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2
I . O L D E R R E S U L T SO N T H E M A R K O F F S P E C T R U M
L r u u e l . S u p p o sae: f e o , a t , . . . , a n , b t , b z , .a. .nl d P : L a y , a r , . . . , a n , ct,C2,...w 1 ,h e r e n ) 0 , a gi s a n i n t e g earn da 1 , . . . , a n , b t , b 2 , . . . , c t , c 2a, r. .€. positive integerswith br I c1. Thenfor n odd, a > B if and only if b1 > q; for n even,a > p if and only if b1 < c1. Lnuun 2. Supposea and B satisfy the hypothesisof Lemma l. Then l"- fl<2-\n-t). Pnoor. Define pr and qi (0 S j < n) by Pilqt :fd,o,ar'...,aif,
@ i , q , ): l .
Then by induction qr > 2U-t)1, (0 < j < n), and the lemma follows from this by a simple calculation. It is easyto prove that the Markoff spectrum contains the Lagrangespectrum (Chapter3, Theorem l). The two spectraare not identical(Chapter3, $l), but they do coincideif we only considerthe numbers( 3 in eachspectrum. Indeed, Markoff [1879] showedthat for any indefinite binary quadratic form /, the inequality fffilm("f) > /5 trotds and rhar there is only a countablenumber of possiblevaluesof rtfilm(/) lessthan 3. Each of these values is associatedwith some form / with integer coefficientssuch that f(x,l):0 has rootswith a purelyperiodiccontinuedfraction. Thus, eachpoint of the Markoff spectrumlessthan 3 is associated with a purely periodic doubly infinite sequence A; for sucha sequence we haveL(A) : M (A), so L and M coincidebelow 3. The situationfor definitebinary quadraticforms is very differentand much simpler. It is classicaland easyto prove (seeCassels[1959,pp. 30-33]) that for any definitebinary quadraticform f , the inequaliW\nA(nllm(f) >,/3 holdsand that any value ) r/3 is taken onby {dffilm(f) for somedefinite form /. The result that ltQt) ) r/5 for all real rr is often referred to as Hurwitz's Theorem, even though it is contained in Markoffs work. This is because Hurwitz [891] proved p(,t) > r/5 directly, whereasthe paper of Markoff U879] approachesthe problem via quadratic forms. In fact, even earlier, Korkine and Zolotareff[873, pp. 369-370]sratedthe resulrJV(f; I mt.f l > r/5 and also statedthat the next largestvalue of ,/V(f; I *f -f1is t/2. Markoff [879] refersto their work as the startingpoint for his own. We give a proof here of the result of Markoff [1879] mentionedabove. Our proof usescontinuedfractions,but the ideasgo back to Markoff. Other accountsof the proof alongsimilar lines havebeengiven by Heawoodt1922) and Dickson [1930. Chapter VII]; theseare more easily readablethan the original work of Markoff. The followingsimplelemma and its corollarieswill be importanl in our proof. Our goal is to characterizeall thosedoubly infinitc sequences '1 for which Ai(A) < 3 for all i. Sincethc digits in such a sequence,,1can only be I or 2, we shall oftcn omit thc commasbetwcenthe partial quolicnls
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4
I. OLDER RESULTSON THE MARKOFF SPECTRUM
then B > o, so Lemma 3 with n : 2 shows that )"i(A) > 3 for some i. Pnoor or Conorllny 4. If the pattern mentioned in the corollary occurs and we define a and B by p a .-l lr22tp+z ll...l2-, then p > o, so Lemma 3 with n : p t2 showsthat Ai(A) > 3 for some i. Lnurra.r.4. If Ai@) 24 0 | 2 2)u 142 2 | wilh u)
| can occurin A.
Pnoor. If a pattern of the form given in the lemma occurs, since 2 | 2 does not occur we can define subsequences a, p, y,6 of I as follows: 6
a
F-l-lZ1l ta EZaI1, w h e rfe : l ( 2 2 1 l ) u - t 2 a . . . 1N. o t e ^_l2y+7 5y+3
^_l2f+7 sB+3'
ByLemma 3 with n = 4 wehaveB < a and 7 < d. Sincethefunction (l2x +7)lQx + 3) increases as.r increases, theseinequalities imply l2Y+7,126+7 l79P+105 B < t' : t r u > 5 d + 3: ' r s p + 4 4 ' This impliesB < lTZl1l (the positiveroot of 75xz- l35x - 105 : 0), w h i c hi s f a l s e b y L e m mI as i n c e P:l(22 | l)u-t2q...1. Tnronru l. If ),i@) < 3 for all i, thenA hasoneof the twoforms (l)
. . . 2 2 l t e q 2 2 t r p t 2 2 t p n l2 2 . . .
or (2)
. . . 1 l 2 * e r tl t 2 r o tl l 2 p n .I 1 . . .
