ON THE A P P R O X I M A T I O N OF S Y M M E T R I C OPERATORS BY OPERATORS OF F I N I T E R A N K BY
SHMUEL KANIEL* ABSTRACT
The following Theorem 1 and Theorem 2 are established. The proof utilizes the elementary properties of symmetric operators. It is shown that the symmetry condition is necessary for these theorems to hold. In this work we shall prove the following theorem which stems from consideration of the minimal iteration method for eigenvalue problems ([1], [2]). TrmOREM 1. Let A be a bounded symmetric operator in Hilbert space H. Let f ~ H f ~ 0 be chosen and let H k denote the span of f, Af,...,Akf. Let e be the orthogonal projection on H k and let B denote the restriction of P A to H k. Then there exists an absolute constant c(k) tending to zero as k ~ co which does not depend on A nor on f so that for some 2 ~ a(A) and some # ~ a(B): 2
[2 - "1--< c(k)I1 All-~ 4(~ 1---7 + IIa II"
(1)
The proof of Theorem 1 depends on the following propositions:
PROPOSITION 1.
if
IIAT-,fll---~llfll,
then there exists 2 ~ a ( A ) s o
that
Ix-.l__<~.
This is a direct consequence of the spectral decomposition theorem. PROPOSITION 2.
Let p, be the eigenvalues of B. Then
I.,I---II a
11"
This is a direct consequence of the minimax principle. It is easy to see that B is symmetric. Hence m - - ' p , ' = sup
211
(BS,f) =
sup
I(eaf, f)l
= sup
I
< sup I(Af'f)[ = IIAI]. - :°,~ I ' - - ~ PROPOSITION 3. I f p(x) is a polynomial having degree at most k, then p ( a ) f = p(B)f. Received November 11, 1964. * This work was sponsored in part by NSF contract 2426. 1
2
SHMUEL KANIEL
Proof.
[March
For i = 1,2,...,k, A ~ E HSO~ by , a simple induction:
The passage to polynomials is trivial. PROPOSITION 4. If B has a multiple eigenualue, then all the eigenvalues of B belong to a(A).
Proof. We shall prove that in this case H is an invariant subspace of A. So B = restriction of A to H, and hence a(B) c a(A). It is sufficient to prove that in this case the vectors f, Af, A% are dependent because then we have for some 15 k - 1 1
~ ' + '= f
C AYE H
~ ,
i=o
and we can prove inductively that
So let gl and g , be two different eigenvectors corresponding to the eigenvalue p, then some linear combination of g1 and g, can be expressed by:
Thus the linear dependence of A% i = 0, I , . . . , k is proved since not all coefficients in the last sum are zero. This leads to a contradiction and the proposition is established. Proof of Theorem 1. In view of Proposition 4 it is sufficient t o consider the case where pi, the eigenvalues of B, are distinct. So let gi, i = 1, k 1be an orthonormal base of eigenvectors corresponding to the eigenvalues pi. Let f = C:L,'fligi.We shall prove that Pi # 0. Indeed, Hk is spanned by ..a,
+
19651
OPERATORS OF FINITE RANK
3
If for some m: f t , = 0 , it follows that A Jr are orthogonal to g,, or that orthogonal to g, which is impossible. Consider now the following polynomials: ~+I
qj(x)= ~
.=, n~j
H
Cnj iCj,n
(x-
Hk
is
~i)
I-I ( ~ . - ~,)' i¢=j,n
j=l,2,...,k+l,
where ]c.j[ = }l/flnl and the sign of c.j is chosen so that for any n and j:
(m -
,,) >0. ~,)
I/~.qj(m)} = 1 ,
n#j.
H
(.J -
fljCn j i~=j'n
H
i~j,n
The polynomials qj(x) satisfy: (2) We shall prove that:
(s)
max I fljqj(ttj)] > k. J
In fact: k+ 1
,+ 1
H
(~j -
~,)
~_~ f l j q j ( p j ) = ]~ fljCn j i* j , . j= l n,j= l H (~n -- ~i) nC:j i¢=j,n
> (k + 1)(k)(k+J)(k)J
2, fli i,j,.
.n, ,~=j ,p.
H i~j,n
I.,,.-71
= (k + 1)(k),
because the expression under the root sign is symmetric in j and n. Suppose that [fl,,q.(#,.)[ > k and consider the vector g = qm(A)f. Since qm(X) has the degree k - 1, it follows that qm(A)f= qm(B)f and: k+l
g = E
qm(fl,)fl,g,.
i=1
By (2) and (3) k+l
IlgII2-- z
[qm(pi)fl, I2 = ~, ]qm(l~i)fli[2 + [fl.qm(#m)[2 >=k + k 2.
i=1
Consider now Ag - #.g. Since the degree of p(x) = (x - l~.)qm(X) is k, we have:
II a g - . . g l l 2 = II (A - p.l)qm(A)f]]2 = I](B - pmI)qm(B)f]12 k+l
=
~.~ I ( p i - pro)" q.(p,)" ,,12 < k max Ifli- . . l 2 ~
l=l
f
4klla]l 2.
4
SHMUEL KANIEL
[March
So: 2
IIa g - grog II ~ ~
IIa II [I g II
which by Proposition 1 means that there exists ;t e a(A) so that (1) holds. Thus the proof is complete. REMARI¢. This estimate has no significance from the numerical analysis point of view. In general the estimate will be much better provided that some additional information about a(A) is given. For so-called "practical" estimates the reader is referred to [2]. Let us now apply Theorem 1 to get: THEOREM 2. Under the conditions of Theorem 1 the eigenvalues of PA can be ordered so that for any #i, J = 1,2, ..., k there exists 2(j)E tr(A):
2
I#J- ~(J)l ~ ,/(k+2-j)I1~11
(4)
REMARK. This means that for large k most of the eigenvalues of B will be close to points in the spectrum of A. As in Theorem 1 the estimates depend on k and II only.
11A
Proof. As in Theorem 1 it is sufficient to consider the case where the eigenvalues of B are distinct. We shall prove the theorem by induction. Suppose that the theorem is true for j = 1,2,...,m and consider the vector
fm= fi
]=1
(A - #fl)f.
Denote by H~' the span offm, Afm,...,Ak-mfm; by Pm the projection on H~','and by B= the restriction of PmA to H~'. By Theorem 1 there exists # ~ a(B=) and 2 ~ a(A) so that: 2
I.-41 --<x/k-m+ 1 Ilall"
(5)
The induction will be complete if we show that # e tr(B)and/~ #/~i,i = 1, 2, ..., m.* Indeed, by Proposition 3 for i = O , . . . , k - m : k+l
:
fI (A - .,0S-j=l
fi j=l
-",')f--
x
l=m+l
m jfI= l
This means that A~fm, i = 0,1, . . . , k - m are spanned by gj, j = m + 1, ...,k + 1. A~fm are independent because A~f, i = 0,1, ...,k + 1 are independent; therefore
* In this case we define /~=/tm+ I, establishing (4) for m + 1.
1965]
OPERATORS OF FINITE RANK
5
by dimension argument H~' is the span of gi, J = m + 1 , . . . , k + 1. Hence B is invariant in H~' and the eigenvalues of B,, coincide with ttm+ 1, "",/~k+ 1 which are different f r o m /~l,...,/am. Thus the p r o o f is complete. The condition that A is symmetric is necessary as the following example will show. Define H to be the space of double sequences ( ' - ' , a - i , ' " , a0, "",ai, "" ) so that ~,~_oola,l 2 < oo. Define A to be the left shift operator, i.e. if q~i is a unit vector ("',0,"-,0, li, "",0,'") o(3 < i < o0, then Ad)i = dPi-r A is unitary so for any 2~¢r(A): [2 t = 1. Let f = ~bk; then Hk will be spanned by ~bo,dpt, "',~bk. The point 0 is the only point in a(B)(~b o only is an eigenvector so [0 - 21 = 1 contrary to the t h e o r e m ' s conclusion. -
-
REFERENCES 1. D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra, Translated from Russian by R. C. Williams, W. H. Freeman and Co., San Francisco and London (1963). 2. S. Kaniel, Estimates for some computational techniques used in linear algebra, to appear. STANFORDUN[VERSITY STANFORD,CALIFORNIA
SOME REMARKS ON NUMBER THEORY BY
P. ERDOS ABSTRACT
This note contains some disconnected minor remarks on number theory. I. Let
Iz l--1, l__
(1)
be an infinite sequence of numbers on the unit circle. Put
s(k,n)= ~
z~,
A, = l i m s u p
j-I
l s(k,n)[
k=oo
and denote by Bk the upper bound of the numbers Is(k,n)l. If zj = e e~tj~ ~ 0 then all the Ak'S are finite and if the continued fraction development of has bounded denominators then Ak < ck holds for every k (c, c t , ' " will denote suitable positive absolute constants not necessarily the same at every occurrence). In a previous paper [2] I observed that for every choice of the numbers (1), lira SUpk= ~Bk---- oo, but stated that I can not prove the same result for Ak. I overlooked the fact that it is very easy to show the following THEOREM. For every choice of the numbers (1) there are infinitely many values of k for which (2)
Ak > c I log k.
To prove (2) observe that it immediately follows from the classical theorem of Dirichlet that if ] Y~I = 1,1 < i < n are any n complex numbers, then there is an integer 1 < k < 10" so that (R(z) denotes the real part of z) (3)
1
R(yk) > T '
16i6n.
Apply (3) to the n numbers z,.+l,'.',z(,+l),, 0 < r < oo. We obtain that there is a k < 10 ~ for which there are infinitely many values of r so that (4)
R
~: 1=1
Received February 10, 1965.
z,,+,
>~-.
SOME REMARKS ON NUMBER THEORY
7
(4) immediately implies A k ~_ n/4, thus by k < 10 n (2) follows, and our Theorem is proved. Perhaps Ak > ck holds for infinitely many values of k*. In this connection I would like to mention the following question: Denote by f ( n , c ) the smallest integer so that if I z~l > 1, 1 < i < n are any n complex numbers, there always is an integer 1 < k < f(n, c) for which
I
t=!
>__c
A very special case of the deep results of T u r i n [8] is that f ( n , 1) = n. R~nyi and I [3] obtain some crude upper bounds for f ( n , c,) if c > 1, but our results are too weak to improve (2). II. Is it true that to every ~ > 0 there is a k so that for n > no every interval (n, n(1 + ~)) contains a power of a prime Pi < Pk? It easily follows from the theorem of Dirichlet quoted in I that the answer is negative for every ~ < 1, since the above theorem implies that to every ~ > 0 there are infinitely many values of m so that all primes Pi --< P~ have a power in the interval (m, m(1 + t/)) and then the interval (m(1 + t/), 2m) must be free of these powers. Let us call an increasing function g(n) good if to every t / > 0 there are infinitely many values of n so that all the primes p~ < g(n) have a power in (n, n(1 + t/)). It easily follows from the theorem of Dirichlet and ~ ( x ) < c x / l o g x that if (5)
[loglog n • logloglog n) g(n) = o ~ ~ g ~ g l o - - ~ n
then g(n) is good. I leave the straightforward proof to the reader. I can obtain no non-trivial upper bound for g(n). Letl<~<2andput (6)
A(n, ct) = ~,'1/ p
where in ~ ' the summation is extended over all primes p for which n < pa < ,m for some integer /~> 1. (5) and ~ p < y l / p = l o g l o g y + 0 ( 1 ) implies that for infinitely many n (7) A(n, a) > loglogloglog n + 0(1). Now we are going to prove (8)
lim inf A(n, ct) = O. B=t:O
To prove (8) we shall show that to every 8 > 0 there are arbitrarily large values of n for which (9) A(n, o~) < 5. • By a remark of Clunie, we certainly must have c __< 1. Added in proof: Clunie proved f(n,c) < g(e) n log n, A k > c k ½
8
P. ERDOS
[March
Let k = k(e) be sufficiently large. Consider ~'A(21,~) where in ~ ' the summation is extended over those l, 1 < ! < x for which the interval (2 t, ~2~) does not contain any powers of the primes Pi, 1 < i _< k. Put D(~, k) =
1
log Pk
i = 2
.
Let ax, "", ak be positive numbers which are such that for every choice of the rational numbers rl, ..., r k not all 0, ]~=lria~ is irrational. The classical theorem of Kronecker-Weyl states that if we denote by x,, 1 < n < ~ the point in the k dimensional unit cube whose coordinates are the fractional parts of n~i, 1 _< i _< k then the sequence x, is uniformly distributed in the k dimensional unit cube. From this theorem is easily follows that the number of summands in ]~'A(2', a) is (1 + o(1))xD(oq k). Thus to prove (9) it will suffice to show that for every suffÉciently large x £
]~ 'A(f, ~z) < -~- D(~, k)x.
(lO)
We evidently have ]~ u(j, x) pk
'A(f, ~) =
where u(j, x) denotes the number of those integers 1 < l < x for which the interval (2t,,t2 ~) contains a power of p j, but does not contain any power of Pt, 1 < i _< k. For fixed j we obtain again from the Kroneeker-Weyl theorem (11)
log(1 + ~) x. log pj
u(j, x) = (1 + o(1))o(~, k)
Put (12)
E'A(Y,~)=
Z
pk
u(j,x)= z~ + Z~ Pj
where in ~ l Pk < Py < T = T(k,e) and in ~2 T < pj < 2 ~. From (11) and (12) we have for sufficiently large k oo
(13)
]~1<(1+o(1)) D(a,k) l o g O + a ) x
]~
1/p~logpj<-~- O(ot, k)x
j=k+l
since ~ 1 /pj logpj converges. To estimate ~2 observe that there are [x log2/log p j] powers of pj not exceeding 2 ~, thus for every j and x
u(j, x) < x log 2 / log pj.
(14)
From (14) we have for sufficiently large T = T(k, epC) (15)
~2 < x l o g 2
~ p;>T
1/pflogp~ < - ~ D(*t,k)x
1965]
SOME REMARKS ON NUMBER THEORY
9
(10) follows from (12) (13) and (15). By a refinement of this method one could perhaps prove that for infinitely many n A(n, o0 < c / logloglog n.
Using the classical result of Hoheisel [6] n(x+xl-~
- n(x) > c x l - 8 / l o g x
we obtain by a simple computation that for all n cl / loglog n < A(n, ~) < c 2 logloglog n. III Sivasankaranarayana, Pillai and Szekeres proved that for 1 < 1 < 16 any sequence of l consecutive integers always contains one which is relatively prime to the others, but that this is in general not true for l = 17, the integers 2184 < t < 2200, giving the smallest counter example. Later A. Brauer and Pillai [1] proved that for every l > 17 there are l consecutive integers no one of which is relatively prime to all the others. An integer n is said to have property P if any sequence of consecutive integers which contains n also contains an integer which is relatively prime to all the others. A well known theorem of Tchebicheff states that there always is a prime between m and 2m and from this it easily follows that every prime has property P. Some time ago I [5] proved that there are infinitely many composite numbers which have property P. Denote in fact by u(n) the least prime factor of n.n clearly has property P if there are primes Pl and P2 satisfying (16)
n - u(n) < Pl < n;
n < P2 < n -4- u(n).
One would expect that it is not difficult to give a simple direct proof that infinitely many composite numbers satisfy (16), but I did not succeed in this. In fact I proved that there are infinitely many primes p for which p - 1 satisfies (16) but the proof uses the Walfisz-Siegel theorem on primes in arithmetic progressions and Brun's method [5]. In fact I can prove the following TI-mOaEM. T h e lower density etv o f the integers having p r o p e r t y P exists a n d is positive. We will only give a brief outline of the proof, since it seems certain that the density of the integers having property P exists and our method is unsuitable to prove this fact; also our proof is probably unnecessarily complicated. To prove our Theorem we need two lemmas. LEMMA 1. For a s u ~ c i e n t l y s m a l l e > O we have ( p 1 = 2 < p2 < ... is the sequence
of
consecutive
primes):
10
P. ERDOS
[March
~,l(Pi+ l -- P~) > clx where in ~1 the summation is extended over those Pi+l < x for which
(17)
e log x < Pi+l - Pi < (1 - e) log x.
