMS 77 (1988) 83-94. 64. Diophantine inequalities for forms in an algebraic number field, J. Number Theory 29:3 (1988) 324-344. 65. On homogeneous additive congruences, Sci. Sin. (A) 32:5 (1989) 524-536. 66. On small zeros of quadratic forms over finite fields, J. Number Theory 31:3 (1989) 272-284. 67. (with Fang Kai Tai) Number theoretic methods in applied statistics, Chin. Ann. Math. 11B (1990) 51-65. 68. (with Fang Kai Tai) Number theoretic methods in applied statistics (II), Chin. Ann. Math. 11B (1990) 484-494. 69. (with Fang Kai Tai) A sequential algorithm for optimization and its applications to regression analysis, in Lecture Notes in Contemporary Mathematics A, eds. Wang Yuan et al. (1990), pp. 17-28. 70. (with Fang Kai Tai) Applications of quasirandom sequence in statistics, Proc. Asian Math. Conf. (1990) 135-139. 71. Hua Loo Keng: A brief outline of his life and works, in Number Theory in Honor of Hua Loo Keng, eds. Gong Sheng et al. (Springer-Verlag and Science Press, 1991), pp. 1-14. 72. (with M. V. Subbarao) On a generalized Waring's problem in algebraic number fields, in Number Theory in Honor of Hua Loo Keng, eds. Gong Sheng et al. (Springer-Verlag and Science Press, 1991), pp. 265-277.
485
73. (with Fang Kai Tai) A sequential number theoretic methods for optimization and its applications in statistics, in The Development of Statistics: Recent Contribution from China (Longman, 1991), pp. 139-156. 74. (with Fang Kai Tai) A sequential algorithm for solving a system of nonlinear equations, J. Comp. Math. 9:1 (1991) 9-16. 75. (with Fang Kai Tai and H. L. Wong) A new method for generating the uniform distribution on the unit sphere, Tech. Rep. Hong Kong Baptist College 15 (1992). 76. Small solutions of congruences, J. Number Theory 45:3 (1993) 261-280. 77. On small zeros of quadratic forms over finite fields II, Ada Math. Sin. (New Series) 4 (1993) 382-389. 78. (with Fang Kai Tai and P. M. Bentler) Some applications of number theoretic methods in statistics, Stat. Sci. 9:3 (1994), 416-428. 79. (with Fang Kai Tai) Uniform design of experiments with mixtures, Sci. Sin. (A) 39:3 (1996) 264-275. 80. (with Pan Cheng Dong) Chen Jingrun: A brief outline of his life and works, Ada Math. Sin. (New Series) 12:3 (1996) 225-233. 81. Pan Cheng Dong: A brief outline of his life and works, Ada Math. Sin. 3 (1998) 449-454.
II. Books and Monographs 1. (with Hua Loo Keng) The Numerical Evaluation of Integrals (Science Press, 1961), in Chinese. 2. (with Hua Loo Keng) Numerical Integration and Its Applications (Science Press, 1963), in Chinese. 3. Prime Numbers (Shanghai Education Press, 1978); (GuangDong Sci. Tech. Press, 1996), in Chinese. 4. (with Hua Loo Keng) Applications of Number Theory to Numerical Analysis (Springer-Verlag and Science Press, 1981); (Science Press, 1978), in Chinese. 5. Goldbach Conjecture (World Scientific, 1984), 2nd edition (2002); (Hei Long Jiang Education Press, 1987), in Chinese. 6. (eds. Pan Cheng Biao and C. C. Yang) Number Theory and Its Applications in China, Cont. Math. AMS, 77 (1988). 7. (with Hua Loo Keng) Popularizing Mathematical Methods in the People's Republic of China (Birkhauser 1989); Some Topics on Mathematical Modeling (Hu Nan Normal Publisher, 1991), in Chinese. 8. (eds. Zhang Gong Qing et al.) Lecture Notes in Contemporary Mathematics (Science Press, 1990). 9. Diophantine Equations and Inequalities in Algebraic Number Fields (SpringerVerlag, 1991).
486 10. (eds. Gong Sheng, Lu Qi Keng and Yang L6) International Symposium in Memory of Hua Loo Keng, Vol. I. Number Theory, Vol. II, Analysis (SpringerVerlag and Science Press, 1991). 11. (with Fang Kai Tai) Number Theoretic Method in Statistics (Chapman and Hall, 1994); (Science Press, 1996), in Chinese. 12. Hua Loo Keng (Kai Ming Press, 1995); (Jiu Zhang Press, 1995); (Jiang Xi Normal Publisher, 1999), in Chinese (Springer, 1999). 13. (with Fong Yuen) Calculus (Springer-Verlag, 1997). 14. Wang Yuan, Selected Papers (Hunan Normal Publisher, 1999), in Chinese. 15. Wang Yuan's Exposition on Goldlach Conjecture (Shandong Normal Publisher, 1999), in Chinese. 16. (with Yang De Zhuang) Mathematical Career of Hua Loo Keng (Science Press, 2000), in Chinese.