wherethe k(i)'s are nonnegativeintegers. Pnoor'. It is clear that there are no singleton I's or 2's in I and we can assumeA + 22-11 (which is of form (l) and of form (2)). Hence there exist three consecutiveoccurrencesof either I or 2 in A. First supposethat2n@> 3) occurs.By Lemma 3, Corollary3, we in fact have n > 4. If I is not of form (2), then at leastone pattern l- (m > 4 by Lemma 3, Corollary 2) occurs. Now consider two patterns 2n @ > 4) and | ^ (m 2 4) which are closestto each other, that is, only doubleton I's and 2's occur betweenthem. The two patterns cannot actually be adjacent by Lemma 3, Corollary l, so we have a pattern (3)
2 a ( 11 2 2 ) u l m 2 2 l ,
u)l,m)4
(using Lemma 3, Corollary I to seethat the rightmost digit is l). In fact, nt : 4 in (3) by Lemma 3, Corollary 4. But now it follows from Lemma 4
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I. OLDER RESULTS ON THE MARKOFF SPECTRUM
Lerraun5. Suppose Ai@) < 3 for all i and A lT, so A is of form (4) with k(i) :Zr(i) for eachi. Thennopattern (5) l p p 1 1 2 .2. . t p s 1 2 2l p p . 2 2l e \ 2 2 l p p 1. . . 2 2 l p Q j + r ) w i l hp ( r ) > p ( 2 ) a n dp ( 3 ) > p ( 4 ) ,o r w i t hp ( 1 ) > p ( 2 ) ,p ( 3 ) : p ( 4 ) ,p ( 5 ) : p(6),...,p(2j- l) : p(zj), p(2j + l) > p(zj +2) for somej ) 2, occurs in A. p(2) : p(l) - 2, Pnoor'.If a patternof form (5) occurs,we canassume for otherwisethereis an i with )"i(A) > 3 by Theorem2, Corollary.Now if we take n : p ( l ) , o : 2 2 l p p 2l 2 . . . , f : 2 2 l p g 2l 2 . . . in Lemma 3, we find thal.),/A) > 3 for some i, becausep > a by the hypotheses of Lemma5. TnsonrM 3. In orderto haveAi@) < 3 for all i and A +1, it is necessary and suficient that A havetheform . . . 2 2 l z , ( - r 2) 2 l 2 , e y2 2 1 2 , g212 . . . , wherethe r(i) are nonnegativeintegerswith thepropertiesi (n) lr(i) - r(j)l < | for all i andj: (B) if r(i + l) - r(i) rs -l or +1, respectively, thenthefirst of the integers : j r(i + + l)- r(i i) (i 1,2,...) whichis notzerois +l or -1, respectively. partexcepttheuniversalinequality Pnoor. All conclusions of thenecessity Theorem 2, Corollary,and Lemma5. r(j)l I follow at once from S lr(t) If the pattern 2 2 l n 2 2 l n , r 22 2 l r a a2 2 l m occursin ,4, then we would havem > n + 2 by Lemma3, Corollary4 and this givesa patternforbiddenby Lemma5. Thus if any two of the r(i) differ by two or more,a pattern (6)
2 2 ln 2 2 lna2 ... 2 2 ln+z2 2 ln+q
must appear somewherein ,4. we supposea pattern (6) does actually occur i n I w i t h , s a y ,2 r ( 0 ) : n , 2 r ( l ) : " ' : 2 r ( k ) : r t * 2 , 2 r ( k + l ) : n * 4 , k > 2. Choosinga pattern in which k is minimal and using Lemma 5 with p ( l ) : 2 r ( l ) , p ( 2 ) : 2 r ( 0 ) , w e h a v e2 r ( - l ) : p ( 4 ) > p ( 3 ) : 2 r ( 2 ) : n * 2 ; lr(- l) -/(0)l ( I thus implies2r(-l) : n *2. The minimality of k combines with Lemma 5 to give contradictory restrictionson r(-k). This proves (l). For the sufficiency part of the theorem, we apply Lemma 3 as follows: Consider any pair 2 2 in the sequence,4 and let A:6lp22lny, where d and y have the form 2 2... andn > k without loss of generality. By (A) we have n: k or n: k * 2. In the former caseLemma 3 immediately gives max([2 ln y7+t0 2 ln 61,12lnd] + [0 2 1^ yl) < 3.
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I. OLDER RESULTSON THE MARKOFF SPECTRUM
r(k + l), we must come to a term r + I on the right and an equidistant term / on the left. Thus the latter term is one of a block of consecutiveterms r which is longer than the correspondingblock of consecutiveterms r on the right. Hencethe first nonzerodifferences(i +7 + l) -s(, - j) is -1. So we seethat (B) holds for the s(i)'s. To finish the proof of Theorem 4, we use the argument of the proof of Theorem 3 to show that since (9) and (B) are valid for the s(i)'s, so is (A). Following Dickson [930, p. 88], we call the sequence{s(i)}, which is obtained from the sequence{r(i)} in Theorem 4, a derivedsequence.We can continue this processand obtain another derived sequencefrom {s(i)}. Indeed, we can repeatedlyform new derived sequencesuntil we reach a sequence of all equal numbers. The fact that the string of derived sequences does terminate after a finite number of stepsis the key idea in the proof of the next theorem, which is the main result obtained by Markoff [1879]. A such Turonru 5. Given any e > 0, the set of doubly infinite sequences that Li(A) < 3 e for all i is finite. Every such sequenceA is purely periodic. The set of numbers lessthan 3 in the Markof or Lagrange spectrum is countableand discrete,with 3 as its only limit point. Pnoor. Given s > 0, supposeI is any sequencesuch that ).i(A) < 3 - e for all i. We excludethe cases.4 = T and A :2, so both l's and 2's appear in A. By Theorems 3 and 4, we know I satisfies A - . . . 2 2 1 2 , 1 _21 1 2 1 2 , Q21 2 l 2 , g 2 2 . . . where the nonnegativeintegersr(i) are either all equal or satisfy
(l 0 )
1,... { r ( i ) } : . . . , r 1 l , f s ( - l ) , r * l , / r 1 6 y , / lf , / r 1 1 ; , r +
r(i) and s(i ) for somenonnegativeintegersr and s(i); further the sequences satisfy conditions (A) and (B) of Theorem 4. It follows from the proof of Theorem 4 that we can continue this process. That is, the integerss(i) are either all equal or satisfy { s ( t ) }: . . . , 1 I l , t u ( - t ) ,It l , t y e 1 ,I l l , t y p 1 ,+t 1 , . . . for some nonnegativeintegerst and u(i), where the a(i) satisfyconditions (A) and (B) of Theorem 4. The integersa(i) either are all equal or make of form (10), etc. We shallprove that the numbersr,s,t,11,... a sequence are all boundedby a number dependingon e, and after a boundednumber (depending r(i ), s(i ), t(i), u(i),... is made on e) of stepsoneof the sequences up of all equal numbers. Working backwardsfrom this sequenceof identical numbers,we obtain only a finite number (dependingon e) of different A, eachof which is purely periodic. This will prove the first two sequences assertionsin the theorem. a - t , e 0 , 4 t , . . . c o n t a i nbso t h I ' s a n d 2 ' s ,w e m a y a s s u m e S i n c el : . . . , : 2 a ie: a - i - 1 f O ri : 2 , 3 , . . . , 1 . S i n C e a n da g : a t : l . S u p p O S A-2: a-t
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' " ' ' ( zz ' z l ) z ' z l | | z z ' z l T z, ' ( z l z z ) " ' :
"''(,zl (, t*'71 T) Z Z t * ' ( ' 7 1Z T ) " '
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'(s)I>ft+t)(7+y) l?ql peceld,(leculeuuKs (7+t)(7+U') lszel lB eJBeJaql s^\olloJ 'rr srrcd lI lenbe 'snql '/ qJearoJ t 'Uel eql ol u:e1ed culetuurfs B r{lLr\ 7 (!)t ol?q e^\ pue (t*u+,t)rcl ... (z+4ttzl ...tzl ZZ Z7 Z Z,el | | Z Z o{rl s{ool tt+t)ttl T,(, Jo lqEu eqt ot pue tuu.orJ V o c u e n b eesq l u e q l ' I + t - ( l + > t ) t ' t : ( l ) r e s o d d nes ^ \ J I ' I * y : /rog lou l n q U I f I t r o J ( . f- l ) t : ( t + { + 7 ) . r ' s r1 e q 1i ( 1 + 7 ) t ' ( 4 ) t r r e d p r l u e c eql ruo{ luelsrprnbeeJ?qcrq^\ s.(l); yenbego srrudJo Joqrunutunturxerueql elouep y loT '(I + )t)t + (r1); ,{es'rnrco s.(l)"rlenbeun leql ,lrou esoddng 'popunoq sr r os '(1) uro; otllJo sl (l)/ ecuenbes '(s)I oqlJI luqt ecuole sA\olloJ ) z-rzaeqt \'(s)I > 1 leqt qcns a uo ,{po Surpuedep(e)/ requnu B sr oroql oroJereql 'oo * ll't : llc'Z'01+l,o'l'l'T,l 1s? f +- @)I-V leql Z eunue1ruo{ s1(olloJ wnun:rds rro)uvw 3HJ No sJrnsEuuEc1o 'l
I. OLDER RESULTS ON THE MARKOFF SPECTRUM
Tl.sI-r l. The ten smallestnumbers in the Markoffspectrum A
M(A)
I
r/5
=
2.236068
.