It is easy to prove the Lemma by the methods used in 14] LEMMA 2. Put Nk = IIe k / 2.
The Lemma can be deduced from 1.6] without any difficulty. Now we can prove our Theorem. It is easy to see that if n does not have property P then it is included in a unique maximal interval of consecutive integers no one of which is relatively prime to the others. Denote these intervals of consecutive integers by I1, I2 "" where/1 are the integers 2184, 2185... 2200. Let I, be the last such interval which contains integers < x. l I I denotes the length of the interval I. To prove our Theorem it suffices to show
(18)
i=l
IIJl <X(1--C2)
Clearly none of the intervals Ij contain any primes. To prove (18) it will suffice to show that for some ca < cI (19)
•311j[ < (ci - c3)x
where cl is the constant occuring in Lemma 1 and in ~3 the summation is extended over those I j, 1 ~ j < r which are in the intervals (pj, p j+ 1) satisfying (17). Let T be sufficiently large and consider in the intervals (17) those integers all whose prime factors are at least T. It easily follows from Lemma 1 and the Sieve of Eratorthenes that the number of these integers not exceeding x is at least (20)
(1 + o(1))clx I I
(1 - 1 / p ) > c 4 x / l o g T
p
IiJl
Further these integers can clearly not be contained in intervals Ij with z 7for otherwise they would be relatively prime to all the other integers in Ij. Thus to complete the proof of our Theorem we only have to show by (20) that for sufficiently large T 1 (21) ,lIJl < T c,x/log r where in ~,~ the summation is extended over the Ij in ~3 for which The Ij in ~4 satisfy
I zJI >
r.
SOME REMARKS ON NUMBER THEORY
19651
11
Z < Iljl < (1 - e) log x.
(22)
Write
~,,lljl
(23)
= IE
~g~ltjl
r
where in ~ ' ) we have (r = 0,1...)
2"r< Itl__< 2'+1T
(24)
if2"+lT>(1-e)logx, then the upper bound in (24)should be replaced by (1 - e)log x. Now we show that for sufficiently large T and every r (25)
V(,)I ,-,4 I I J 1I < 2x / (2'T) ½.
From (25) and (23) (21) easily follows for sufficiently large T. Thus to prove our Theorem we only have to show (25). The integers in the Ij of ~ ' ) can not be relatively prime to N2,+I.T (Nk is the product of the primes not exceeding k) therefore if Ij is in an interval (uN2,+,. T, (U + 1)N2r+l.r ) Ij must lie in an interval (ai + uN2,+~.r, ai+ ~ + uNz,÷~.T) where 1 = a 1 < ... < %(Nz,+l.T ) =
N2,.+I.T --
1
are the integers relatively prime to N2,.+ ~.T" Since 2 "+ 1T =< (1 -- e) log x, it follows from the prime number theorem that N 2 . . . . r = O(X), hence we easily obtain from Lemma 2 for sufficiently large T ,,,
<
(ix])
+ 1
g2.+,.r/(2"Z) '/2 <2x/(2"Z) I/2,
thus (25) and hence our Theorem is proved. Unfortunately I can not handle the I Ij I > log x and thus can not prove that the density of the integers having property P exists. COROLLARY. There are infinitely m a n y composite integers satisfying (16). By % > 0 there are infinitely many composite integers having property P, and if there would be only a finite number of integers with property (1) then for sufficiently large i in the set of integers p~ < t < Pt+i no one would be relatively prime to the other, thus only a finite number of composite integers would have property P. This contradiction proves the corollary. Let us say that the primes have property P0, the composite integers satisfying (16) have property P~. By induction with respect to k we define: An integer n has property Pk if it does not have property Pj for any j < k, but both intervals (n, n + u(n)) and (n - u(n), n) contains an integer having one of the properties
12
P. ERDOS
P j, 0 < j < k. It is easy to see that for every k > 0 the integers having property Pt have property P too, and conversely every integer having property P has property Pk for some k > 0. It is easy to show by induction with respect to k that the integers having property Pk have density 0, hence from ~p > 0 we obtain that for every k there are infinitely many integers having property Pk. REFERENCES 1. A. Brauer, On a property of k consecutive integers, Bull. Amer. Math. Soc. 47 (1941), 328-331: Sivasankaranarayana Pillai, On m consecutive integers III, Prec. Indian Acad. Sci. Sect. A, 12 (1940), 6-12. 2. P. Erd6s, Problems and results on diophantine approximation, Compositio Math., 16 (1964), 52-65, see p. 52-53. 3. P. ErdiSs and A. R6nyi, A probabilistic approach to problems of diophamine approximation, Illinois J. Math., I (1957), 303-315, see p. 314. 4. P. Erd/Ss, The difference of consecutive primes, Duke Math J., 6 (1940), 438-441. 5. P. Erd6s, Amer. Math. Monthly, 60 (1953), 423. 6. G. Hoheisel, Primzahlprobleme in der Analysis, Sitzungsber., Berlin (1930), 580-588. 7. C. Hooley, On the difference of consecutive numbers prime to n, Acta Arith., 8 (1962-63), 343-347. 8. P. Turfin, Eine neue Methode in der Analysis und deren Anwendungen, Budapest 1953. TECHNION-ISRAELINSTITUTEOFTECHNOLOGY, HAIFA
COMPACTNESS AND SEMI-CONTINUITY BY
S. P. F R A N K L I N ABSTRACT The purpose of this note is to point out that a recent result of Ceder yields easily converses t o well known theorems of Wallace and Birkhoff and thus provides two new characterazations of compactness as well as sl~ecifying the class of spaces for which the theorems are true. A very slight extension of Ceder's theorem is also obtained, as well as new and simple proofs.
Let X be a non-empty topological space and P a partially ordered set. A function f : X ~ P is an upper-(lower-)semicontinuous function iff x~ ~ x 0 and eventually f ( x , ) > c( < c) implies that f(xo) > c ( < c). The theorem of Birkhoff ([1], p. 63) asserts that when X is compact, (B) every upper-(lower-)semicontinuous function to a partially ordered set assumes a maximal (minimal) value. A quasi order < (reflexive and transitive)on a non-empty topological space X is upper-(lower-) semicontinuous ifffor each x ~ X, {y E X lx < y} ({y e x ly <=x}) is closed in X. The theorem of Wallace ([4]) asserts that when X is compact, (W) each upper-(lower-)semicontinuous quasi order on X has a maximal (minimal) element. (See [5] p. 146 for a proof.) That (B) and (W) are equivalent is part of the folklore of the subject and is not difficult to verify. It will emerge from Ceder's theorem that they hold precisely on the class of compact spaces. Let X and Y be non-empty topological spaces, ~o(Y) (~c(Y))the colloction of non-empty (closed) subsets of ¥. A set valued function f : X ~ Po(Y) is an upper(lower-)semicontinuous carrier iff for each open U~_ Y, {xlf(x) _ u) ((x [f(x) n U 4 ~b}) is open in X. Ceder's theorem ([2]) asserts that a necessary and sufficient condition for X to be compact is that (C) each upper-(lower-)semicontinuous carrier into the closed subsets of a T 1 space assumes a maximal (minimal) value with respect to set inclusion. It is possible to strengthen (C) by deleting the word closed without destroying the result if one restricts attention to the upper-semicontinuous case. Call this strengthened form (C')A proof of the necessity of (C') to compactness is essentially Received Febuary 14, 1965.
13
14
S.P. FRANKLIN
contained in the first part of the proof of Theorem 1 of [2], the basic idea being that a maximal element for a chain {f(x~)} of images is provided by the image f(xo) of any cluster point x0 of the net (x~} giving rise to the chain. The sufficiency is immediate since (C') implies the uppersemicontinuous case of (C). A simple proof of the necessity of (C) can be had from the theorem of Birkhoff by noting that each upper-(lower-)semicontinuous carrier is an upper-(lower-) semicontinuous function with respect to set inclusion as the partial order on ~o(Y)(~c(Y)) This fact also implies the sufficiency of (B) for compactness. More precisely it shows that (B) implies ((2) which by Ceder's theorem implies compactness. The upper-(lower-)semifinite topology on ~o(Y) is obtained by taking the family of all sets of the form {A IA _~ U} ({A I A n U ~ ~b}),where U is open in Y, as a subbasis. One sees immediately that f : X ~ ~o(Y) is an upper-(lower-)semicontinuous carrier i f f f is continuous with respect to the upper-(lower-)semifinite topology on ~o(Y) (see Michael, [3], p. 179). Still another easy demonstration of the necessity of (C) can be had directly from the theorem of Wallace by noting that set inclusion is an upper-(lower-) semicontinuous quasi order on ~3c(Y) with the upper-(lower-)semifinite topology and that continuous images of compact sets are compact. The sufficiency of (W) follows from its equivalence to (B). (It is also easy to verify directly that (W) implies (C).) Hence we may assert that each of the conditions (B), (W), (C), (C') is necessary and sufficient for compactness. Also, Ceder's construction for the second part of the proof of his Theorem 1 can be modified to yield simple direct proofs of the sufficiency of (B) and (W). For example, to show that (B) does not hold in a non-compact space (for the lower-semicontinuous case), let {Ya},~o be a net in X with no cluster point. Following Ceder, the collection of nonempty open sets Vb = X \ {ya [ a >- b} - cover X. For each x ~ X, let f ( x ) = {b [x ~ l/b} ~ D. Suppose in X a net {x,} converges to xo and eventually f ( x , ) ~_ C c D. If a ef(xo) I C, eventually x~ e Fo contradicting f ( x , ) ~_C. Hencefis a lower-semicontinuous function into ~0(D), partially ordered by set inclusion, which assumes no minimum. REFERENCES
1. G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloquium Publications, 25, rev. ed. (1948). 2. J. G. Ceder, Compactnessandsemicontinuous carriers, Proc. Amer. Math. Sot., 14 (1963), 991-3. 3. E. Michael, Topologieson spaces of subsets, Trans. Amer. Math. Sot., 71 (1951), 152-82. 4. A. D. Wallace, A fixed point theorem, Bull. Amer. Math. Soc., 51 (1945), 413-16. 5. L. E. Ward, Jr., Partially ordered topologicalspaces, Proc. Amer. Math. Soc., 5 (1954), 144--61. UNIVERSITY OF FLORIDA AND CARNEGIE INSTITUTEOF TECHNOLOGY
UNION CURVATURE OF A VECTOR FIELD IN A SUBSPACE* BY
R. N. KAUL
ABSTRACT
The differential equations of union curves on a hypersurface V~ immersed in a Riemannian V,+I have been obtained by Springer [1]. These results were generalized later for a subspace in a Riemannian space by Mishra [2]. The author [3] has defined the union curvature of a vector field with respect to a curve on a hypersurface If. of a Riemannian V~+t. The purpose of this paper is to consider union curvature of a vector field with respect to a curvein a subspace V. of a Riemannian V,..
1. Subspaces of
Vm .
Totally indicatrix variety of a vector field.
Consider a subspace Vn of coordinates x~(i = 1 , . . . , n ) (1.1)
and metric
ds 2 = g~jdxtdx J
immersed in a Riemannian V, of coordinates y ~ ( a = l , . . . , m ) and metric (1.2)
ds 2 = a~t~dy~dy p
Let N~/(tr = n + 1,..., m) be the contravariant components in the y ' s of (m - n) independent mutually orthogonal unit vectors normal to V~, then the following relations hold [4] (1.3)
~ p = 1 a~pN~/N~!
(1.4)
• P = 0 ao,aN~/N~!
(It # a)
(1.5)
a~pN~/ya, i = 0
(tr = n + 1, ..., m)
(1.6)
y;~j = •
I'~,/,IN~*/
where the coefficients fl,/~ are symmetric covariant tensor of the second order. Let Vbe a field o f unit vectors in Vn such that if x t = xt(s) defines a curve C in V~, there is associated a unit vector with each point of the curve. I f v~ and p~are the components of Vin the y ' s and the x's respectively, then v~= y ~,~p. • The author is indebted to the referee for helpful suggestions. Received March 5, 1965. 15
16
R.N. KAUL
[March
We have the following relation for the derived vector T ~ o f Fin Vmwith respect to the curve C [3,4,5]
/
i dxj\
•
I o~
T ~ = Xo [PG/ijP---~-s)No: + ok,q y.,
(1.7)
As an analogue to the osculating variety of a curve C we define the totally indicatrix variety of a vector field I/with respect to the curve C as that determined by the vector field V and its derived vector in Vm with respect to C. Further let us consider a set of (m - n) congruences of curves one curve of each of which passes through each point of In. Let 2,~ be the unit vector in the direction of a curve of the congruence which is in general not normal to V, and therefore may be specified by i • 2 a, / = t,/y,~ + ~,
(1.8)
a co~/N~/
o
where tr/i and c,,/ are parameters. It is easy to verify that [2] i j gijtr/t~/ + ~
(1.9)
2 cot /= 1
o
1.10
cos Oo,/= co,~
and
(1.11)
cos ~r/ = gijt,/P
j
where Oo,/ and ~,/are the inclinations of the vector 2, to the vectors No~ and V respectively. 2. Union curvature of a vector field. If we consider an ndicatrix in V m with the above direction ,t~/, then the condition that it is an indicatrix variety of the vector field V requires 2~/= u,/y,~pi + w,/T ~
(2.1)
where u~/ and w~/ are parameters to be determined. From (1.7), (1.8) and (2.1) it follows that i
Ct
CI
-~-
•
l
t~/y,~ + ~o cot~No~ ur/y.~p + (flo/ijpi dnY \ ~ + vkaq iY,~] _~_s )No/ and therefore we obtain
(2.2/
u,/= Pd~
1965]
UNION CURVATURE OF A VECTOR FIELD IN A SUBSPACE
(2.3)
1
dx! t)°/~JP~ ds
W ~I
Co~I
17
and dx b
(2.4)
~k,q h +
C~,l
Since (2.3) implies that
. dx j 1
D'/'sP' " - ~
W~I
DalUP
C~rr/
,dx J ds
Ca~/
therefore (2.4) may also be written as e(~kn)
h
#~! = vk"qh + ( z, ~, )X~pc2v'l/2a'/=0
(2.5)
where e(vk,), the normal curvature of the vector field V with respect to C in V, of V,, and the vector tr~ are defined by
,,k~ = ~", ( D~/"bp" --~s dxb'~2 ] and h
t h
lh
a,! = Pi(t,/P - P t,/). h The vector with components #~/will be called the union curvature vector of the vector field V with respect to the curve C relative to the congruence 2~/. Thus union curves of a vector field Vrelative to a congruence 2~/are curves along which the union curvature vector #h/is a null vector. Its magnitude which is the union curvature of the vector field V with respect to C relative to the congruence 2,/is given by =
#~/#~lh
e(~k.) h ~ e(~k,) a ~ =(vkuq h+ (~,c2,)i/2aqJ (vk.qh + ( ~ , , h ] V
*2
or
(2.6)
oeu( ) = ok; +
2~k,e(~k.)
qh ,h +
vk 2 V
If e(vk,)= 0, then (2.6) gives ~k~(~) = ~k.. Hence:
h
18
R.N. KAUL
T h e union curvature of a vector field V relative to a n y congruence )t,/ with respect to asymptotic lines of the field V is the associate curvature of the vector f i e l d V. From (2.6) we find that vk,(z) vanishes if vko and ~k, are both zero. Thus: I f a curve in V, is an indicatrix as well as asymptotic line of the vector field V then it is also a union curve of the vector field V relative to a n y congruence A,/. If the congruence be one of normals to V, in which case t~/= O, it folIows from
(2.6) that Therefore:
~k.(,') = ok,.
A union curve of a vectorfield V in V, relative to any normal congruence 2,/ is also an indicatrix of the vector field E
Making use of (1.10)and (1.11)in (2.6)we obtain for the magnitude vk,(z) of the union curvature of a vector field relative to the congruence 2,/the following relation in terms of the inclinations 0 , , / a n d ~t,/ rsin%t,/- ~cos20,,/]'/2 (2.7)
~k,(t) = vko - e(vkn) [.
~os2-~,/
•
.