,/6
:
2.828427
2 2 t l
TZIIIa z z z l
| |
T;--ia L 1 6
JTaoo1n
5----iL
,f;tlu1ts
a
t g
T11 ffi L
a
,/nt 1s J tsn 1n '/7sasps
|
|
z
L
,Z tn
1 4
2.9'132t4 2.996053 :
2.99920j 2.999423
:
2.sessr6
t/znr'ns1rcs: : JwSaopt
2.sses77
t/c88,5ntztt=
2.ssss88
2.eeee82
In his secondpaper, Markoff [f880] made a more detailed study of the sequences,4such that M(A) < 3. A thorough accountof this work is given in Dickson [1930,pp. 92-107]. Our next theoremis the main result in Markoffs 1880paper. Before stating it, we remind the readerthat two quadratic forms .f(x,y) and g(x,y) are said tobe equivalentif there exist integersa, b, c, d suchthat ad - bc: *l and f (ax + by,cx * dy) : g(x,y). Tnronnvr 6. The Markof (or Lagrange) spectrum below 3 consistsof the numbers t/9fr - alm, where m is a positive integer such that (ll)
m 2+ m l * m ] : 3 m m 1 m 2 , f t 4 11 m , m 2 1 m ,
holdsfor somepositive integersm1 and m2. Given such a triple m, fttt, t7t2, defineu to be the leastpositive residueof *.m1lm2 mod m and defineu by
(t2)
u2 + | :,tJr/t.
If wedefinethequadratic form "f*(x,y) by (13)
f^(x,y) : mx2 + (3m - 2u)xy+ (u - 3u)yz,
then f^(x, l) :0
(14)
has a root a suchthat ttla) :
9m2- 4lm.
Further, given any a such that (14) holds for some m, there exist positive integersm1 and m2 suchthat (ll) holdsand suchthat a is a root of f (x, I ) : 0, wheref (x,y) is a quadraticform equivalentto theform (13), with u and u as defined above. We postponethe proof of this theorem until Chapter 2. Table 2 lists the ten solutionsof (ll) with the smallestvaluesof m, and the corresponding
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'
* r) : (,t'x)l rt(hpI - z(,tu
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r 'r€'68
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I. OLDER RESULTSON THE MARKOFF SPECTRUM
Tunonrla 7. Supposef(x,y): (x * cry)z- Lyz, L > 0, has m(f) : Then either L > 2.21 or one of thefollowing holds:
l.
L:1,o:
modl; A,:2.21,o: +t modl. i modl; L:2,ct=0 Pnoor. We can suppose0 < o S j by making a unimodular transformation x : !x, I Ay,, y : y,. Since z(/) : l, for all integersx, y not both zero we have either Ly2 < (x + ay)z - |
or
Ly'> (x + ay)2 + 1.
Let P(x,y) denotethe first inequality and N(x,y) the second.If P(0, l) holds then A > 0 is contradicted,so N(0,1) must be true. Now if P(1,1) is true, we get successively (l+o)2-l>A>o2+l; ">i:
o:*,
a:i.
This is one of the possibilitieslisted in the theorem. Now we assumeN( l, I ) is true; if also P(3,-2) is true, we get successively ( 3 - 2 a ) 2- l >
4L>4+4(l+a)2; 0>20a; o:0;
L:2.
This is another one of the possibilitieslisted in the theorem. Now we assume N(3,-2) is true; in conjunctionwith N(1, l), this gives 4 4 > m a x ( l + ( 3 - 2 a ) 2 , 4 +4 ( l + 0 ) 2 ) . The two quadraticson the right are both equal to 8.84 when o : fr; also, the first quadratic is decreasingand the secondis increasingas o increasesfrom 0 to j. Hence 4A > 8.84, with equality if and only if o : t.a. This completes the proof of the theorem. We assumedin the hypothesesof Theorem 7 that the form in question attains its minimum. This involves no loss of generality, because if is any elementof M, there existsa form /* such that the mint/Ml^6
imumm(f.) is attainedandfif\lm(f.)
: t/Vffi1m(/) (seeLemma6
below for an equivalentassertionin terms of doubly infinite sequences).One can even simply omit the assumptionthat the form attains its minimum in Theorem 7, and give a (somewhatmore complicated)proof along exactlythe samelines; Cassels[1959,pp. 38-a0] gives the details. L s i l , I l , l6 n . L e t A b e a n y d o u b l yi n f i n i t es e q u e n c e . . . , a - t ,e o , a t , . . . . I f M(A) isfinite, then thereexistsa doubly infinite sequenceB suchthat M(A) : M(B): Ao(B). Pnoor. If 1n@) : M(A) for somen, we can chooseB to be ... ,b-t, bo, b1,... where bl : aian for all i. Otherwisethere is a sequenceof integers ---+ n(l), n(2),... such that ).n1;1(A) < M(A) for all i and )"n1i1(A) M(A) as i * oo. Sincethe integers4, are bounded when M(A) is finite, there exist an integer,say b6, which is equal to infinitely many of the ar1;;'s.Similarly, for any integer k > 0 there exist integersb1 (k S j S k) such that b - 0 , . . . , b - t , b o , b t , .b. .t ,
(sssog'e' orrgv' E)= eJl' IJI) 'L loiJalulaqJ vhu ra'I .(V)W -*{rull : (g)oT: (g)n ,srleql
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'trAIur eJ? slurodpue esoq^\ e eugep 0,,!\ 'tr^truI eJeslutodpue esoq,ndeE e ul do? de8 e eq ol trAtr Tounxnut ur deEfue 'rueroeql eldruts Eut,nollogeql Jo esn?ceg o1 peErelueeq uec 141 'Apnls pelrelepeJoru 01rr'(g reldeq3) uo re1e1 'ereq sdeEeseql Sutu:ecuocsllnser ? o{eru il?qs 'tr{i go slutod ou ul?luoc qclqrt\ sp,uelut 'st lseldruts eqr Jo ,nege errtt e.&\ '€ lerll eloq? 141wsdn?ere eJeqlleql [qIZ6Il uorJod oculsul\oull ueeqseq lI 'releanoll '€ A\oleq eql urn:lceds lnoqe uorl?uuoJul eloqe eql ueql elalduoc uI sJeqlunu oql Eururecuoce8pelaroul rng ssol qcnu sr u?ql -releert trAI € '[Ogg'd'7'6 ureroeql'It6I] II€H ,(q ps1e1slsru sel\ 9 ?rutue1 w llnsor InJesn1nq elduts eql leql sreodde11 'g ecuonbes elrugu,{1qnopeql os pu?''.''t-Q 'lq urclqo os pue onulluoce,n'/ (,(ueu ,(letpgul) esoql ,(po epnlcul ot {tV} 8ur1ecuru1 '0q ,(?s 'elues eqt tl rr!, qcql( rog / ,(ueu ,(1elrugureJ? eJeql 'punoq uolutuoc e eABq 11jr eqt ecuts 'roC '9 eturuel ur ue,ut fpeeryu uollctulsuoJ eql sI goord eq1 'Jooud 'ut>
'1 tol I > ut11o
'Q:
,t,,tr',,
'l stslxa aUu{u! uo araql < I 11oto!'pW q)ns {(rt)t} acuanbasqns 'lDllt q)ns "''tq '0q 't-q' ''' - (I acuanbas 'l a71u{u1 tlqnop o 7 tu [.uo tol s$lxa aaqt uaqt 'papunoqan gio ary g 'sta&atutaurtsodto acuanbasa71u{u7 "' - {v q'{ tlroanf '^uv'I'IodoJ tlqnop D apuap "''(;D'dr't6o' ur poprulsuoc "''rq'uq't-Q'
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qll/r\ Isslluepl s! I InUIJAdS IJO)UVW
EHI NO SI'InSaU U3C1O
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14
I. OLDERRESULTSONTHEMARKOFFSPECTRUM
is a maximal gap in the Markoffspectrum; M(A) : ,m ody for A:3 and M(A) : t/T2 only for any A made up of I's and 2's, and containingpatterns (21), for arbitrarily large n. Pnoor. In order to prove that there is no doubly infinite integer sequence A - ... ,a-t, do,at,... such that *[12 < M(A) < tfB, we may assumethat ai < 3 for all i, since obviously M(A) > 4> {l:t if any ai exceeds3. If no 3 occurs in ,4, then clearly by Lemma I
M(A)
I
M ( A )> l ,(A ) 2 [3 ,l , l J] + tO,3Jl:3.82202...> ,/t:t. Similarly, if an - 3, ar+t :
2 is true for any n, then M(A) > 1,(A) >
. t=f 3 , t h e M n ( A ) : t 3 l + t o , 3=l [ 3 , 2 , T ; 3 ] + [ 0 , 3=J3] . 6 2 2 0 2 . . . > r / T 3A
JT1.
Lemma 7 goesback to Perron [1921b];our proof is a little simpler. Another simple proof was given by Wright 11964l,using an idea of Forder U9631. Various other authors have rediscoveredLemma 7 without being aware of Perron'swork; for example,Davis U9501and Fay [1956]. Before proving our next result about gaps,we need the following preliminary lemma. s L n u u a 8 . I f a d o u b l yi n f i n i t es e q u e n cAe- . . . , a - 1 , a o ,e t , . . . c o n t a i n a pattern br, b2,...,b*, wherek > I is an odd integer,then
M (A)> lE,:u-; - Al + t lt6r4z,. - ,il] and M ( A ) > t b r , b r - t , . . . , b t +\ | l l b k , b k - r , . . . , b r 1 . Pnoop. For each integer m, definexlct^-t,am-2,... ]. When x^ ) l^ we have
:
Lam,em+1,...fand y^
:
1 ^ ( A ) : x mI Y ; ' 2 x m+ x ; l and when l^ ) x^ we have A^-r(A) : xm-t + y;t_r : xi' + y^ ) x^ I x^t.
Thus for everym, M(A) >- xm + x;t holds. Now choosen so that Xn-k : [,bt,bz,...,b*,xn|.Sinceftisodd,wemuSthavemax(xn_k,X,)> t6". -,il1. This provesthe first lower bound on M(A) given in the lemma. The other lower bound on M(A) follows by a similar argument,using the fact that M(A) > lm + lir holds for everym. Lemma 8 goesback to Perron ll92la, Satz 5, pp. 8-lll. Bumby 11976, Proposition 2, p. 3001has given a generalresult which is related to Lemma of his proof. Perron[l921b, Satz3, pp. 8; our proof aboveis a specialization 8-l2l usedLemma 8 to prove the following:
'{JO^\
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2. MARKOFF NUMBERS AND MARKOFF FORMS
l8
hasthe rootsrn and m' : 3mtfttz-rn. If (m, m1,m2) is a nonsingularsolution with, say, n41) 1142, then f(mr) : (mr - m)(mr - m') = 2ml + m] - lmlm2 < O so that max(m1,m) is strictly between m and m'. lf m is the largest of flI, fll | , t/12 wA Obtain
m>max(mt,mz))m' and we also have tlt't : 3mm2 - mt ) 3m - m > m, m\:3mmr-n4z)3m-m>m. Thus for a nonsingularsolution one of the neighbors(2) has a smaller maximum element and the other two neighborshave larger maximum elements. Hencegiven any nonsingularsolution we can work backwardsfrom it through successiveneighborswith smaller maximum element. This processmust terminate in a singular solution. In fact, the singular solution (1, l, 1) has only one neighbor,(2,1,1), whoseonly other neighboris (5,2, l). Thus the solutions can be arrangedin a tree, as shown in Figure l. We shall call this the Markof tree. To prove the last assertionof the theoremwe observethat if in any solution (m,mt,rn2) sometwo of the integershavea commondivisor d,thenby (l) d also divides the third integer in the solution. By working backwardsthrough t h e n e i g h b o r s o (f m , f t . l r , m 2t)o ( l , l , l ) w e f i n d t h a t d = L Theorem I goesback to Markoff [880]. Hurwitz [907] made a study of the more generalequationxl + x] + " ' + xl : kxtxz' .. .' xn. In particular, he showed that x2 + y2 + z2 : kxy z has positive integer solutionsonly when k : I o rk : 3 . Now supposewe take m to be the largestof m, m1, m2 in any solution of (l). For each such m, we define a quadratic form as follows: Define z to be the least positive solution of !m2x : nt1 rrtodrn (this is sensiblesince m and rfi2 ara relatively prime by Theorem l) and define u by u2 * | : unr (note m divides uz + | becauseof ( I )). Now definef-(x , y) by (4)
f^(x,y) = mx2 + (3m - 2u)xy+ (tt - 3u)y2
We call these the Markof forms. Note that since m2, = -m3. mod z, the integersu,u and the form f^(x,y) are unchangedif we interchangem1 and m2 in any solution of (l). The first ten Markoff forms are given in Table 2 of Chapter l. The simplestpropertiesof the Markoff forms are given in Theorem 2. We need the following lemma for the proof of that theorem. Lrvvn l. Supposem is the largestinteger in a given nonsingularsolution (m,mr,m) of (l). Let u be the leastpositivesolutionof *m2x: mt modm.