Taking m = n + 1, N,~/= N ' a n d the vectors of the field Vtangent to the curve C, we find that the formula (2.7) for ~k,(~) agrees with the known formula for k, given by Springer [1]. In this case we obtain cos20\ 1/2 c~0 -) .
. /sin2a--
(2.8)
ku = k s - £,~
If the congruence be one of normals to V,, then cos0 = 1, and cos~ = 0, therefore (2.8) yields the relation k, = k s which is the known result [1] that corresponding to a normal congruence, the union curves are geodesic curves. REFERENCES 1. C.E. Springer, Union curves o f a hypersurface, Canadian of Maths., 2 (1950), 457-460. 2. R. S. Mishra, Sur certaines courbes appartenant aun sous-espace Riemannien, Bull. Sei. Maths., 19 (1952), 1-8. 3. R. N. Kaul, Union curvature of a vector field, The Tensor (New Series), 7, No. 3 (1957), 185-189.
4. C. E. Weatherburn, An introduction to Riemannian geometry and tensor calculus, Cambridge University Press, 1950. 5. T. K. Pan, Normal curvature of a vecwr field, Amer. of Maths. 74 (1952), 955-966. UNIVERSITY OF CALIFORNIA BERKELEY, CAUFORNIA
THE AFFINE IMAGE OF A CONVEX BODY OF CONSTANT BREADTH BY G . D . CHAKERIAN ABSTRACT A convex body is said to have constant diagonal if and only if the main diagonal of the circumscribed boxes has constant length. It is shown that an n-dimensionalconvex body, n ~ 3, is the affine image of a body of constant breadth if and only if it has constant diagonal. Affine images of bodies of constant breadth are also characterized by the property that the orthogonal projection on each hyl~rplane is the affine image of a body of constant breadth in that hyperplane.
A convex body in n-dimensional Euclidean space E. is a compact, convex subset with non-empty interior. A convex body has "constant breadth" if the distance between parallel supporting hyperplanes is constant. A set K' is the "affine image" of K if there exists an affine transformation f: E, -* En such that f ( K ) = K'. A rectangular parallelopiped will be referred to as a "box". A convex body has "constant diagonal" if the main diagonal of the circumscribed boxes has constant length. In this paper we are concerned with characterizing affine images of bodies of constant breadth, a problem raised by S.K. Stein. Our main theorem is, THEOREM 1. In E,, n >=3, a convex body is the affine image of a body of constant breadth if and only if K has constant diagonal. We shall also prove a related theorem, THEOREM 2. In E,, n > 3, a convex body is the affine image of a body of constant breadth if and only if its orthogonal projection on each hyperplane is the affine image of a body of constant breadth in that hyperplane. The proofs of these theorems depend on the following lemmas. LEMMA 1. In E~, n >=2, a convex body is the affine image of a body of constant breadth if and only if K + ( - K) is an ellipsoid. Proof. This follows immediately from the observation that K is a body of constant breadth if and only if K + ( - K) is spherical. Received April 2, 1965.
19
20
G.D. CHAKER1AN
[March
LEMMA 2. In E,, n > 3 a convex body is an ellipsoid if and only if its orthogonal projection on, each hyperplane is an ellipsoid in that hyperplane. Proof. If K is an ellipsoid, then all its orthogonal projections are ellipsoids. The following proof of the converse is an adaptation of the p r o o f given by Siiss [3] for the case of E3. Now suppose each orthogonal projection of K is an ellipsoid. Then each supporting hyperplane of K intersects K in just one point. Denoting the points of E, by x = (xl, ".-, x,), assume that the segment joining a = (0, ..-, 0,1) to a' = (0,...,0, - 1) is a maximum diameter of K. Then the hyperplanes x, = 1 and x, = - 1 are supporting hyperplanes of K. Let K* be the orthogonal projection of K on the hyperplane xl = 0. Then K* is an ellipsoid in xi = 0. aa' is one axis of K*, since the intersections of x, = 1 and x, = - 1 with xl = 0 are supporting (n - 2)-planes of K*, orthogonal to aa' and passing through a and a ' respectively. Let H be a supporting hyperplane of K which is orthogonal to xa = 0. Then H intersects x~ = 0 in H*, where H* is a supporting (n - 2)-plane of K*. If, in particular, H is also chosen parallel to aa', then H* is parallel to aa', and hence H* n K* lies in x, = 0. It follows that H n K lies in x, = 0. But this argument could have been applied to any supporting hyperplane parallel to aa'; hence, any supporting hyperplane parallel to aa' intersects K in a point lying in x, = 0. From this it follows that the intersection o f K with x, = 0 is identical with its orthogonal projection on x, = 0, which is an ellipsoid K**. It is easy to show that K** must be centered at the origin. Now let f : E. ~ E, be an affinity which keeps aa' fixed and maps K** onto a sphere S in x, = 0 centered at the origin. Then f ( K ) intersects x, = 0 in the sphere S. Each diameter of S is orthogonal to the supporting hyperplanes o f f ( K ) through its endpoints.Also, x, = 1 and x, = - 1 are supporting hyperplanes o f f ( K ) , orthogonal to aa' and passing through a and a' respectively. Finally, all the orthogonal projections of f ( K ) are ellipsoids (here one needs to use the fact that not only the orthogonal projections, but all projections, of K are ellipsoids). In the argument above, the only property o f aa' we actually used was that the supporting hyperplanes through a and a' were orthogonal to aa'. From this it followed that the hyperplane through the origin orthogonal to aa' intersected K in an ellipsoid. Thus if bb' is any diameter of S, the same arguments can be applied to show that the hyperplane through the origin orthogonal to bb' intersects f ( K ) in an ellipsoid; moreover, this ellipsoid has aa' as an axis. It follows that every 2-plane containing aa' intersects K in an ellipse having aa' as one axis and a diameter of S as the other. Thus f ( K ) is an ellipsoid of revolution, and K is an ellipsoid. This completes the proof. LEMMA 3. In E,, n > 3, a convex body is an ellipsoid if and only if all its circumscribed boxes have their vertices on a fixed sphere.
1965]
IMAGE OF A CONVEX BODY OF CONSTANT' BREADTH
21
Proof. The sufficiency of the condition is proved, for n = 3, in [1]. We proceed to the general case by induction. Suppose K is a convex body in E~, n > 3, all of whose circumscribed boxes have their vertices on a fixed sphere S, and assume we know the lemma for Ek, 3 < k < n. Let H be any supporting hyperplane of K, and let K* be the orthogonal projection of K on H. Then any box B* in H circumscribed about K* is a face of a box B circumscribed about K. The vertices of B lie on S; hence, the vertices of B* lie on the sphere S ~ H . Thus K* is an ellipsoid in H. It follows that the orthogonal projection of K on any hyperplane is an ellipsoid, so by Lemma 2, K is an ellipsoid. The result follows, for all n, by induction. The converse, namely that all circumscribed boxes of an ellipsoid have their vertices on a fixed sphere, is a simple matter of analytic geometry. This completes the proof. Proof of Theorem 1. 1. Let S"- 1 be the unit sphere centered at the origin in E,. A "direction" in E n is a point u ~ S n - I . N ow suppose K is the affine image of a body of constant breadth, so K + ( - K) is an ellipsoid E centered at the origin. Let p(u) be the support function of E measured from the origin, and let b(u) be the breadth function (distance between parallel supporting hyperplanes orthogonal to direction u) of K. Then p(u) = b(u) for all directions u. But if u 1,u2,'", u, are any n mutually orthogonal directions, then )ST=l[p(ui)] 2 is constant, by Lemma 3. Hence, ~ ' : l[b(ui)] 2 is constant, which is precisely the condition that K have constant diagonal. Conversely, if K has constant diagonal, then ~,'~:l[b(ui)-I 2 is constant, so ~7=l[p(ui))] 2 is constant, where p(u) is the support function of K + ( - K ) . Thus all the boxes circumscribed about K + ( - K) have their vertices on a fixed sphere. By Lemma 3, K + ( - K) is an ellipsoid, so K is the affine image of a body of constant breadth. This completes the proof of the theorem. Proof of Theorem 2. For each direction u let Eu be the hyperplane through the origin orthogonal to u. Let K~ be the orthogonal projection of K on Eu. If K is the affine image of a body of constant breath, then K + ( - K) is an ellipsoid. Hence, [K + ( - K ) ] u - - K ~ + ( - Ku) is an ellipsoid in E,, so K u is the affine image of a body of constant breadth in E~. Conversely, suppose K~ is the affine image of a body of constant breadth in E~, for each u. Then [K + ( - K)]~ = K~ + ( - K~) is an ellipsoid in E, for each u. By Lemma 2, K + ( - K) is an ellipsoid, so K is the affine image of a body of constant breadth. This completes the proof.
REMARK. An interesting characterization of affine images of curves of constant breadth in E 2 is to be desired. While Theorem 1 is true in one direction in the plane case, viz. an affine image of a curve of constant breadth has constant diagonal, the converse is false. Blaschke, in [2], gives examples of centrally symmetric
22
G.D. CHAKERIAN
convex curves with constant diagonal which are not ellipses. Such a curve could not be the affine image of a curve of constant breadth, since the only centrally symmetric curve of constant breadth is the circle. REFERENCES 1. W. Blaschke, Eine Kennzeichnende Eigenschaft des Ellipsoids und eine Funktionalgleichung aufder Kugel, Berichte der Leipziger Gesellsch. d. Wiss. math.-phys. Klasse 68 (1916), 129-136. 2. W. Blaschke, Uber eine Ellipseneigenschaft und iiber gewisse Eilinien, Archiv ftir Math. und Phys. 26 (1917), 115-118. 3. W. Siiss, Eine Elementare Kennzeichnende Eigenschaft des Ellipsoids, Math. -Phys. Semesterber. 3 (1953), 57-58.
ON
CLIQUES
IN GRAPHS
BY J.W. MOON A N D L. MOSER ABSTRACT A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph with n nodes is determined. Also, bounds are obtained for the number of different sizes of cliques possible in such a graph.
§1. Introduction. A graph G consists of a finite set of nodes some pairs of which are joined by a single edge. A non-empty collection C of nodes of G forms a complete graph if each node of C is joined to every other node of C. A complete graph C is said to be maximal with respect to M if C _ M and C is not contained in any other complete graph contained in M. If the complete graph C is maximal with respect to G then C forms a clique. Some time ago Erdtis and Moser raised the following questions: What is the maximum number f(n) of cliques possible in a graph with n nodes and which graphs have this many cliques? Erd6s recently answered these questions with an inductive argument. In §§2 and 3 we determine the value off(n) and characterize the extremal graphs by a different argument. In §23 and 4 we obtain bounds for g(n), the maximum number of different sizes of cliques that can occur in a graph with n nodes. It follows from these results that g(n)~ n - [log2 n]. §2. Determining the value off(n). THEOREM 1.
I f n > 2, then f(n) =
3 n/a, 4.3tn/3j-~, [ 2.3 t"/3j,
if n = 0 (mod 3); if n = 1 (rood 3); if n = 2 (mod 3).
Proof. The theorem is easily verified if 2 < n _< 4. Let G be any connected graph with at least five nodes and which contains c(G) cliques. The set of nodes joined to any particular node x of G will be denoted by F(x). Suppose there are ~(x) complete graphs contained in F(x) that are maximal with respect to G - x, the graph obtained from G by removing x and its incident edges. Also, suppose there are [3(x) complete graphs contained in F(x) that are maximal with respect to F(x) but not with respect to G - x. From these definitions it follows that X(x), Received April 7, 1965
23
24
J.W. MOON AND L. MOSER
[March
the number of cliques of G containing x, and c(G - x), the number of cliques of G - x, are given by the following equations:
(1)
z(x) = ~(x) +/~(x);
(2)
c(O - x) = c(6) - [~(x).
Suppose nodes x and y are not joined in G. Then, if G(x; y) denotes the graph obtained by removing the edges incident with x and replacing them by edges joining x to each node of F(y), it follows that (3)
c(G(x, y)) = c(G) + Z(Y) - X(x) + a(x).
To prove this, let fl(x) = fl(x, y) + if(x, y), where fl(x, y) denotes the number of complete graphs in F(x) n F(y) that are maximal with respect to G - x - y. In transforming G into G(x; y) the contribution of these complete graphs to the total number of cliques is not affected. There is a loss, however, of the cliques counted by if(x,y). In adding the edges joining x to the nodes of F(y) it is not difficult to see that a new clique is formed for each of the complete graphs counted by ~(y) and fl'(y,x). Hence,
c(G(x; y)) = c(G) - i f ( x , y) + ~t(y) + fl'(y, x) c(O) - / r ( x , y) -/~(x, y) + X(Y), c(G) + Z(Y) - Z(x) + ~(x),
using (1) and the fact that fl(x, y ) = fl(y,x). Now let G be any graph with n nodes (n > 5) and having a maximal number of cliques. A simple argument shows that G is connected and has no node joined to every other node. If nodes x and y are not joined in G then it must be that X(x) = Z(Y), for if )~(y)> Z(x), say, the graph G(x; y) would gave more cliques than G, by (3), and this would contradict the definition of G. It also follows from (3) that g(x) = 0 for all nodes x of G. For some arbitrary node x of G let a, b,-..,f be the other nodes with which x is not joined. We may replace G tl)= G by G(2)= G(a; x) without affecting the number of cliques in the graph or the properties described in the preceding paragraph. We now replace G(2) by G (a) = G (2) (b; x) and, continuing this process, we eventually obtain a graph which has the same number of cliques as G, satisfies the properties in the preceding paragraph, and in which none of the nodes x, a,b. i.,f are joined to each other but all are joined to all the remaining nodes. We may now apply this procedure with respect to some node y in F(x). By continuing to make these transformations it is clear that we will ultimately obtain a graph G* which has as many cliques as G and which has the following simple structure: The nodes of G* may be partitioned into disjoint subsets such that two
1965]
ON CLIQUES IN GRAPHS
25
nodes are joined if and only if they do not belong to the same subset. If these subsets have Jl,J2,"',Jz nodes, where Jl +J2 + "'" +Jl = n, then it follows that (4)
c(G*) =JlJ2 ""Jr
Simple calculations show that this product assumes its maximum value when as many as possible of the subsets have three nodes and the remaining ones have two or four nodes. Since G was assumed to have a maximal number of cliques and since c(G)= c(G*), it follows that f ( n ) is given by the above expressions. §3. Characterizing the extremai graphs. Let Hn denote the graphs havingf(n) cliques described at the end of the preceding section. THEOREM 2. n>2.
I f the graph G has n nodes and f ( n ) cliques then G = Hn, if
Proof. The theorem is easily verified when 2 < n < 4. Suppose G is some graph with n nodes (n > 5) and f ( n ) cliques which is not one of the graphs H.. By the preceding argument it is possible to repeatedly modify the graph G until a graph H . is obtained, without affecting the number of cliques it contains. Let G' denote the last graph in this sequence before H,. That is, G' has f ( n ) cliques and contains two nodes x and y which are not joined to each other such that G'(x; y)
Sn. Let us suppose that n = 31, in which case H~ consists of l triplets of nodes such that two nodes are joined if and only if they do not belong to the same triplet. Since x and y are not joined it follows that they belong to the same triplet of un joined nodes in H.. Let z be the third node of this triplet. If, in G', x is joined to t, of the nodes in the i'th remaining triplet of unjoined nodes, i = 1 , 2 , . . . , l - 1 , then it is not difficult to see that (5)
Z(x) = tlt2 "'" tt- 1"
NOW X(X)= Z(Y), by the earlier argument, and it is easily seen that X(Y) _- 3l- 2. Since each t~ < 3 in (5) it must be that t~ = 3 for i = 1, 2,..., 1 - 1. That is, F(x) __ G' - x - y - z. If x is joined to z in G', then c(G') = 2.31-1. But this is less than f(n), a contradiction. Hence x is joined to every node in G' except z and y. This implies that G ' = H,, by definition. The proof of the theorem may be completed by applying a similar argument in the cases when n is congruent to 1 or 2 modulo 3. §4. A lower bound for g(n). It is not difficult to construct a graph with n nodes which contains cliques of sizes 1,2,...,[½(n + 1)]. This shows that g(n) >=[½(n + 1)] for all n. When n > 26, an improved bound is given by the following result. (In what follows all logarithms are to the base two.) THEOREM 3.
g(n) >_ n - [log n] - 2[log log n] - 4.