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20
2. MARKOFFNUMBERSANDMARKOFFFORMS
D e f i n ee ( m , m 1 , m z ) : t :
* l a n d i n t e g e r us 1 , u 2s a t i s f l i n g 0< u i l m i - |
f o r i : 1 , 2b y t/11
mDdm,
= t7t2
rtodntl,
t/4l2lt: (5)
tilnt1
tt7l1l12:
tul
mOdm2.
Defineintegersa,u1,u2 bj) (6)
u2+l:'t)t/t,
ul+t:'t)ttnt,
ul+l:1)2t7t2.
Then (7)
ftIzlt - l7tlt2: 67nr,
wlut - tltllt:
wherem' :3mtm2 - m, and (8) m 2 u* r f i a z- 2 u u 2- 3 m 1 ,
tl/12,
lltzltt - l/l1lt2:
67nt,
f f i u 1* t n r u - 2 u u 1: 3 v n r .
Pnoor. We have lltZtl - //ltl2:
t/l2tl = 0,t/11 mO,dm
by (5) and m2u - trtlt2= -tllUz: -em2 lmt : ttTlt
modm2
bv (5)and (l), so fttzu- r/tu2= tn41 modmm2
(9)
since rn and m2 are relatively prime by Theorem l. lf u2 > 0, we have tTlzu - tvluz - tftl1 1 tllzu - m I mr 1 ft|ffi2
and also r/tzU- rfiUz- 8m1) m2 - m(m2- l) - t\tl ) -t/1t712. Thus (9) gives the first equation in (7) if u2 > 0. When Lt2: 0, then (5) implies fri2: l, e : I and u: tltt, so the first equationin (7) holds trivially. The other equationsin (7) are proved similarly. Finally, squaringthe first two equationsin (7) and using (6) and (l)gives the equationsin (8). THeonsu 2. (A) The discriminantd(f*) of .[,n6,y) is 9m2 - 4. (B) Theform.f.(x,y) is properlyequivalentto -f,,(x,y). (C) The minimum of lf^(x,y)l taken overall integersx,y not both zero is
f^(l,o): m.
(D) If rt^ and B,ndenotethe rootsof f,*(x,l):0 with u,n> p,n, -l -41m. then | ) rt^ ) 0, B,n< and Fkv,n): t/9m7 (E) If m is odd, then the cofficientsof f^(x,y) haveno commonfactor. If m is even,then |f*(x,y) has integercofficientswith no commonfactor.
Pnoor. That the discriminantof f,,(x,y) is 9m2- 4 followsdirectly from the definition (4) and the first equation in (6). The rest of the theorem is
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(f t )
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(q t )
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22
2. MARKOFF NUMBERS AND MARKOFF FORMS
and (18)
mf^(x,y):y2+3myz+22.
I f w ed e f i n e x t : t t x o + ( 3 u - u ) y o y, t : l / t x o + ( 3 m- u ) y o : 3 m l o * z s a n d Zt : rltxt - ttlr : -ys, then (17)and (18)give 0 < mq : mf^(xo,!d : m.f^(xt,!t) - yfi + 3myszo + zf,: y3 -l ytzo, so !1zs > -yE > -2fi, wherethe lastinequalitycomesfrom (16). We also h a v ey 1z s : 3 m y s z o + z E < t B b y ( 1 6 ) s, o l y r l< l z o l .T h u sl y l l + l z l : the minimalityof lyol+lzol. If we lyrl+ lyol < lyol+lzol, contradicting supposethat yszs < 0 and lyol > lzol,then by definingx1 : (3m - u)xs +
( u - 3 u ) y 6 l,t : - t t t X o * u y O : - z g A n dZ t : r l l X t - u l t : 3 m z g * . / O w e obtain a contradiction in the sameway as above. Therefore,we may assume that yszs > 0. Since g(x,y) ) 0 for all integersx,y (becauseg(x,y) has discriminant -4), it follows from the identity .f^(ux - uy, mx - uy) : g(x,y) - 3y(mx - uy) and (14) that lf^(uxo - u!0, ,dl : lg(xo,yd - 3yozol< g(xo,yd t 3lszs : q. The above inequality must actually be an equality by the minimality of 4. Thus yez6 : 0 and part (C) of the theorem is proved. In part (D), the inequalities | ) o^ > 0 and 0^ < -l are implied bV (a) and the inequalities 0 < \/gm2 4
- 3m + 2u < 2u < 2m < Jgm, 4
* 3m - 2u,
which follow immediately from the definition of u. Inthe theory of indefinite binary quadratic forms, the inequalities for a^ and B^ mean that the form is reduced(seeAppendix l). The formula for pt(a*) in part (D) follows from the fact that p(a^) is equal to Jd(fr divided by the minimum of lf^(x,y)l (seeAppendix l). Part (E) follows from the equation u2 + | : um in (6), which implies that u and m are relatively prime, and that u and u are odd when m is even. We recallthat the continuantK(a1,a2,...,an) is definedto be the numerator of the continued fraction lar, e2,. . . , enl (seeAppendix 2 for properties of continuants). Our next theorem u'ill show that the Markoffnumber m can be expressed as a continuant whosedigits are essentiallythe period of the simple continued fraction for the root dm of the Markoff form f^(x,l). It will be convenient to first have the following lemma, which goesback to Frobenius [1913, pp. 610-61ll. Lnuur 2. Supposethe positive integersm,u and u satisfy m > u and mu - u2 : l. If we expand mlu and ml@ - u) as simple continuedfractions with an evennumber of partial quotients,then theseexpansionsare symmetric.