26
J.W. MOON AND L. MOSER
[March
Proof. We temporarily restrict our attention to the case where n > 47. To establish the lower bound for g(n) we exhibit a graph L. which has cliques of at least n - [log n] -2['log log n] - 4 different sizes. Let m be the unique integer such that n = 2" + 2m + ['log m] + (l + 3), where 0 _< 1 -< 2 " + 1 + [log(m + 1)] - [log m]. When 0 _< 1 _< 2", let L, be the graph with n nodes illustrated in Figure 1, where, for convenience, we let t = [log m] + 1
and h = 2 m-1 - 2 ' - t + 1.
(The restriction that n > 47 was made to insure that h > 0.) The symbol < k > denotes a complete graph of k nodes. We refer to the nodes in the first, second, and third columns as the A, B, and C nodes, respectively; in addition, the 2"- 1 + 1 encircled B nodes will also be referred to as D nodes. The edges of the graph L, are as follows: Each A node is joined to every other A node, and similarly for the B and C nodes. Each A node a is joined to every B node not contained in a complete graph connected with a by a dotted line in the diagram. (The D nodes are encircled to indicate that they are all joined to all the A nodes except the one indicated.) Finally, each C node c is joined to every D node not contained in a complete graph connected with c by a dotted line in the diagram.
<1>
<2'-* + 1>
(I>
(2+
<1>
<1>
1>
t.(1 + 1>
<2 "-~+ i>
(1>
<4 + 1>
<1>
<2 + 1>
O>
....
<1 + 1>
Figure 1 There are a total of 2" + 1 cliques in L. involving only A or B nodes since the
1965]
ON CLIQUES IN GRAPHS
27
nodes of one and only one of the complete graphs connected by each of the first m + I dotted lines can belong to any one such clique. The smallest of these cliques consists of the m + 1 A nodes and the largest consists o f the 2" + rn + l B nodes. From the nature of L,, the fact that 0 < l <- 2", and the fact that every positive integer can be expressed as a sum of distinct powers of two, it follows that there is a clique involving only A and B nodes of every intermediate size as well. Similarly, among the cliques involving only C or D nodes there are certainly cliques of every size between t + 1 and 2 t + t. (There may be still larger cliques of this type which contain the top complete graph of h + 1 nodes, but for our present purpose we need not consider these.) From the definition of t it follows that 2 t + t > m, so L, contains cliques of all sizes between t + 1 and 2 " + m + l, inclusive. Thus, the number of different sizes of cliques that occur in L, is 2" + m + l - t = n - m - 2[logm] - 4. Since m < log n, this suffices to complete the proof of the theorem under the above assumptions. The cases where 1 = 2" + 1 or 2 " + 2 can be treated very easily. First set aside the one or two " e x t r a " nodes and form the graph described above on the remaining nodes. Then adjoin one of the " e x t r a " nodes as an isolated node to form a clique of size one and if there is a second " e x t r a " node attach it to any other non-isolated node to form a clique of size two. It is not difficult to check that, for the values of n under consideration, these two new cliques will increase the total number of different sized cliques to the required amount. It can be shown, using an example that differs from the graph in Figure 1 in that there are no C nodes, that (6)
g(n) > n -
2[log n] - 1,
for all n. This is weaker, of course, than theorem 3 when n is large. However, for n < 47 this result is at least as strong as the one we are trying to prove, and hence the truth of Theorem 3 when n < 47 follows from (6). We omit the proof of (6) because it is similar to and simpler than the proof given for theorem 3 when n>47. Somewhat sharper lower bounds could be obtained by using more complicated examples, but the improvement does not seem to be worth the effort.
§5. An upper bound for g(n). THEOREM 4. I f n > 4, t h e n g ( n ) < n - [log n]. Proof, Consider any graph G, with n nodes, where n > 4. If a largest clique T in G has t nodes then we may as well suppose that
28
J.W. MOON AND L. MOSER t_~ n -- [ l o g n ] + 1,
since the number of different sizes of cliques occurring in Gn cannot exceed t. Let S denote the set o f s = n - t nodes not belonging to T. Since each node of Tis joined to every other node of T it is not difficult to see that if A and B are two cliques with A ~ S = B r~ S then it must be that A = B. Thus, the number of different sizes of cliques occurring in G~ is certainly no greater than S, the number of subsets o f S. But, 2 s < 2 tzog .1-*
and this last quantity is less than or equal to n - [log n] if n >_-4. This completes the proof of the theorem. §6. Concluding remarks. The maximum number of edges a graph nodes can have without containing a clique with more than l nodes is ediate consequence of a theorem of Tur/m [1]. The maximum number of a graph with n nodes can have without containing a clique with more than is equal to
with n an imcliques I nodes
max j t j 2 " " j , where Jt + J 2 + "'" + j r = n and t < I. This follows from the fact that the transformations used in the proof of Theorem 1 do not increase the size of the largest cliques. If a node x is joined to precisely k other nodes in a graph then it is clear that the maximum possible number of cliques containing x is f(k). A bipartite graph consists of two disjoint sets of nodes, A and B, such that no edge joins two nodes belonging to the same set. Let A and B contain a and b nodes, respectively, where 2 _< a < b. The definition of a clique in a bipartite graph is similar to the definition of a clique in an ordinary graph except that in the bipartite case we require that if a clique does not consist of a single isolated node then it must contain nodes from both A and B. It is a simple exercise to prove that the maximum number of cliques possible in such a bipartite graph is 2 a - 2. We have been unable, however, to obtain good analogues to theorems 3 and 4. REFERENCE
1. P. Turin, On the theory of graphs, Colloq. Math. 3 0954), 19-30. UNIVERSITYOF ALBERTA EDMONTON, CANADA
ON THE PRODUCT OF THE DISTANCES OF A POINT FROM THE VERTICES OF A POLYTOPE BY
BINYAMIN SCHWARZ ABSTRACT
Let xl .... , xm be points in the solid unit sphere ofE n and let x belong to the m
convex hull of xl ..... x,~. ThenI- [ I x-x,[I <=(1 - Ilxll) (1 + tlx II)"-1. l=l
This implies that all such products are bounded by (21m)m(m - 1)'~-i. Bounds are also given for other normed linear spaces. As an application a m
bound is obtained for I P(Zo)l where p(z)= 1-[ (z -z,), H < 1, i = l ..... m, i=l
and p'(z0)=O. Introduction. In § 1 we consider m, not necessarily distinct, points x~, ..., Xm belonging to the solid unit sphere (unit ball)of the real n-dimensional Euclidean space E,. Let x belong to the convex hull H = H(xl,...,Xm) of the points xi. The main result of this paper (Theorem 2) states that under these conditions
i=l
llx-x,il _-<(,-i)x
+ )ix ii)
(ll x II is the
Euclidean norm of x.) We obtain this bound by a simple, but lengthy, geometric argument, which proceeds by induction on the dimension of H (Theorem 1). An immediate corollary to Theorem 2 states that tlx--x't I < i=1
m
( m - - l ) m-l, ~-
for all sets ( x , x , . . . , X s ) satisfying It x~ll --< 1, i = 1,...,m, and x E H ( x l , " ' , X m ) . In §2 we show that this corollary can also be deduced from Szeg6's maximum principle. This principle asserts that the product I-Ii~ll x - x, ll attains, for fixed points xi, its maximum in any bounded closed region only at the boundary of this region. For our purpose it is convenient to formulate a consequence of this principle for the convex polytope H ( x , .'.,xm) (Theorem 3). Received April 13, 1965. 29
30
BINYAMIN SCHWARZ
[March
In §3 we consider the solid unit sphere of an arbitrary normed linear space N. We denote by h,,(N)the supremum of ~7= ~1]x - xi [] for 11 ]1x~ ][ __<1, i = 1, ..., m, and all x e H(xa, ..., xm) and obtain bounds for h,,(N). The lower bound is attained in the Euclidean case (Corollary to Theorem 2) and the upper bound is attained for spaces with the supremum norm and for spaces with the L~ norm. We conclude this paper with a simple application of Theorem 2 to complex polynomials having all their roots in the unit disk. As the title of this paper may be misleading, we stress that the m (not necessarily distinct) points are in an arbitrary position and that x is always restricted to their convex hull H. We consider the product of all the m distances ~ x - x~ ][, i = 1,..-,m, and not only the product of the distances from the vertices of the convex polytope H. OC and int C denote the boundary and the interior of the convex set C relative to the flat (linear variety) of smallest dimension containing C. This dimension is denoted by dim C. An edge is a one-dimensional face of the convex polytope H (belonging to OH). The author wishes to thank Dr.A. Ginzburg of the Technion, Haifa, for his help in the preparation of this paper. 1. Polytopes in euclidean space. THEOREM 1. Let Sn be a solid sphere of En and let x, xl,'",Xm, (m >=2), be (not necessarily distinct) points of S~ such that x belongs to the convex hull H = H(x~,---,xm) of xl, "",Xm. Then there exists points xl', ...,x,~ in S~ such that x lies on an edge of their convex hull H' = H(x~,'",Xm) and (1)
i~ i=1
IIx-
m
x, IL iI1 IIx- x; II =1
Proof. The left hand side of (1) vanishes only if x coincides with one of the xt. ¢ t l ! In this case we choose x2 = x3 . . . . . Xm = x and take as xl any point of S, different from x. We shall in the sequel disregard this trivial case and always assume that x v~ xi, i = 1,..., m. This and x ~ H(xl, ..., Xm) imply x e int Sn. In the conclusion of the theorem x will be an interior point of an edge of H'. Let k = dim H and let Pk be the k-flat carrying H. S k = S n n Pk is a solid k-sphere containing all the x i. We shall find points xi', i = 1,...,m, belonging to Sk, and hence also to Sn, satisfying the requirements of the conclusion. The proof will be by induction on the dimension k of H and we disregard E~ and S,. We denote the center of Sk by Ck. For x e int Sk, x # Ck, we denote the point of Sk farthest away from x by ak(X) and we denote the point of ask nearest to x by bk(X); i.e. ak(x ) and bk(X) are the endpoints of the diameter through x. ak(Ck) and bk(ck) are the endpoints of an arbitrary diameter of Sk. For the induction it is convenient to prove more than stated in the theorem; we show that H ' can always be chosen as a segment or as a triangle. Precisely, we prove the following strong version of Theorem 1:
1965]
ON THE PRODUCT OF THE DISTANCES
31
Let k = dim H ( x l , "',xm), let S k be a solid k-sphere such that xi e Sk, i = 1, ...,m, and let x e l l , x # xi, i = 1 , . . . , m . W e can choose the m points x; o f S k which satisfy (1) and f o r which x lies on the segment x'lx'2 in the f o l l o w i n g way: t ! i x 1 # x 2, x 1 # ag(x) and ( f o r m => 3) xa' = " " " = Xm' = ak(X). We prove the strong version by induction on k. For k = I we choose x~' = b~(x) a n d x 2' = Xa' = "" = x ', = a,(x). (1) is obvious and H ' = Sl, h e n c e x e H'.
For k = 2 we have three possibilities. (a) x = c 2. Set x~ ,
X2
,
=
X3
,
----" " ' "
~
Xm
~" a2
(c)2
=
b2(c2) and
•
(b) x e all. x is thus an interior point of the segment xxx2. (We avoid subsubscripts. A subset of p points of the given set will always be denoted by Xl,..., x r ) I ! I l Set X1 = Xl( ~ a2(x)) , X 2 = X 2 and Xa . . . . . Xm = a2(x). (C) x e i n t H , x # c2. Let r be the radius through x(r = c2b2(x)) and let d be its intersection with dH. If d is a vertex o f H or, more general, if d is one of the l ! l l points x~, then we set x 1 d and x 2 = x 3 . . . . . x m = a2(x ). If d is not a vertex of H, then it is an interior point of the side (edge) xax2 of H. Let l be the line through x parallel to this segment x,x2. Clearly, c2 ¢ I. Denote the intersection points of I and aS2 by x~ and x~, choosing x'i on the same side of r as x~, i = 1,2. (Cf. Figure 1). Then we have for i = 1,2 =
oL=Jx)
b;~tx)
x~ Figure 1
(2)
IIx- x, II < IIx
- x;ll.
Indeed, let T , i = 1,2, be the triangle bounded by the segment x x / ( o n l), by the segment x b2(x) (on r) and by the smaller arc o f aS2 between b2(x) and x{. As T1 u T2 is the segment o f $2 cut off by I which does not contain the center c2, it follows that the angle of T~ at x~' is smaller than n / 2 . T( is thus contained in the disk with center x and radius this implies (2). We choose I ! again x3 . . . . . Xm = a2(x) and thus proved the strong version for k = 2.
IIx- x;l[.gs
32
BINYAMIN SCHWARZ
[March
We now assume that it holds for all H with dim H < k and prove it for dim H = k(k > 3). We have again three cases. (a') x = ck. As in (a), we put the x / i n the endpoints of an arbitrary diameter. (b') x e H~, where H l = H(xl,...,xp) is of dimension l with 1 < l < k (hence I < p < m). Let Pl be the/-fiat carrying Hi and St = St ~ P t . By the assumption of our induction we have p points Yi in the solid/-sphere Si, Yl ~ Y2, Yl ~ at(x) and Y3 . . . . . yp = al(x), such that x lies on the segment YlY2 and such that P
(3)
P
11 I!x - x, II z i11 IIx =l
i=1 t
-
yi
II. !
I
r
We set x'l = Y l , x 2 = Y2 and put x'a . . . . . Xp=Xp+l . . . . . xm=ak(x). (Only if Ck~Pl, then at(x)= ak(x); but always x ' l ( = y l ) ~ a k ( X ) . ) (3) and the definition o f ak(X) imply (1) and the strong version is established for this case (b'). Note that this covers the case x ~ dH. (c') x e int H, x --/:Ck. Let again r be the radius through x (r = Ckbk(X)) and let d be its intersection with OH. If d is a vertex of H or, more general, if d is one of the t ! l l points xi, then we set xl = d and x2 = x3 . . . . . Xra = ak(x).Ifd ~ Xi, i = 1, " " , m , then it is an interior point of a /-dimensional face Hi of the convex polytope H(H l c t~H), 1 < l < k. Let x l , . - . , x p (l < p < m) be those points of the original set (xl, "",x,,) which lie in H~. Then H t = H(xl,...,xp). Let Ql be the/-flat carrying H I. As d is an interior point of H~, it follows that d is the only point on the radius r and in Qt: ckq~Q~, x~Q~. Let Pl be the /-flat through x parallel to Ql;ck~Pt. For i = l, ...,p let ri be the radius of S k going through x~ and denote Pl (3 r i = z t. We thus project H t = H(x~,...,xp), Ht c Q~, from Ck into the l-polytope H~= H(z~,...,zp), H,'~ Pl. d e i n t Hi implies x ~ i n t H~. We project once more. This time the p points z i are projected from x onto aS~; i.e. let l* be the ray from x through z~ (l* c Pl) and denote OSk n l*, = x~'. tr Clearly, H['= H(x';, "",Xp) is a convex l-polytope, H; ~ H/' and x e i n t H lt l . (Figure 2 illustrates our construction for k = 3, 1 = 2 and p = 3. The parallel triangles H2 and H~ are not necessarily normal to r. H~ and H~' lie in the same plane P2 but are in general not similar.) For each i, i = 1,...,p, let P i b e the plane (2- fiat) defined by the radii r and ri. c~, x and d are on r; Ck, Zi and xi are on ri; d and xl are in Ql and therefore on the line pi n Qi; x, z~ and x[' are in Pi and therefore on the parallel line li = pi n Pt. (li contains the ray l* from x through z~.)All these points lie in the disk $2i = Sk n pl i i with the center c2 = Ck. C2, X and d lie in this order on the radius r of S~ ; xi ~ r and x~' is the intersection point of l* with t3S~, where l* is the ray from x parallel to the segment d x~. x~ and x~' are on the same side of r. We thus have the same situation as in case (c) of k = 2. (We use now x~' instead of x/.) In analogy to (2), it follows that (4)
gill <
11 -
II,
19651
ON THE PRODUCT OF THE DISTANCES
33
II
X3
b3(Xl
II I
II
H~
Figure 2 (In Figure 2 we assumed that P i is the plane of drawing. Compare with the lower half of Figure 1.) Consider now the m points x~', ..., xp', xp+ l, "", xm. They all belong to Sk, x lies in their convex hull and, by (4), the product of the distances o f x from these points is larger than the original product I-[,~=1 IIx - xi I1" If, by chance, tP
dim H(x'~, ...,xp, xp+l, "",xm) < k (i.e. if a vertex of the original H = H(xD'", xm) lies in Pz), then the strong version follows by the assumption of our induction. If not, then we only note that now p of the points, namely x~,...,x~, belong to the /-flat Pt (1 =< I < k) and that x e H(x~,..., x~') ( = HI'). We therefore reduced this case (c') to the former case (b') and thus completed the proof of the strong version. Theorem 1 is thus established. This theorem implies our main result, which we formulate only for the solid unit sphere U . = {x: tlxll-<_ 1} (c. = 0) of En. r!
THEORI~M 2. Let x,x~,...,xm(m >=2) be (not necessarily distinct) points of the solid unit sphere U. of E. such that x belongs to the convex hull of xl, ...,xm. Then
34
BINYAMIN SCHWARZ
~5)
Irl
[March
tl x - x, 11--< <1 - I1x II) <1 + I1x 11)--1
i=1
For 0 < llxl[ < 1 equality holds in (5) only under the following conditions: 11x,[l-: 1, i = 1, , m , m - 2 of the x, coincide with the point a(x) of U, farthest away from x, and x lies on the chord bounded by the two remaining points. Proof. If x coincides with one of the xi, then there is nothing to prove; this must happen if II x 11-- 1 if x = 0, then (5) is obvious and equality holds only if !Ix, tl = 1, i = 1 , , m Let 0 < 11x 11< 1, x ~ x,, i-- 1 , , m By Theorem 1 we can assume that x is an interior point of the edge xtx2 of H = H(xt,...,xm). If we do not already have the situation mentioned in the equality statement, then we move x~ and x2 into the endpoints of their chord and, for m > 3, move xa, ...,xm into a(x). This increases the product and yields the conditions of the equality statement. But now l-It=311x - x , l [ - - I [ x - a < x ) l l --~ =(1 + Ilxll)- - ~ The product 1tx - x l II tl x - x~ tl of the segments of a chord through x depends only on x. Hence, denoting the point of t U n nearest to x by b(x), we have
IIx - x~ I1 IIx - x= II -- 11x - b<x)II 11x
-
a(x)II
=
(1
-
II x II) ~1 + IIx It).
completes the proof of Theorem 2 and shows also that the strong version of ¢ Theorem 1 can be further strengthened: 11 = bk(x), x2 = x3 . . . . . xm = ak(X). The function (1 - II x 11)~1 + I1x I1)"-1 attains its maximum only at IIx II = ( m - 2)/m. We thus obtain from Theorem 2 the following
q-his
l
l
!
COROLLARY. The assumptions of Theorem 2 imply (6)
H
I l x - x , ll--< m
(m-1)m-l"
i=1
Equality holds in (6) only under the following conditions: tlxll = < m - 2 ) / m , 11x, 11-- 1, i-- 1 , , m, m - 2 of the xi coincide with a(x), and x lies on the chord bounded by the two remaining points. 2. Szeg6's maximum principle. We show that the above corollary is also a consequence of the maximum principle for the product I'I,ml IIx-x, II <See P61ya-Szeg/5 [4, section III problem 301]; and [5], [3].) The proof in [4, p. 328] not merely shows that this product attains its maximum in any bounded closed region D of En only at dD, but establishes that for any plane P the maximum in P ¢3 O is taken only at d(P ¢3 D). For convex polytopes this implies THEOREM 3. Let xt,...,x,, be points in E, such that at least two of them are distinct. For x varying in the convex hull H = H(xl,...,xm ), H~'=I I!x-x, II attains its maximum only at interior points of edges of H. For completeness we outline the proof. For dim H = 1 there is nothing to prove. For dim H --- 2 identify the plane carrying H with the plane of the complex num-
1965]
ON TIlE PRODUCT OF THE DISTANCES
35
hers and apply the (ordinary) maximum principle to the polynomial ]-[~'=1 (z - zi). For dim H = k, 2 < k < n, it suffices to consider the k-fiat Ek carrying H. Let e > 0 and set H ~ = H - [,.)m=l { [ I x - x,[ I < e}; for small e the maximum of the product will be taken in H,. Let P be any plane such that P n H~ ~ ~ . Choose coordinates (~1, "", ~k) in Ek such that P satisfies ¢i = ci, i = 3, ..., k. For x e P set log l-[~---, IIx - x, It = f ( ¢ ' , ¢2). If x e P n H~, then f(¢x, ¢2) e C 2. Using that not all the x~ lie in P, we obtain ~2f/d~2 + ~2f/d~2 > 0. This excludes the possibility of a maximum at interior points of P r3 H,. Varying the c~(i = 3,-.., k), it follows that l-It'-- 1 II x - x, 11takes its maximum only at ell. If this maximum were taken at an interior point of a/-dimensional face of H with l > 1, then we would again obtain a contradiction by intersecting (for l > 2) this face with planes or by considering (for 1 = 2) the plane carrying this face. Theorem 3 implies the above corollary. Indeed, to find max l-I~= 1 11x - x, [I for all sets (x, xl, ..., x,,) satisfying IIx, It x and xeH(xl,...,x,,), it suffices by Theorem 3 to consider only those sets for which x is an interior point of an edge of H. We continue as in the proof of Theorem 2 and show that for such a set the product is bounded by ( 1 - Ilxll) (1 + llxll:-', Varying I[xlt, we obtain (6). Theorem 2 itself does not follow from Theorem 3 and seems to require a geometric proof. The property stated in Theorem 1 for the solid spheres is not valid for all convex bodies C of E,. Indeed, let C be a regular simplex with center c and let x~, i = 1,..., n + 1, be the vertices of C. As for any y e C, y ~ x~, i = 1,..., n + 1, we have [Ic - y It < 11c - xl II, it follows that for any set of n + 1 points x / o f C, such that c belongs to an edge of their convex hull H', the relation n+l
n+l
IIc- x, II 1-I [Ic-x; [I < i l-I i=1 =1 holds. 3. Normed linear spaces. Let x l , ' . . , x , , be points in the solid unit sphere of n-dimensional unitary (complex Euclidean) space. As this space is just the real Euclidean space of dimension 2n, it follows that Theorem 2 and its Corollary hold also for this space. Moreover, as they deal only with the convex hull of m points, it follows that (5) and (6) and the corresponding equality statements remain valid for (real or complex) Hilbert space. For normed linear spaces the following result, related to the Corollary, holds. THEOREM 4. Let N be a (real or complex) normed linear space and let U = {x; Ilxll < I} be its solid unit sphere. For m > 2 set (7)
hm(N) = sup f i
I1x - x, ll,
=1
where the supremum is taken over all sets (x,xl,...,xm) satisfying x i e U , i = 1,...,m and xeH(xt,...,xm). Then
36
BINYAMIN SCHWARZ
(8)
m
< hm(N) <
Moreover,
[March (m - 1)m.
(2)-
(9)
hm(N) = -~
(m - 1)m
for spaces with the supremum norm (l°°(n), n > m; l°°; L °°) and for spaces with the LI norm (P(n), n > m; 11; 12). The Corollary implies that in the Euclidean case (/2(n) = En, any n; 12; L2) hm(N) attains the lower bound of (8).
Proof. To obtain the first inequality sign of (8) it suffices to consider any diameter of U:let IIa II -- land set ~ l = - a . x ~ ..... x.= a and ~=((2- m)/m) a To prove the second inequality of(8) we note that the assumptions on(x, xl,...,xm) are: tlx, ll__o,
(10)
i=l
~ ti=l,
i=l,...,m.
i=1
This implies
fix- x, ll -- I[(t,- 1)x, + jE~ t t,x, II < ( 1 - , , ) (11)
<(l-t,)+
~
tj=2(1-t,),
IIx, II + jx~ l tjIIx, tl i=l,...,m.
j:#i
Hence,
(12)
(1 - ,,) <=2.
IIx - x, II < 2 . i=l
1-
=
(8) is thus established. If, under the assumption (10),
(13)
~I
11x - x, II = m
(m
- 1) m,
i=l
then, using (11) and (12), it follows that
(14)
IIx, ll = 1,
ti = --,1
i = 1,...,m.
m
To prove (9) we give sets (x,xl, "",Xm) satisfying (14) and (13). For l~°(n) and P(n), n > m, we denote xi=(~il,'",~in), i = 1,...,m. For l°°(n)we choose ~k = 1 -2cSik (i = 1, ...,m; k = 1,...,n). Then
k
m
j#t
m
1965]
ON THE PRODUCT OF THE DISTANCES
37
This proves (13) and the same example holds for l =~. For L°°(0, 1) we may e.g. take m distinct Rademacher functions. For l l ( n ) we choose ~k = 6~k (i = 1,-..,ra; k = 1, ...,n). Then 1
6
2m - 2
k=l
I'D,
The same example holds for l l and for U(0, 1) we may e.g. take xi(t) = mSik for (k - 1 ) / m < t < k / m ( i , k = 1, ...,m). This completes the proof of Theorem 4. Note that in the just considered cases the dimension of the maximizing H ( x l , . . . , x,,) is necessarily m - 1; this is in contrast with the Euclidean case. We add the following remark. If N is finite dimensional, then, owing to the compactness of the solid unit sphere, the supremum in (7) has to be attained. If (9) holds for a finite dimensional space, it follows therefore that there exist m points x,, ]Ix, !I = 1 satisfying
i \ rtl
/
Fn
.i g i
\
lcgl /
Fn
j ¢: i
As for 1 < p < ~ equality in Minkowski's inequality implies linear dependence, (15) cannot hold for m > 2 and N = lP(n) with 1 < p < oo. More general, by a result of Achieser and Krein (quoted in [1, p. 82] and [2,p. 112]) it follows that (15) cannot hold for rotund spaces (if m > 2): for finite dimensional rotund spaces and m > 2 the second inequality sign of (8) is strict. 4. The absolute value of a polynomial at critical points. We conclude with a simple application of Theorem 2. THEOREM 5. L e t Pro(z) = I-L=1 " (zi = 1,..., m. T h e n p~,(zo) = 0 i m p l i e s
zi), m > 2, a n d a s s u m e
I p,.(Zo) I -_ (1 -izol)(1
(16)
that
z~ II
< 1,
÷ Izol) "-1
For m > 3 and zo ~ O, l Zo I ~ t equality holds in (16) only if m-2 z o - --
In
e ~" a n d p,,(z) = (z - e i') (z + el') " - l
Proof. By the theorem of p ' ( z o ) = 0 implies Zo • H ( z l , . . . , Let now m > 3 and 0 < only if I z i ] = 1 for all i and
Izol
Gauss and Lucas [-4, section III, problem 31] z,,). Inequality (5) of Theorem 2 (for E2) gives (16). < 1 By Theorem 2 we can have equality in (16) if m - 2 of these roots coincide, say
Z 3 =
...
~_ Z m --. __ e i~.
If dim H = 2, then it follows from the Gauss-Lucas theorem that the two critical points different from - e i~ are in the interior of the triangle H and Theorem 2 excludes equality in (16). There remains thus only the one dimensional case:
BINYAMIN S C H W A R Z
38
zl = e i`, z2 = z3 . . . . . zm = - e~'; the only critical point different f r o m - e ~" is zo = ( ( m - 2 ) / m ) e ~ and in this case equality holds in (16).
For
m = 2 equality
holds in
(16)
if lz 1 [= [z2 l = 1 (zo = (Zl + zD/2). For
m > 3 the other cases o f equality are trivial: I f p.~(0) = 0 and all I z~] = 1, then both sides o f (16) equal 1, and at multiple roots on the unit circle both sides vanish. REFERENCES 1. D.F. Cudia, Rotundity, American Mathematical Society, Symposium on Convexity, cattle, Proc. Symp. Pure Math. 7 (1963). 2. M.M. Day, Normed linear spaces (2rid printing), Springer Verlag, Berlin, (1962). 3. Obreschkoff, N., L6sung tier Aufgabe No. 10, dber, Deutsch. Math.-Verein., 33 (1925), 30. 4. G. P61ya and G. Szeg/5 Aufgaben und Lehrs~itze aus der Analysis, 1, Dover, New York (1945). 5. G. Szeg6, Aufgabe No. 10, Jber. Deutsch. Math.-Verein, 32 (1923), 16. TECHNION-ISRAELINSTITUTEOF TECHNOLOGY,HAIFA
LOCAL CONVEXITY AND STARSHAPED SETS BY
F. A. VALENTINE ABSTRACT
Previously [7] we proved among other results that a closed connected set in E,, which has a unique point of local nonconvexity is starshaped. Here we characterize a fairly large class of plane sets whose points of local nonconvexity are so arranged that starshapedness follows. This theory determines as a special case the simple closed polygonal regions which are starshaped. In order to proceed simply we utilize the following notations and definitions. NOTATIONS. The interior, closure and boundary of a set S in Euclidean n-space E n are denoted by intS, clS and bd S respectively. The closed segment joining points x ~ En, y ~ E~ is denoted by xy. Set union, intersection and difference are indicated by L), ~ and ,-, respectively. The symbol conv S denotes the convex hull of the set $. The symbol 0 denotes the empty set. DEFINITION 1. A point x ~ S is a point of local convexity of S if there exists a neighborhood N of x such that N n S is convex; otherwise x is called a point of local nonconvexity. The set of all points of local nonconvexity of S is denoted by Q. DEFINITION 2. A set S is starshaped with respect to a point p if px c S for all points x e S. DF2INITION 3. The two closed rays of a line L which have only a point x ~ L in c o m m o n are called complementary rays. I f R(x) is a ray with endpoint x, its complementary ray is denoted by R'(x). DEFINITION 4. A ray R(x) with endpoint x e bd S is an external ray of support to the int S if R(x) n i n t S = 0. DEFINITION 5. I f X is a boundary point of a set S, then K(x) is the union of all the external rays of support to int S at x. A boundary point x e S is called a onesided point of external support of int S if
K(x)
O,
and if K(x) lies in a d o s e d half-space which contains x in its boundary. The set K(x) is called an external cone of support. The union of all the complementary rays R'(x) where R ( x ) c K(x) is denoted by
r'(x). Received May 4, 1965 The preparation of this paper was supported in part by NSF Grant GP-1988. 39
40
F.A. VALENTINE
[March
The following theorem is one of Krasnoselskii type [2], and it is also related to results developed by Eugene Robkin in his thesis 15]. For another kind of theorem of Krasnoselskii type for polygonal regions see Molnar [4]. THEOREM 1. Suppose S c E 2 is a bounded plane set which is the closure of an open connected set. Then S is starshaped if and only if both of the following conditions hold. (a) Each point of local nonconvexity x e S has a nonempty cone of external support K(x) to intS at x. (b) I f xl,x2,x3 are three points of local nonconvexity of S (they need not be distinct) which are also one-sided points of external support to int S, then there exist three external rays of support R(xl),R(x2),R(x3) to int S at xl,x2, x3 respectively whose corresponding complementary rays R'(xO, R'(xz),R'(x3) are concurrent and meet in S, so that (1)
S ~ R'(Xl) N R'(x2) N R'(x3) ~ 0.