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. ( t - u 2 4 , . . . , r 4 ) X: * 'arcwlaqunr
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0z)
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n
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CINV SUAShINN IIO)UVW
2. MARKOFF NUMBERS AND MARKOFF FORMS
24
k2,...,k2n-3 We also havek1 - kzn: 2, kzn-z : kzr-t : I and the sequenc€ is symmetric. Pnoor. SinceI > om > 0 by Theorem2(D), we can let a^ : lO,kr* ,lrl, where kt,.,.,k; is the period of the continued fraction for a^. We have - 4 by Theorem2(A). rf m is odd, then f*(x,y) has coeffid(f^):9m2 cients with no common factor by Theorem 2(E). The least positive solution of the Pell equation x2 - d("f^)y2 : 4 is x : 3r,ft,! : l. lf m iseven, then has coefficientswith no common factor. The leastpositive solution )f*(x,il 4 is x :3nt, ! :2. In eithercase,by applyingthe theory of x2 Id(f^)y': of the Pell equationto f^(x,y) or jf*(x,y), respectively(seeAppendix 3, especiallyTheorem 2), we obtain ())\ \--/
u:K(k2,...,ki-t),
3u-u:K(k2,...,ki),
m:K(kt,...,ki-t),
3m-u:K(kr,...,k).
< 3uandm > 2u B y T h e o r e m2 ( C )w e h a v em < - f ^ ( 0 , 1 ) : 3 u - a follows from the definition of a. Since mu - u2: I by (6), we obtain (23)
2u<m<3u
and 2u
It follows from (22) that the continued fractions for mf u and (3m - u)lm are m f u : I h , . , , , k 1 - i a n d ( 3 m- u ) l m : l k i , k i - r , . . . , & r l . Therefore, (23) implies h : kj: above and inverting gives (24)
2. Subtracting2 from the secondequation
:m - u- : l k i - 1 , k i - 2 , . . . , k i .
If 7 were odd, then by Lemma 2 this continued fraction would be symmetric, so we would have kj-, : kr : 2. But by (22) and the fact that k1 : 2 we have
t#
: lkr.kr-r,...,k2 a fn d h
: lki-y.k1-2,....k21.
lf k1-t: 2, then the secondequationabovecontradictsthe inequalityu > 2u in (23). Therefore7 must be even,sayj :2n. Using the fact that kr : 2, we can write (24) as m : l k z r - r , k z r - 2 , . . .k,2 ,l , l ] m-u and this continuedfraction has an even number of terms. Thus Lemma 2 kz,...,kzn-3 applies,and we find that k2n-z: k2,-t : I and the sequence : gives This K(k2,...,ft2r-t). formula u the is symmetric.Lemma 2 also completesthe proof of Theorem 3. Now we are going to associatewith each Markoff number m a pair of Frobeniuscoordinatesp,u, which were introduced by Frobenius [f913, $8]. These coordinatesare relatively prime positive integers,defined as follows
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lEql aunss? ,(eu a,n ,fuesseceu '| 'g secrpuroql Eurlnured fq e^JesqolsJg 016 'seJrpurJo 1eq1 J! T, rrud ,(ueroJ sploq (td wr$ qcns (7'l '0 : t) (t)n '(l)ri setuurprooc snluoqoJC qpt zw'llu'ut sra8alutaerql uarrf aJgoA\osoddps'osJa,ruoc aql ertordo1 ',{urrrsrql ur '(17) sogsrlesooJl er{l ur uorlnlos rep8ursuou ,fuo^e luql oose^\ Surpeecor4'(17) ,{gsrlesoslp IIrl( zut'ILu'LuJo sroqq8reuee.rqlIIe leql (92) ruo{ Jeolcsr 1rueql '(17) ,(;sr1es1 arnSrgJo aeJleql ur suollnlosrelnEutsuou f'uest (2t11'tta'ut)JI '(SZ) EutpecerdIIUleql qcns(1) go uotlnlos.teln8utsuou 'rooud Jo osnuceq(17) segsrles(l)Jo (l'7,'g) uorlnlosrelnEursuoueql 'nqwnu anbrun autos tol [ot1to1,,t1 tmd a1outptooJsmuaqotl aLfis1 n'rl sta&alutautud rQadtyolatlottod qcog '(t){o |Lu'ttu'w uaql uouryos nln8utsuou o aa8 'sanput nnd {uo tol splotq '7 'l '0 : t) (t) rr'(t)rl Q) totqtqcns (tQar4cadsat lo sapurptooJ smuaqotl tlltfir Tur'tur'w aa&a|ul aatql [uo uau? '{1asta^uo) 'tT:
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Qe)
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SI IUOJ CCO)UVI^I CINV SdaSI InN JIO)UVI^I
Z
2. MARKOFF NUMBERS AND MARKOFF FORMS
m(0,1), m(1,0), which we know is in the tree from (25). Thus the triple fr\,tnr,tfl2 filuStalsobe in the tree. To prove the final assertionin the lemma, we consider any pair of relatively prime positive integers1t,u. We can find unique nonnegativeintegers and F ) x, u > y. Now the pairs (pl,u), (x,y), x,y suchthat py - ux :*l (lt - x,u - y) satisfy (27) when consideredas Frobeniuscoordinates.Therefore, we can apply the procedure in the above paragraphto trace this triple of coordinatesback to the triple (1, l), (0, l), (1,0), which doescorrespondto a triple of Markoff numbers in the tree. Hence there is a Markoff number in the tree with Frobeniuscoordinatesp,u. That this number m(p,u) is unique follows from the uniquenessof our choice of x,y. It is not known whether m(p,v) : m(!',2') is possiblefor distinct pairs p,u and 1t',u'. Of coursethis is just the problem of the uniquenessof the Markoff numberswhich was mentioned in Chapter l. Let m > 2be a Markoffnumber, and let p,u be the Frobeniuscoordinates of m. It follows from Theorem 3 that m ( p , u ) : K ( 2 , k 2 , k 3 ,. .. , k 2 r - 3 , 1 l,) : K ( 2 , k 2 , k 3 , . - ' , k z n - v 2 ) , where the sequencokz, . . . , k2n-3 is symmetric. We define S ( t t , u )- k 2 , . . . , k 2 n - 3 to be the symmetric sequenceassociatedwith m(p,z). Our next goal is to give a description of S(p, z) in terms of p and u. 4. Let m = m(il,u), mt = r/'t(pr,ut),mz: fl(ltz,u2) be a nonLrrurra,t singular solution of (l) with largestmember m. Let e = *l be definedas in Lemma I and definesequences711'. Tr:1122,
T-t:2211.
Then provided min(m1,mz) ) 2 we have
(28)
u1). uz): s(ltz,v2)T-rs(1t1, s(tt, u) : s(rr, v1)T6s(p2,
If min(m1,mz):2, thenwe musthavep:
(2e)
I and
S(l,z):22,-2.
If min(m1,n12): l, then we must haveu : I and (30)
S(P, l) : l2u-2.
Pnoor. For brevity we put S : S(1, /), S, : S(pi,ui) (l : 1,2). To prove (28), we first suppos€0 : l. Then bV (3) and (7)
(31)
m = 3mrrflz- rn' : mr(2mz* u2)'t (mt - ut)mz.
We have from Theorem 3 (which is applicablesincemin(rer,mz) > 2)
2m2+ uz: 2K(2522)+ K(Sz2): K(22522), t r l r- t t r : K ( 2 S rI l ) - K ( S r 2 )= K ( 2 S r1 ) '
(t)2: (tn)2
'(l - tn'"''(,'I : ! ) l t n l r n ( l- / ) l - U n l t n t ) : ( t ) 2 0Jeq^\
( t)))?Z2l .1g: s ( t t ) t t T r r r ( t * i n \ t t t l . . . 7 7 ( t ) t t z l e r ( t t ) ? z r c Z (. 7 t -7t a
(lg)
senrEsrsaqlodfquorpnpur eql os 'z5a; tg : 5 o^BqaA\ euual ,{g 7 'l: rl qlyr\ (Zt) sr qJrq,r'z-nzZ: (/t'l)g se,tr8(69) ecurs'1 < rl etunss? osle ,{eura16 'z-nzl s\ (t'l)S leq1 sarldur (gg) ecurs'l < n aunssu ,{eura16 'S JoJ ueJoaql eql a,tord o,r pu? 'zS'pue IS JoJonJl sr ruoJoeqloql ounsse em os 'y ern8rg ur eeJleql uo uotpnput ue ,{q rueJoeqleql enordlleqseAA('5 euuel ut uentSse 'lT : rltTl- I/td wrp]ceJ aqt ,{1uoSursn'no1eqluerun8reeqt turtroylogfq (Zg) enord uec am ',{1pn1cy) '?:
/IIfl -
Inrl
(ee)
senrB saleurpoorsnruoqoJCeql Jo pue s Jo suorlruuepoql SutsnuollJnpul pJe.t\JoJ -lq8rerls V 'l BtuIuo'I uI peugep Jequnu eqt oq IT : g le'I 'zS'ocuenbos oql uer{l .reiuol s1 Ig ecuenbesoql luql ,{lryereua8 Jo ssol lnoqll^\ ournsse,(eru ol[ ',{larrlcedser'zw pue tw'ut ttrl\Mpeleroossusecuenbesculauu,(s eq1 e l o u o pz S ' p u et 5 ' ( n ' t l ) g : S ' l e . 1 ' ( t ) ; o ( ( 7 n ' e i l ) u ' ( t n ' t r l ) w' t t t ) u o t l n l o s eql leT 'dool{d reyn8ursuouor{l ur Joqunu lsa8reyaql aq Z < @'rl)w ueiut:.? ' 7 - n 7 ls r ( n ' r l ) S u a l f i ' l - n I t .(r)'zlz1(I-n)'zl.
7 7 ( z ) x z 1 7 7 \ t t :t 1 Q t , r l ) g
Qt)
tq uau7 st (rt'l)g acuanbascutawuts aLfluaqJ
'0-'t'"''(,'l:!)
l ,tl rt(t- r ) l - In ln tl: ( t ) ) t
au{aq 'l < /t quw raqunu [ot1to1,tgD aq Z < Qt'rl)ut p7 'V hrauoaHl reqrunultteuoqlC eqt st (1'rl)w'etolateql'S
's/t\olloJ (Og)Pue (z'dzl)X : ( I'l )ut pue Z : (l'g)ra qtt,tl
'l 0 ' Z - r l ) u t- ( I - r l ) u g :
0'n)w
ot peel (gZ) pue (g) 'elduruxeroJ 'soteulproor snlueqorCoql Jo uorlruuep oql luo{ ,(pseemo11o;(og) pue (67) suorlenbeletcedseq1 'ctrleruurfs are zS 'snoEopue ,(1pexe sl I - : 3 es?ceql uI (gZ) Jo ;oord eq1 os?Jaql pue IS'S: leql lceJ eql tuor; s^\olloJf,tllenba puoceseql pu? 'l:3 ul (82) ur ,(lqenbe lsrg eql se,rr8srql '(7 xtpueddv uI elnluroJ 1se1eq1ees)
: Qzsdx(rtsdx+ Qcszdxfittsz)x: w Qzszztr'sz)x sr IS esneceq)(ISZ)X:
sa,rf (1g) ur suorlenbeeloq? eqt Sursn '(culeurur,ts (ZIS)X leql lrBJ eql sosnfllenbe lsBI eql ereq^\
SWUOJ llO)UVW
LZ
CINV SUaSWnN JCO)UVhI
'Z
28
2. MARKOFF NUMBERS AND MARKOFF FORMS
and n Q ) : l i p z l u z l- t ( , - l ) p z l u z l ( i : l , 2 , . . . , u 2 l ) ,
4 Q 2 ): q ( l ) .
It followsfrom (33) that P
(35)
U
F Ut
r
e UUI
sosinceu)u1
1 , f l : 1 , # ) r oir: 1 , 2 , . . . ,tu. ,
(36) Therefore,
r c ( i :) ( ( i ) f o r i : 1 , 2 , . . . , u-r l .