Proof. We will prove that conditions (a) and (b) imply that S is starshaped. Let Q1 denote the set of all points of local nonconvexity of S which are also onesided points of external support to int S (see Definitions 1 and 5). First, observe that if S is convex then Q1 = 0; however, in this case S is also starshaped, being convex. Hence, suppose S is not convex. Since int S ~ 0, when int S is not convex a theorem of Leja and Wilkosz [3] implies that a point q of local nonconvexity of S exists which is the midpoint of the bounding diameter of a closed semicircular region which, except for q, lies in int S. Clearly such a point q is in Qt, so that QI ~ 0. Now define C(x) as follows,
C(x) - ( c o n v K'(x)) n conv S where x E QI (see Definition 5). Since conv K'(x) is closed, and since S is compact, the set C(x) is compact. By hypothesis, every three of fewer members of the collection of compact convex sets
{C(x),xe 01} have at least one point in common. Helly's Theorem [1] then implies that (2)
M-
(']
C(x) # O.
xEQI
Let p 6 M. We will prove that S is starshaped relative to p. Firstly, if pC S, then since S is compact and connected there exists a cross-cut of the complement of S, say uv, such that (3)
u e S, v e S, S n (uv ~ u ~ v) = 0, (p, u, v) are collinear.
Secondly, if p e S and if there exists a point y ~ S such that
1965]
LOCAL CONVEXITY AND STARSHAPED SETS
(4)
41
p y q~ S
there again exists a cross-cut of the complement of S, uv, such that (3) holds. Hence, to prove p y ~ S for each y e S, we prove that (3) cannot hold. Suppose (3) does hold. Since S is closed and connected, the segment uv divides a component of the complement of S, say K, such that K .,. uv contains at least one bounded component. Let K 1 be such a bounded component of K .~ uv. This component K 1 of K ~ uv abuts uv from one side of uv. Let H be that closed half-plane whose boundary contains uv and in which K t abuts uv. Consider the set K* - cl conv(H n K1). Since K* is a compact convex set with int K* ~ 0, and since int K* c H, there must exist a point x e bd K *
and a line of support L to K* such that (5)
Lt~ K* = x, x ~ S
and such that p and uv lie in the same open half-space bounded by L. Since u e S, v e S, u # v, since S = cl int S, and since int S is open and connected, for each e > 0 there exists points u 1 ~ int S, vl~ int S such that lIul-ul]
< e, ] l v l - v i i < e
and such that a polygonal path P ( u ~ , v l ) exists in int S joining u 1 and v i. Now, clearly x is a point of local nonconvexity of S otherwise the convexity of S r~ N for some neighborhood N of x would violate condition (5). Furthermore, one can choose a bounded component Ks of K .,. uv and an e. > 0 (above) so that K ( x ) must be in the closed half space bounded by L which contains p and uv, because K ( x ) N P(Ul,Vl) = O.
However, in this case, no ray
R(x) in
conv
K(x) exists
whose complementary ray
R ' ( x ) contains p. This, however contradicts (2), since p E M. Hence, we have
arrived at a contradiction. Therefore, S is starshaped. Since the converse situation is trivial, we have proved Theorem 1. Theorem 1 has an interesting form when the boundary of S is a simple closed polygon in E2. To state it we first note the following. DrrINmON 6. A vertex x of a simple closed polygon P is called reentrant if x is a point o f local nonconvexity of the closed bounded polygonal set whose boundary is P. TrmOR~M 2. L e t S be a bounded closed set in E2 whose b o u n d a r y is a s i m p l e closed polygon. S u p p o s e that f o r each three r e e n t r a n t vertices o f bd S, x t, x 2, x a
42
F . A . VALENTINE
there exist three e x t e r n a l r a y s o f support at x l , x z , x 3 respectively to int S whose corresponding c o m p l e m e n t a r y r a y s are concurrent (see (1)) and meet in S. T h e n S is starshaped. Proof.
T h e o r e m 2 follows f r o m T h e o r e m 1 since c o n d i t i o n s (a) a n d (b) a r e
a u t o m a t i c a l l y b o t h satisfied. BIBLIOGRAPHY 1. Math. 2. 3. Math. 4. 5. 6. 7.
E. Helly, Uber Mengen konvexer KOrper mit gemeinshafilichen Punkten, Jber. Deutsch. Verein 32 (1923), 175-176. M.A. Krasnoselskii, Sur un critdre pour qu' un domaine soit dtoile, Math. Sb. (61) 19 (1946) F. Leja and W. Wilkosz, Sur une propridtd des domaines concaves, Ann. Soc. Polon. 2 (1924), 222-224. J. Molnar, Uber Sternpolygone. Publ. Math. Debrecen. 5 (1958), 241-245. E.E. Robkin. Characterizations of starshaped sets, Doct. Thesis, Univ. of Calif. (1965). F. A. Valentine. Convex Sets, McGraw-Hill Book Co. (1964). F. A. Valentine. Local convexity and Ln sets. (To appear in Proc. Amer. Math. Soc.)
UNIVERSITY OF CALIFORNIA, Los ANGELES, CALIFORNIA
VARIETE SUR
UN
DE NON-SYNTHESE
GROUPE
ABELIEN
SPECTRALE
LOCALEMENT
COMPACT
PAR
M. FIL1PP1 RESUME
On demontre quelques r6sultats concemant les ensembles de synth~se ou de r~solution spectrale dam les groupes abeliem localement compacts.
§1. Introduction. Soit F un groupe localement compact, la loi de groupe de F sera not6e additivement, et soit G son dual (groupe des caract6res g de 1"). Un caract~re de F est une fonction continue born6e sur F, et nous noterons, par abus de langage, de la m~me fa¢on un 616ment de G e t la classe dans L°°(F) de la fonction sur F qu'il d6finit. Soit V un sous-espace faiblement ferm6 invariant par translation de L~°(F), nous appellerons spectre de V l'ensemble a(V) des 616ments de G qu'il contient. Le probl~me de la synth6se spectrale est le suivant: notant par Vt la sous-vari6t6 de L°°(F), faiblement ferm6e, invariante par translation, engendr6e par les caract~res appartenants a or(V), V1 est la plus petite sous-vari6t6 de spectre a(V). Nous dirons que Vest une varidtd de synthdse spectrale si V = V~. Alors "la synth~se est possible" pour tout 616ment de V, c'est ~ dire que tout 616ment de V peut &re approch6 dans L°°(F) par des combinaisons lin6aires des caract~res appartenant a(v). Une caract6risation des vari6t6s de non-synth~s¢ spectrale semble diflicilement abordable; dans [2] P. Malliavin a montr6 l'existence des vari6t6s de non-synth~se sur un groupe ab61ien localement compact, non compact (la synth~se sur un groupe compact &ant toujours possible). On est ainsi amen5 h travailler sur une notion plus restrictive que celle de synth~se spectrale: on appeUe "varidtd de rdsolution spectrale'" une vari6t6 faiblement ferm6e V, invariante par translation, telle que toute sous-vari&6 de V ferm6e, invariante par translation, soit de synth~se spectrale.
It
Notations. Nous noterons comme d'habitude tl~ et 11"'" II1 les normes dans L~°(F) et L'(F) resp., MI(F) notera l'alg~bre des mesures born6es sur F, et nous noterons encore II "'" I11 la norme darts MS(F). Le produit de convolution dans le groupe F, sera not6 par un simple point, Received May 8, 1965.
43
44
M. FILIPP1
[March
et nous 6crirons 6galement e"a pour l'exponentielle de convolution de a dans F, e t a "2 pour le carr6 de convolution de a. Le produit de convolution dans le groupe additif des r6els sera, lui, not~ par une 6toile ( * ). Enfin, soit (I) un 61~ment de L~°(F), nous noterons ((I)) la sous-vari~t6 f e r m ~ engendr6e par (I) et ses translat6s. 1.1. Uaieit6 et r6salation speetrale. Soit V une vari6t6 ferm~e invariante par translation dans L~°(F), nous 6tudierons la synth~se spectrale dans cette varlet6 en relation avec la d6croissance gt l'infini des 616ments de V. DI~FINITION 1. Nous appellerons L~ (F) l'espace des fonctions de L°°(F) qui "tendent vers 0 ~t l'infini", c'est h dire l'espace des fonctions (I)~ L°°(F) telles que quel que soit e > 0 donn6, il existe un compact K de F, tel que, si hx note la fonction caract6ristique de K, 11(I) (1 - hr)ll ~o< ~. Nous aurons alors un th6or6me qui relie la notion de r6solution spectrale celle, mieux connue, "d'ensemble d'unicit6". Rappelons que, soit F u n groupe localement compact et G son dual, on appelle ensemble d'unicit6 un ensemble ferm6 E c G, tel que si (I) e L~(F), et que le spectre de ((I)) est dans E, alors (I) est nulle. Nous dirons qu'un ensemble ferm6 E c G, est de r6solution spectrale si il n'est le spectre que d'une vari6t6 de L°°(F), et que cette vari6t6 est de r6solution spectrale:
THI~ORf~ME1. Duns un groupe localement compact G, tout ensemble de rdsolution spectrale est un ensemble d' unicitd. Ce r6sultat a 6t6 ddmontr6 par P. Malliavin dans [3], dans le cas off le groupe G est le tore ~t une dimension C. 1.2. D6eroissanee h l'intini. Pour 6tudier la d6croissance 616ment de L°°(F), nous utiliserons les notions suivantes:
~ l'infini
d'un
D~F1NmON 2. On appelle "suite d'unit6s approch6es de/.1(1) '' une suite ~ de fonctions de LI(F), de normes 1, positives, telles que pour toute fonction sur F, a, support compact et continue, j'ra~ld~ (d~ mesure de Haar sur F) tende vers a(0) quand i tend vers l'infini (0 6tant l'616ment neutre de F). Nous savons que de telles suites existent si et seulement si F a une base d6nombrable de voisinage de l'616ment neutre. Etant donn~e une suite d'unit~s approch~es, ~ une fonction • e L®(F), et ~ tout > 0, on associe les ensembles (1.1)
A , [ ¢ , {~o,}] = {7 e r/lim sup (¢. 9,)(~) > ,}
(1.2)
B~[O, (~,}] = A,[¢, {T,}] - A,[O, {91}].
1965]
VARIETE DE NON-SYNTHESE SPECTRALE
45
Une autre mani6re canonique, mais moins fine, d'6tudier la d6croissance /~ l'infini de • est: 6tant donn6e une base d6nombrable ~ de voisinages ouverts de l'616ment neutre de V, on pose: dO(y) = lim sup [I O(x)hv(x - 7)[1~o V¢~ll
(ot~ hv est la fonction caract6ristique du voisinage ouvert V). Nous consid~rerons alors les ensembles:
(1.1')
.4,(O) = {7 E F/dO(y) > e}
(1.2')
/~,(O) = L ( O ) - L ( O ) .
Remarquons que si ia fonction • est continue ces deux notions coincident, (c'est ~ dire que A~(O, {%}) = .4~(O)), et que les 616ments de L~(F) sont les fontions • de L~(F), dont les A~(O,(%}) et les ,~'~(O) sont relativement compacts. 1.3. Enonc6 ties r6suitats. PROPRII~TI~ (P). Un ensemble B darts F poss~de la propri6t6 (P), si, quel que soit le nombre entier N, et quel que soit un nombre fini d'616ments Yi de F, on peut trouver un 61~ment y e F dont tousles multiples non nuls d'ordre inf6rieur en valeur absolue ~t N n'appartiennent pas ~ Wi(B + y~).
TrI~,ORi~ME 2. Soit F un groupe localement compact, ayant une base d(nombrable de voisinages de l'origine, soit • ~ L°°(F); supposons qu'il existe une suite d'unitds approchdes % telles que les ensembles Be(O,{9i}) possddent pour tout e > 0 la propridtd (P), alors (0) n'est pas de rdsolution spectrale. THI~ORI~ME 2'. Soit F un groupe localement compact, ayant une base ddnombrable de voisinages de l'origine, soit OEL°°(F); supposons que pour tout e > O, les ensembles B,(O) vdrifient la propridtd (P), alors (0) n'est pas de rdsolution spectrale. THI~OR[ME 3. Soit dans un groupe localement compact F, une varidtd V fermde invariante par translation de L°°(F), telle que V ~ L o ( F ) # {0} alors V n'est pas de rdsolution spectrale. Le th6or~me 1 6nonc6 plus haut n'est, bien entendu, que la forme (un peu moins pr6cise) que prend le th6or~me 3 si on le traduit, par dualit6, en termes d'ensembles dans le dual G de F (en regardant les spectres de nos vari6t6s). C'est sous ia forme du th6or~me 3 qu'il sera d6montr6. THi~OR~.ME 4. Considdrons le groupe Z"; soit O~L~°(Z ") telle que les ensembles ~ ( 0 ) soient pour tout e > 0 de "densitd in/drieure" nulle, alors (0) n'est pas de rdsolution spectrale.
46
M. FILIPPI
[March
Rappelons que si B c Z ~, on appelle "densit6 inf&ieure" de B la limite inf&ieure du rapport [B N K p l l -~ o~ est le cube des 616ments de coordonn6es, dans Z ~, toutes inf6rieures h pen valeur absolue, et }A I le nombre d'616ments de A c Z'. Enfin nous d6montrerons, comme corollaire du th6or~me 4, le THI~OR~ME 5. Dans le tore d u n e dimension C, l'ensemble de Cantor n'est pas de rdsolution spectrale. Ce r6sultat (qui a 6t6 montr6 ind6pendamment par Kahane et Katznelson dans [1]) donne alors un exemple d'ensemble d'unicit6 (l'ensemble de Cantor 6tant d'unicit6, cf. 15]) qui n'est pas de r6solution spectrale.
§2. D6monstration du th6or~me 2. La d6monstration de ce th6or~me se fera en deux temps: - - un premier lemme montrera que si il existe une fonction ¢I)E L~°(F)et une fonction a ~ U ( F ) telles que + soit int6grable (en la variable r6elle u), alors (~) n'est pas de r6solution spectrale. - - nous construirons ensuite explicitement, lorsque • v&ifie les hypotheses du th6or~me 1, une fonction a ~ L~(F), qui avec • v6rifie les hypotheses du lemme pr6c6dent.
11 (1 lul)
II
2.1. LEMME. 1. Soit F un groupe localement compact, supposons qu'il existe une mesure bornde a ~ MI(F), et une fonction dp ~ L~(F), telles que (2.1)
f
+oo
lie "s... --00
* I1 (1 + lul)du < + oo
alors il est possible de trouver une constante c r~elle, telle qu' en posant b = a + c6 (6 ~tant la mesure de Dirac ?t l'dldment neutre d e F), la fonction de L°°(F) ¢ + ~ o / "iub qs = J-~ote • O)udu vdrifie
(2.2) (2.3)
b . ~P ¢: 0 b.b.'I'
= 0
(dp) n' est pas de rdsolution spectrale.
D~mostration. Remarquons d'abord que la relation (2.1)est aussi v&ifi~ en rempla~ant a par b, car e "~'b= el'Ce "l', et el'%st un nombre de module 1. Soit F(x) une fonction de la forme (2.4)
F(x) = P(x)e -x2,
ofa P e s t un polyn6me; alors F(u) (transform6e de Fourier de F(x)) d~roit
1965]
VARIETE DE NON-SYNTHESE SPECTRALE
47
/t l'infini comme ]u [%1" n &ant le degr6 de P). Cette fonction &ant enti~re, nous pouvons d6finir la mesure born6e F(b). D'autre part, la fonction de R /t valeur darts l'espace de Banach MI(F), e "iubl~(u) est continue et de norme int6grable ([le'~Ublll <el~lllbll~), elle est done int6grable; soit I son int6grale, et g un caract6re de F, on a:
(1,g) =
(e "Ub,g)P(u)du =
e
--
- - QO
qui est 6gal d'apr~s la formule d'inversion de Fourier h 2zF((b, g)). Nous avons d0nc:
F(b)= l/2n f_~e "b P(u)du
(2.5)
u e .l~b. • est une fonction /t valeur vectorielle dans L°°(F) continue et de norme int6grable, done int6grable dans L°°(F), soit ~,+~ q / = | (kl,b .O)udu,
(2.6)
ona:
J - - - O0
F(b) " • = 1/2n
"iubl~(U)du
(d ub " ¢)t dt. co
Or la fonction de R 2 dans L°°(F), (u, t) ~ e "ttu+t)b • ¢, est continue et sa norme est major6e par le produitl[e'"bl],l]~ ''b "O lifo, done ~(u, O e(u)t est int6grable, et d'apr~s Lebesgues-Fubini on a:
F(b) " W = 1/2nL2 (e "'`u+°b • ~ ) P ( u ) t d u d t soit en posant v = u + t et en appliquant encore Fubini
F(b)" • =
2.7) f(O
~ ) ( P * f)(v)dv
= v. --X2
1) Prenons F(x) = xe , P(u) = u e-"'/4( ~ ) et son produit de convolution avec la fonetion f est une constante ~gale ~ 27r. Done
b" e "-~" • =
(e "~b .O)du, oo
soit:
b'e'-b~'~
= f ~ " ~ (e"'u"'O)e d u . _
Mais e "v'°- • &ant une fonction continue non nulle ~ l'origine ( 4 &ant suppos6e non nulle), elle ne peut pas ~tre orthogonale h toutes les fonctions e ~u~pour c r6el
48
M. F I L I P P [
[March
quelconque, d o n c il est possible de trouver c tel que b • e "-b2 • tF ne soit pas nul ce qui entraine: (2.2) b . • # 0. 2) Prenons maintenant F ( x ) = x2e - ' ' , sa transform6e de Fourier est r ( u ) = x/n-72(u2/4 - 1)e-"21+ d o n t le produit de convolution avec f est nul, d o n e b • b • e "-b2 • qJ est nul, et en convolant par e "b2, on a: (2.3)
b.b.
W = 0.
3) b • W 6tant diff6rent de z~ro, il existe une fonction ~ e L I ( F ) telle que • b • W # 0; par contre (2.3) entraine que l'id~al engendr6 par ~. b • ~ • b dans L~(F) est orthogonal b. W, done que les caraet~res de F contenus dans (W) sont o r t h o g o n a u x ~ 8 • b • e • b e t d o n c ~ e • b: ceci entraine alors que ces caract~res ne peuvent pas engendrer (qa). Enfin tF appartenant/t (q~), (~F) est inclus dans (q)) et:
(oo) n' est pas de r~solution spectrale. 2.2. LEt,rME 2. Soit
• un Jldment de L°°(F), vdrifiant les hypothdses thdordme 2, alors il existe une mesure born& a sur F, telle que:
f T,e +
++
D6monstration. N o u s allons construire cette mesure a en choisissant venablement une suite (?k) d'616ments de F, et en prenant (2.8)
du
con-
a = ~ k-2(1/2i)(3~k -- cS_ak) k
ofl si 7 e F, 6~ note la mesure de Dirac au point ?. N o u s serons amen6s, p o u r v6rifier que notre mesure a satisfait (2.1)/t utiliser des • e n t de e .t,/2) tea-e -a ) en s o m m e de majorations des coefficients du developpem mesures discr~tes de masses 1, c ' e s t / t dire les coefficients du d6veloppement en s~rie de Fourier de e +"~r, , que nous allons d o n n e r maintenant.
2.2.1. Majorations de fonetion de Bessel. Soit ~, un 616ment de F, nous noterons par r(7) l'ordre de ~,, c'est ~ dire le plus petit entier positif n tel que n • ? = 0. La fonetion e ~'jm~r'~> se d6veloppe en fonction uniquement des caraet~res de G appartenant au sous-groupe engendr~ par 7, et dont les coefficients de ce d6veloppement se calculent en se pla~ant dans le sous-groupe engendr6 par ?, art lieu de F, et son dual au lieu de G, ils ne d6pendent d o n e que de l'ordre de 7. N o u s avons les d6veloppements suivants: (2.9)
e +'°~ =
~
Ps.,(u) (mr, g )
m ~R(~) (2.10)
e "(u/2)(~r-#-r) ---
~ Pm,r(U)t~mr rn +R(O
1965]
VARIETE DE NON-SYNTHESE SPECTRALE
49
en posant R( + oo) = Z = groupe des entiers
R(r) = Z/rZ
si
r < + oo
(par abus de langage, nous noterons par la mSme lettre un 616ment de Z/rZ et la classe des entiers modulo r correspondants). Nous noterons P,,co par P~; les coefficients sont donn6s par les formules:
P.(u)
=
1
~
~ 2~ e l .
sin x e - inx
Jo
dx
k=r-1
em,r(u) =
~ (1/r)e ~.a~(2~k/,) e-~2~k/',
v e m.
k=O
Soit C le tore ~ une dimension, dont le dual est Z, le sous-groupe de C form6 par les e ~2~/' a pour dual Z/rZ; en calculant le d6veloppement de e tusinx dans le groupe C aux points x = 2nk/r on a:
e~. sin(2~klr)
-~
2
P~(u)e
12~vklr
veZ
=
~ m eZ/rZ
[~
P,(u)le '2~k'/"
kvcm
J
d'o~t la formule:
P.,,(u) = E
(2.11)
P,(u).
Nous avons les majorations suivantes:
IP..Xu) l _-<1 quel que soit u, r et m ~ R(r),
(2.12)
majorations qui sont 6videntes d'apr6s les formules du calcul explicite de ces coefficients. La d6riv6e seconde de e ~"'si"x e s t -eiU'sinx(u2cos2x÷iucosx) dont les coefficients de Fourier sont major6s par (lu12 ÷ I.I), or ces coefficients sont Pfln z, donc on a:
Ip,(.)l z (1.12 + lul)/n ~ pour
(2.13)
n ~ O.
On tire de cette majoration deux cons6quences immMiates dont nous aurons besoin:
IP..,(u)l <_ r, IP.(,)l ___1 -I- A(I,I 2 ÷ I~1).
~
(2.14) m
(r)
n e Z
Soit d un nombre entier positif, soit m 8 R (r), nous dirons que m est sup6reur en valeur absolue h d et on note Iml > d si quel que soit v~mlvl > d. D'autro part, Iml---d notera la proposition contraire de I ml > d. Alors pour R > 0 donn6, quel que soit r, fini ou infini, et quel que soit I u [ < R, pour tout e > 0 donn6, il existe un entier positif d tel que:
50
M. F I L I P P I
(2.15)
•
[March
[Pm,,(u)[
Iml>a
Enfin quel que soit r > 2, nous avons si 1 ¢ m (m 616ment de R(r))
Ie.,.(u>l =< Ieo(u)l + x IP.(u)l
(2.16)
Ivl _~ 2
Comme nous avons de plus Po(0)= 1, et P~(0)= 0, il existe un nombre positif ¢o tel que I P,(,01 < Ieo(u) l si l u I < ¢o, donc l'in6galit6 (2.16) est valable pour tout 616ment m de R(r) mais en imposant cette fois-ci la condition [ u[ < Go. Consid4rons alors la somme ~l~l-> 31P~(u)l: pour Ivl >__3, les deux premi6res d6riv6es de Pv(u) ~ l'origine sont nulles, et la s6rie des modules des d&iv6es troisi6mes des Pv(u) est une s6rie normalement convergente (s6rie des coefficients de Fourier de - i sin3x e~'i"x); nous avons donc pour [u I < 1:
x
IP~(u)l < lul ~ 8 (oa 8
est une constante positive).
Ivl > 3
Les d6riv6es ~ l'origine de Po(u) et P 2 ( u ) sont nulles, la d6riv6e seconde en u = 0 de Po(u) est - 1, et celle de P2(u) est 1/4; commelPo(O) I+ XI~I _ 21P~(O)I= 1, il existe un nombre ~ < 1, et un nombre ¢, 0 < ~ < ½Go tels que: (2.17)
I P , , . , ( u ) l < ~ p o u r r > 2 , mquelconque, e t ~ < l u l < 2 ¢ .
2.2.2. Estim~e de normes par r6gularisation.
Soit (~3 la suite d'unit6s approch6es de L~(F), nous devrons d6duire une majoration de normes de certaines expressions d6pendant de ~,/L partir de majorations sur les normes d'expressions analogues oh q) est remplac6 par l'une de ses r6gularis6s q~. V~ (q~- V~ sera toujours not6 dans la suite par ~3; c'est pourquoi n ous d6montrerons ici le lemme suivant: LEMME 3. Soit (~i) une suite d'unitds approchdes de U(F), soit ~ e L ~ ( F ) et supposons queen tout point ~ de F on ait lim a lors
supl.,(r~ I ___1
I1"~ II-- 1
D6monstration. Soit ~ une fonction continue d support compact K c F, cp~• converge, en norme dans U(F), vers ~: en effet, soit V un voisinage compact de l'616ment neutre 0, et soit 9,v les restrictions ~t V des fonctions 9,, II ~,- ~,. I1~ tend vers z6ro quand i tend vers l'infini, doric (9~ - 9~v) " a converge vers 0 darts LI(F), et 9~ir. • ~ converge dans LI(F ) en norme vers ~({9~v " ~} ~tant un ensemble de fonctions uniform6ment 6quicontinues ~ support dans un compact fixe, convergeant ponctuellement vers ~). La norme dans L°°(F) de ¢ est la borne sup6riere pour toutes les foctions continues ~ support compact de norme 1 dans L~(F) de [ ( ~ , , c ) l ; de plus ( ¢ , ~ ) est la limite de (¢~,~), quand i tend vers l'infini, car
VARIETE DE NON-SYNTHESE SPECTRALE
(O,,ct) = (O,- ~)(0) = (O. ~,. ~)(0)
51
(o/l ~ d6signe ~(y) = ct( - y))
et Oi " 0t converge dans L1(F) vers ~. Notons alors Sp(e) l'ensemble des y ~ F tels que quel que soit i > p; < 1+ e (e > 0 arbitrairement donnO. La r6union des Sp(e) est tout F. Soit une fonction continue h support compact de norme 1 dans L~(F), la mesure de K c3(S~,(e) tendant vers 0 quand p tend vers l'infini, il existe Po tel que l'integrale de ~ sur Spo(t) soit inf6rieure h e. Nous avons:
=(
+( ,J(Spo(~))
J Spo( O
o~ la premiere int6grale peut ~tre major6e, en module par le sup de O~ sur Spo et la deuxi~me p a r e multipli6e par t[ O,{I°o (qui est inf6rieure it O o0; doric si i>->_po, ( O i , ~ ) [ < l + e ( l + • oo). Comme ( O i , ~ ) t e n d vers (O,~), nous avons (O,~) < l , e t d o n c • ~ < 1 . 2.2.3. Construction de la suite (Yk)" Soit (t/k,~) une double suite de nombres positifs tels que: 2
(2.18)
r I (1 + qk.q) - 1 < e - ' k>q
nous savons que quel que soit r, fini ou infini, il existe (cf. (2.15)) pour tout q, une suite (dq(k)) d'entiers croissants tels que: (2.19)
~ Iral>dq(k)
[p.,,(u) I < qk,,
pour [u [ < q, et quel que soit r,
nous prendrons les entiers dq(k) croissant en fonction de q 6galement. Soit alors e(k) = It • tloo/8~. ]-Ii__
On prend ?~ de telle faqon que ses multiples non nuls jusqu'/l l'ordre 2d1(1) n'appartiennent pas h B~. On prend alors Y2 de telle faqon que ses multiples non nuls jusqu'it l'ordre 2d2(2) n'appartiennent pas h la r6union prise pour ]nl I < 2d2(1) de (B2 + nlYl). On prend alors y~ tel que ses multiples d'ordre inf6rieur (en valeur absolue) it 2d,(p) n'appartiennent pas it la r6union pour I n, ] £ 2d,(i) et i < p de (Op + ~,, ny,). Cette construction est possible car tous les Bp poss~dent la propri6t6 (P). Nous noterons r k = r(Yk) = ordre de Yk
R~ = R(rk) = t Z = le groupe des entiers si r~ = + oo [Z/rkZ si rk < + ~ .
52
M. FILIPPI
[March
Nous noterons D(q) l'ensemble des 61gments 0 de D tels que O(k) = 0 si k < q et F(q) l'ensemble des 616ments de D(q) de k i~me-composante infdrieure en valeur absolue it dq(k) pour tout k ([0(k)[ < dq(k)). Soit N u n entier positif plus grand que q, nous noterons D(q, N) et F(q, N) les ensembles des restrictions aux k < N des 6ldments de D(q), respectivement F(q). Un entier q 6tant fix6, quel que soit ~ e F, il y a au plus 4" l--I dq(k)
(2.20)
k
416ments O(k) de F(q) tels que l'414ment ~ + ~k 0 (k) Yke A r En effet la diff4rence de deux tels 614ments doit appartenir ~ Bp, et est de la forme ; / = ]~k nk~k (O4 on a pos4 nk = 01(k) - 02(k), 01 et 02 4tant deux 414ments de F(q)) avec nk nul pour k < q et Ins ] < 2dq(k). Nous aurons le d6nombrement (2.20), si nous montrons que nk est nul pour k > p: supposons qu'il existe des nk non nuls avec k > p, et soit n u celui qui ale plus grand iudice (alors N e s t sup4rieur/t p, et aussi/t q, car nk = 0 si k _<-q). Nous aurions alors ~ , nk]~ k ---
(2.21)
~' ~ Bp = BN (car N > p) et
k
nN'Ne (BN -- k~N nk'k) or [nkl < 2d,(k), et comme N > q, Inkl < 2dN(k), ce qui est en contradiction avec (1.21) d'apr~s la construction de notre suite 0k). 2.2.4. Majoration de [1e .i,,. tI)I1~ Nous noterons par z(?)~ la fonction translat6 de ~ de ?, c'est g dire fir " ~ (2.22,
e""= kFI • [ L ? , , , k ( k -
2u)6,,k ]
ofl I-I " repr6sente un produit de convolution. 2.2.4.1. a est la somme d'une sdrie convergente, soit N un entier, notons par an la somme des N premiers termes de cette s6rie, et r N la somme des autres termes, pour u donn6, il existe un entier N Otel que IIrN H' < 1/lu I pour tout N > N o. Choisissons alors N sup6rieur h la fois ~t No, et ~t x/I ul/2~ oa ~ est le hombre intervenant dans la majoration (2.17) (l'utilit6 de prendre N > ,4rl u[/2~ sera rue dans 2.2.4.3.). Posons
a=aN+r N e "Juan_ e "luaN" e "iUrN
et on a : (2.23)
Ue " ' " I[' < e.
VARIETE DE NON-SYNTHESE SPECTRALE
1965]
53
2.2.4.2. Prenons l'entier q tel que qa<[ul< (q + 1) 3, alors pour tout k > q, on a
I k -Zu I < q"
(2.24)
Nous allons d~composer alors e "i"Nde la fa~on suivante"
k
m
k
Lm e R k J (N 6tant au moins de l'ordre de lul 1/2, e t a 6tant de l'ordre de lut l/a, d~s que lul q
est assez grand on a N > q~ ce qui donne un sens/l notre d~composition). Nous savons d'apr~s la majoration (2.14) que quel que soit r on a:
E
<
m ~ R(r)
donc (2.27)
+ :(lvl 2 + tvl)
1[Lqlll <(1 + A(lul2+lul))q
(c et p 6tant deux constantes positives convenablement choisies.) Et nous avons 6galement:
Ile"".~,fl~<=lle",,"ll,llgol!lIlM,,..[l~
(2.28)
c'est cette derni~re norme que nous avons maintenant/~ majorer. 2.2.4.3. Nous avons
M,,,q " ~ =
l-[ q
d'ofi, en d6veloppant ce produit: (2.29)
Mq, N. . =
]~
[ q
q eD(q,N)
~,k
Posons alors: (2.30)
Sq,u =
]~
[
EF(q,N)
(2.31)
Tq,N =
~' ot~E(q,N)
1--I q
[
1-I q>k
off E(q, N) est le complfmentaire de F(q, N) dans D(q, N); alors (2.32)
Mq,N "
~ = Sq,N + Tq.N.
Pour majorer la norme de S#,N nous allons trouver une majoration ind6pendante de i (d6s que i sera assez grand) du module de
54
M. FILIPPI
(2.33)
S~(~) =
~
[
1--IP~(k),rk(k-2u)~i(~-
q ~ F ( q , N ) L q
[March
~ ~(k)~)] k _l
(ofa (I)~est • • ~p~); cette majoration ne d6pendant, de plus, pas de ?, en utilisant le lemme 3 nous obtiendrons une majoration de la norme de S~. s dans L°°(F). Soit ~ e F donn6, soit q~ un 616merit de F(q, N), soit p~ = p(% y) l'entier tel que - ~,kq~(k)~kappartienne/t Ap~ n ".Ap~,-1, il existe un indice i~ = i(tp, y) tel que pour tout i > i~, ~(~ - ~k ~(k)~k) soit compris entre e(p~) et e(p~,- 1). F(q, N) est un ensemble fini, doric quel que soit i sup6rieur ~ tous les i, et quel que soit ~0 appartenant ~t F(q, N), ~0' - ~,k ~(k)~k) est compris entre e(p~) et e(p~ -- 1). I1 exist6 an plus 4p 1-Ik
- 1). Soit s(u) le nombre d'entiers k tels que k > q, et ~ < [k-2u] < 2~ (cf. majoration (2.17)), nous pouvons majorer le produit I-[~
Is,(e)l <¢'" z e(p)4~ I-[ d,(k)< I1 11 4( ~ 1/2") p
soit
k
I Si(~)l < c~ e x p ( -
fl~lul ~/~)
oO c~ et fl~ sont des constantes positives bien choisies; et done (2.34)
II S~,r¢ IIoo< ci e x p ( - ~, I u I"~)
2.2.4.4. I1 nous reste alors ~t majorer la norme de T~,s. Soit G(q,N) l'ensemble des fonctions ~b(k) de D(q,N) telles que ~b(k) soit ou nul ou sup6rieur en valeur absolue h d~(k). Alors tout ~l~ment O(k) de E(q, N) se d6compose en somme d'une fonction ~(k) de F(q, N) et d'une fonction ~,(k) de (q,N), cette d~composition &ant unique si on impose de plus la condition d'orthogonalit6 ~(k)@(k)= 0, on prend pour cela
[-- O(k) si ~(k)
~
IO(k)l<=d~(k)
0 sinon.
O(k) = O ( k ) -
Nous d6composerons toujours les 616ments de E(q, N) de cette faqon. Nous allons alors calcuIer la somme (2.31) qui donne T~,n en regroupant les termes, correspondant aux fonctions O(k) ayant mSme composante sur F(q,N), ~0(k), et en utilisant la formule
1965]
VARIETE DE NON-SYNTHESE SPECTRALE
~(k)~0
q
nous avons:
~(h) =0 q
[ I-I Pq,(k),,k(k ,,+,]
T,.N
(2.36)
55
oh S(~k) est d+fini par: (2.37)
S(~,)%+r~.m[ ~,ol~)=°
~¢=0
P,(h).,,(h-2u)zC~.,~b(k)~'k+ ~.,~(h),h)ep]
q
S(~b) peut 6galement s'~crire:
~l~(k)?,
,(h)=o ~0=0
~(h)T4)@) ]"
q
Nous majorerons alors ce terme comme nous avons major~ la norme de Sq.N dans 2.2.4.3., avec la diff6rence que nous ne pouvons ici majorer la valeur absolue du produit
l'-I P,(h).,h(h-2u)
~(h) ffi o q
que par 1, et que la sommation ne portant que sur une partie de F(q, N), notre m6thode de majoration de la norme de la somme de translates de (I), sera ~ fortiori valable; nous obtenons ainsi le r6sultat: (2.38)
IIs(~)ll + < 8 II* II+.
Enfin nous avons
x
[rI
#/eG(q,N) I ~/(k),#O
=[ rI
I
q
E1 +
YI
lml >d,tk)
Ip.,.~(k-'.)13]- 1
I
et d'apr~s le choix de q (1 q- 2u < q), et les in6galit6s (1.18) et (1.19), nous obtenons (2.39)
IIT+ ~II+ < 811+11oo e-q' ~ czexp(- ,= I. I~,~>.
2.2.4.5. La majoration de la norme de T~,N est n6gligeable devant celle de la norme de Sq,~, et nous avons done pour
(2,40)
[le
""°. *
[[o~une majoration
du type suivant:
IIe.`-. +11~ < Cexp(~l u 1'/3 L°gl u I- ~'[u[ 1:2)
ce qui entraine, le terme en ]u I 1/2 ~tant d'ordre sup6rieur au premier que
tdgrale
Fin-
56
M. FILIPPI
[March
f +~ lie"fua"4)]]oo(I + ]uJ)du converge. 2.3. D6monstration du th6or6me 2'. Ce th6or~me n'est qu'un affaiblissement du th6or~me 2, car nous avons la propri6t6: (2.41)
,4"~((I))= A2,(4)).
En effet, • 6tant donn6, soit ? E A 2 , , et soit V un voisinage arbitraire de 7, soit W un voisinage de l'origine de F, tel que W + W c V - ),; d'apr~s la convergence des 9i vers 6, il existe i o tel que pour i > io, la norme dans U(F) de q~ restreint au compl6mentaire de W soit inf6rieure h e/114)1[o~; comme ),~ A2~ il existe un i > i o tel que 19i' 4) 1(?) > 2e, et 91 " • 6tant continue il existe une fonction continue ~t ~t support compact inclu dans W, de norme 1 dans LI(F), telle que I ~ " ~0~• (I)](7) > 2~. Alors (la restriction de ~i au compl6mentaire de W 6tant de norme inf6rieure h e/II 4)II ~o)la fonction fl, produit de convolution de et de la restriction 9~ ~ W, est continue, a son support dans V - 7 , et on a I fl" 4)[" (7) > e, ce qui entraine que II4)v IIoo > ~. Ceci &ant vrai quel que soit re voisinage V de ~, d4)(7) > e. §3. D6monstration du th6or6me 3. Soit 4) un 616ment de L~(F), pour tout 8 > 0 donn6 nous noterons A~(4)) le plus petit ferm6 tel que la norme de • restreint au compl6mentaire de A~(4)) soit inf6rieure ~ 8. Quand le groupe F a une base d6nombrable de voisinage de l'origine, les ensembles ainsi d6finis sont les .~'~(4)). Nous poserons B~(4))=A~(4))- A~(4)), et la propri6t6 de d'appartenir Lo(F) se traduit par le fait que les ensembles A~(4)) et B~(4)) sont compacts. Nous d6montrerons alors un lemme, qui, avec le th~or6me 2' d~montre le th6or6me 3 darts le cas of1 F a une base d6nombrable de voisinage de l'616ment neutre. 3.1. LEMME4. Dans un groupe localement compact non compact F, tout compact poss~de la propridtd (P). Nous allons utiliser le r6sultat classique suivant: Soit F un groupe locaIement non compact ab61ien, et soit U un voisinage compact sym6trique de l'unit6. Soit F' le sous-groupe engendr6 par U, alors F' contient un sous-groupe discret D engendr6 par un nombre fini d '616ments tel que F'/D soit compact. Nous allons traiter les deux cas suivants: a) F contient un 616ment ~'o d'ordre infini qui engendre un sous-groupe discret. b) Si on n'est pas darts les conditions du a), D est un groupe compact (engendr~ par un nombre fini d'616ments d'ordre fini), donc aussi F'. F' 6tant engendr6 par u n
1965]
VARIETE DE NON-SYNTHESE SPECTRALE
57
voisinage de l'origine, il est ouvert donc F/F' est un groupe discret, soit H, dont tousles 616ments sont d'ordre fini. La r6union d'un nombre fini de compacts 6tant compact, nous allons d6montrer que pour un compact K et un nombre entier positif arbitraire N i l existe un 616ment ? de F tel que tous ses multiples non nuls d'ordre n avec [n I < N, soient hors de K. a) Soit (Yo) le groupe ~o " Z engendr6 par ~o; ou bien K ne rencontre pas (Yo) et le probl~me est r6solu en prenant y = Yo, ou bien K rencontre (Yo) et K n (~'o) est compact done fini; soit alors A l e plus grand des entiers tels que A. ~o E K n (~'o), il suffit de prendre pour y l'616ment (IAI + 1) b) l'image K ' de K par la projection F ~ F / F ' = H est compacte done finie, si t E F / F ' est tel que tous ces multiples non nuls et d'ordre ] n [ < N ne sont pas dans K', un repr6sentant y dans F de la classe t r6pond ~t la question. I1 suffit de traiter le probl6me dans le cas d ' u n groupe F discret dont tous les 616ments sont d'ordre fini. Soit K une partie finie d'un groupe discret infini; soit N u n entier positif donn6, supposons que pour tout 616ment ~ F, 3 n < N tel que ny :P 0 et n~ ~ K; en d'autres termes, soit K . l'ensemble des y ~ F tels que n y # O et n T ~ K : nous supposons que F = Un__
Nous utiliserons ici, pour d6signer les quantit~s analogues ~ celles intervenant dans le lemme 2, les m~mes notations (aucune confusion n'~tant possible). Nous d~finirons une suite (~k) d'~l~ments, de F, comme dans 2.2.3., les ensembles A,(q~) jouant iei le r61e des ensembles A,(q~,(~0i}) du lemme 2 (nous noterons encore Ak et Bk pour A~(k)(~) et B~(k)(~). Nous prendrons alors pour mesure a la mesure
a = ~, k-2(1/2i)(6rk - ~-rk). k
58
M. F I L I P P I
[March
II
Nous allons alors majorer la norme e"U° '~ en proc6dant comme dans 2.2.4: la suite d'unit6s approch~es, et la dffinition des ensembles Ak n'intervenaient que dans les deux majorations (2.34) et (2.38) o/1 les quantit6s S~.N et S(~k) sont d~finies par (2.30), et (2.37); nous allons montrer que, sous nos nouvelles hypotheses, ces majoratios sont encore valables. Soit ? e F, et soit 9 un 616ment de F(q,N): notons p, = p(y, cp) l'entier tel que - ]~k 9(k)~k appartienne ~ Ap n ...A o_ x. I1 existe alors un voisinage V(~,, tp) de tel que la norme de la restriction ~ V(~, 9) de ~( ~k 9(k)~)* soit comprise entre e(p,) et e ( p , - 1). Soit V(?)l'intersection des voisinages V(?, tp)quand q0 parcourt l'ensemble fini F(q, N). Nous majorons comme dans 2.2.4.3. le module produit I II~<~< s P,
If ,
11 P,(h).,,(h- ~ u)
¢~(h)= o q
en module par 1 ; ce qui nous donne bien la majoration (2.38). §4. Cas particulier du groupe Z" 4.1. D~monstration du th~or~me 4. Ce th~or~me se d6duit du th6or~me 2/t l'aide du lemme suivant: LEMME 6. Tout ensemble de densitd infdrieure nuUe dans Z", poss~de la propri~td (P). Nous pouvons donner une d6finition de la densit6 inf6rieure d'un ensemble discret B dans R" de la fagon suivante: On appellera "densit6 inf6rieure" de B la limite inf6rieure quand x tend vers + oo du rapport I B n Jx ] (2x)-", ot~ Jx est le cube des points de coordonn6es en valeur absolue inf~rieure ou 6gale ~ x, et ] A I est le nombre d'61~ments de A, si A est une pattie de R". Si nous injectons Z" dans R", les deux d6finitions de densit6s inf6rieures d'un sou-sensemble de Z" coincident. La propri&6, pour un sous-ensemble discret de R" d'etre de densit6 inf6rieure nulle est 6videmment stable par translation et par r6union finie. Soit x un nombre r6el, non nul et B un sous-ensemble discret de R ~, nous noterons xB l'ensemble des 616ments xb ofa b parcourt B; la propri~t6 d'&re de densit6 inf6rieure nulle est aussi stable par "multiplication" que un r6el non nul x. Soit alors B u n sous ensemble de Z" de densit6 inf6rieure nulle, N u n entier, la
1965]
VARIETE DE NON-SYNTHESE SPECTRALE
59
r6union pour tout p < N du "quotient" par p d'un hombre fini de translat6s de B est darts R" de densit6 inf6rieure nulle, doric ne peut pas contenir Z,~ donc B poss6de la propri6t6 (P). 4.2. D6monstration du th6or6me 5. Pour d6montrer ce th6or~me, nous utiliserons la mesure dF sur C, oi~ F est la fonction de Lebesgue (Zygmund, Trigonometric series, 2~me 6dition, vol. 1, p. 196); cette mesure a pour support l'ensemble de Cantor; l'ensemble de Cantor est isomorphe, en tant qu'espace topologique au produit d'une infinite d6nombrable de Z/2 Z, et la mesure dF consid~r6e n'est autre que la transport6e par cet isomorphisme de la mesure de Haar de ce groupe produit. La transform6e de Fourier de dF est la fonction de L°°(Z) oo
(4.1)
~(n) = ( - 1)n(2;r) -1 I-I cos(2~3-kn) k=l
qui n'appartient pas/t L°~ (Z)(car ~(3n) = ~(n)), et dont le spectre est l'ensemble de Cantor. Le th6or~me r~sulte du lemme suivant: LEMME 7. Pour tout ¢ > 0 donn6, soit A l'ensemble des entiers tels que, IcP(n)l > ~, et soit B~ = A~ - A~, B~ est de densit~ nulle. Nous utiliserons l'6criture des entiers dans la base 9. Soit n un 616merit de A~ (oh e > 0 donn6), soit (4.2)
2n = %ctp_ l " " 0~1~o
l'6criture dans le syst~me/i base 9 de 2n, la fonction ~(n) s'6crit: i = p
oO
oO
(4.3) q)(n) = l-[ cos(n~q 0ti_ 1 ... %) 1-[ c°s(2nn3-(2k+1) 1-I cos(21rn3-2h) • i=0
k=O
h=p
Donc n ne pent appartenir /l A~, que si le premier produit est d6j/i de valeur absolue sup6rieure /t e; soit N(e) le plus petit entier tel que Icos(Tc/9)[ N(~) soit inf6rieur A e, s i n appartient ~ A~ l'6criture darts la base 9 de 2n ne contient au plus que N(8) chiffres diff6rents de 0 et 8. Soit R un entier, posons (4.4)
B~,R = B e ~ [ - 9 R, + 93].
S i n appartient/l B~,R, on a (4.5)
n - - n 1 --n2
oh n~ et n2 appartiennent/t A~.
(4.6)
n=nl-n2
oh l n i l et [nzl sont les testes de la division par 9 R+I
de I ,1 et 1 21 (4.5) provient de la d6finition de B~, et (4.6) du fait que nous supposons n ~ [ - 9a, + 93]. De plus 2n~ et 2he ont dans leurs 6critures en base 9 au plus
60
M. FILIPPI
N(~) chiffres diff6rents de 0 et 8, donc 2n~ et 2nz' ont dans leurs 6critures en base 9 au plus N(e) + 1 chiffres diff&ents de 0 et 8. Le nombre d'616ments de Be,a sera donc inf6rieur au carr6 du nombre ha(e) entiers appartenant/t [ - 9 a+ t, + 9a+ t], dont l'6criture en base 9 contient au plus N(e) + 1 chiffres diff6rents de 0 et 8. O r n o u s avons."
nR(e) < 2
(4.7)
(N(e) + 1'~ ~ R+ I]
2R_N(O9mO+,
o6 ( ~ ) est le nombre de combinaisons de p 6Mments pris n ~t n . k- J
Le hombre d'~16ments de [ - 9 R, + 9 R] tend plus vite vers l'infini que nR(8)2 donc B~ est de densitd nulle.
BIBLIOGRAPHIE 1. J.-P. Kahane and Y. Katznelson, Contribution d deux probldmes, concernant les fonctions de la classe A, Israel J. Math., 1, (1963). 2. P. Malliavin,Impossibilitd de la synthdse spectrale sur les groupes abdliens non compacts, Publ. Math. de I'LH.E.S., 2 (1959). 3. P. Malliavin, Ensemble d'unicitd et Synthdse Spectrale (Congr6s Stockholm) (1962). 4. W. Rudin, Fourier Analysis on Groups, Interscience Tracts in Pure a n d A p p l i e d Mathematics, no 12. 5 A. Zygmund, Trigonometric series, 26me edition, vol. 1. FACULTEDES SCIENCES, DEPARTEMENTDES MATHEMATIQUES, ORSAY (S.ET0.) FRANCE