(37) Also,(35)gives
V ' f ) : u r+ l @ r 7
and (36)gives
- r#]: F,-lt)- r, t)#l:[r,, Ir,,so
(38)
r c ( u 1 l)+: l . t + j ( e- l ) : € @ + t )j ( e+ r ) . L u rJ
By (37), (38) and the definition of 7n,,the right-hand side of (3a) agreeswith the right-hand side of (32) up to and including the sequence7"'. Since Sr is longer than 52 and sinceboth right-hand sidesare symmetric sequences,(34) must be the sameas (32). This proves Theorem 4. Our final step in establishingTheorem 6 of Chapter I is to show that there S(p, u) and the doubly betweenthe sequences is a one-to-onecorrespondence infinite sequencesI of Chapter I with Ai@) < 3 d for some d > 0 and all I is the sameas integersi. In fact, we show that the set of all suchsequences the set of all doubly infinite periodic sequenceswith period 2, S(lt,u), l, l, 2 for some 1t,u (seeTheorem 5 below). It will then follow from the results of Chapter I (in particular, Theorem 5 of Chapter l) that every number < 3 in M has associatedwith it at least one pair of Frobenius coordinatesF,u; if the conjectureof Chapter I on the uniquenessof the Markoff numbers is true, then there will be just one pair p,u for each number < 3 in M. In order to prove Theorem 5, we use the following two lemmas. In stating them, we use the following terminology: A doubly infinite sequence (39)
. . . , r ( - 2 ) , r ( - l ) , r ( 0 ) , r ( l )r, ( 2 ) , . . . ,
where the r(i) are nonnegativeintegers,is said to be Markoff balancedif it satisfiesthe conditions: (A) lr(i) - r(i)l ( I for all integersi and 7; (B) if r(i + l) - r(t) is -l or +1, respectively,then the first of the integers
,Q={
s=t
' ( . 0 s 3t , p * , o : ( t ) s 3 * p a p : | * u .
J
0v)
J
,(;srlesqclqA\sreEolurureuac eJep ' ,J' ,Q',D'p'J'q'D eleqm a=t
'n,p+n(.r)s + (t + n),o- (ut+ {)nf 3
(sr)
u+l
,)
pue
n p + n ( t ) s 3 * t+r n ) D = 0 b Z
ftv)
e^uqe^\0
(r.v)
ecuenbesfue eq ot {(l)s} 1o acuanbaspa^IrapuuouB eugop oA\
ftv) ecuenbesB uelrg 'pacu"leq sI sJequnu lunbe ge e go ecuenbes u,(lsnor,rqg'"'' a' a' a' "',tes'srequnu pnbe 1e 3o scuenbes olur peuo^uoc sr (69) ecuenboseql suolleluep ollssoccnsJo Joqlunu ollug B rno l?ql 1 rotduq3 relJe leql (ydtur (17) ocuenbeseql Euturecuocseseqtod,{q '1 reldeq3 uI pougeps3'(6€) Jo g rueroeqlJo;oord eqt uI u^\oqss?ArlI go secuenbespe^rJepolrsseccnsaql Jeplsuoca.tr 'uoluessupuocoseql errord oJ 'suorlrugepeql ruo{ J3el3sI BIuIueleql uI uoluoss?lsJg aql 'Joodd 'piluulaq sl (6€) acuanbasSutpuods -ailn aqt uaqt'! aa&agt 1o tot puo 0 < g autostot I - t > (V)!Y sa{sltas . . . , 7 , 7 i ) t 2 1 . 7 . 7 \ 0 ) t 7 1 , 7 , 7 ( t - ) t 2 1 , .2. ,-2 , . V
(rr)
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I
r>l@+r)/3-(0,31
(0r)
u+tl I
unl
I
paJuDlDqeq ol ples sI (69) ecuenbesy segsrl?s tt Jl ' , ( l e , r r p o d s e r ' Ir-o {) ( -t)t -(t + {+t)t I + s r o r e zl o u s I q c l q / r( " ' ' Z ' l : oc
(INV suSswnN JJO)IUVW 'Z
sl^tuoJ llo)uvn
2. MARKOFF NUMBERS AND MARKOFF FORMS
Our assumptionthat (a2) is balancedimplies l c - b - ( c ' b ' ) l< |
and la- a'l< l.
Using these inequalities and (44), (45), (46), we find that (43) is balanced after some elementarycalculations.This completesthe proof of Lemma 5. We remark that one can show directly from the definitions that any Markoff balancedsequencemust be balanced.This is more than we proved in Lemma 5; but we do not needthe strongerresult, and a direct proof of it seemsnecessarily to involve rather tedious combinatorial arguments. Thus we preferred the approachvia derived sequencesin Lemma 5. If we knew that Markoff balancedimplies balanced,then the secondassertionin Lemma 5 would follow at once from Theorem 3 of Chapter l. Lnuun 6. Let S(tt,u) with u > | be the symmetric sequenceassociated with the Markof number m(1t,v). If we write the periodic doubly infinite sequenceA with period2, S(tt,v),l,1,2 in theform (41), then thecorresponding sequence{r(i)} rs balanced. Conversely,if a sequenceA ofform (41) satisfies Ai@) < 3 - 6 for some 6 > 0 and for all integersi, then A is periodic and the period is of theform 2, S(lt,r),1,1,2 for somepair p,v of relativelyprime integers. Pnoor. If we write the sequence,,4with period 2, S(lt,u),1,1,2 in the form (41), then it follows from Theorem 4 that we can put r(i): rc(i):liplrl - t(i - t)ttlul for all integersi (we use the fact that rc(u): r(l) i l, sinceu > l). Using this formula for the integersr(i) and the elementaryinequalities
l xl + [y] < [x +y] s [x]+ [y]+ I, wecaneasilyverify(40),so{r(r)i is balanced.
To prove the secondpart of the lemma, we first observethat A is purely periodic by Theorem 5 of Chapter I and that {r(i)} is balancedby Lemma 5 above.Let atdenotethe period of {r(i)}. r(i + m) for eachk : 1,2,.. . , u. Since{r(i)} is Considermin6<,r.,Dl periodic and balanced,there existsan integerm for which this minimum is attained for all k. If we let s denote this value of m anddefine a(i) = r(7 + s),
(47)
7 anYinteger,
then k
(48)
k
I/-ta ( i ) : j=t
m i n lu r Q + m )
O1m
fork:1,2,'..
J=l
By the definitionsof balancedand of the integers, k
(4e)
k
k
D o ] l < Dj = lo U + m ) < r + l a Q ) t-l
J-l
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[r(t - u)l-lx:tu]: (u)o os'rz 4 u tollaul:
'n
\29)
(u)o el?q e^\ (tS) puu (OS)fg
I-Du
(rs) SI
stql (n)o7: (n,r)oecurg reqt'n i ryttenJ )tl(t + Q)o) > nl@)o - p sa,rt8 'ttlfi+Qt)o)>ul@)o 1< u le pue 1 ,{uero3 ' | - ( u' 1 ) x e r u ) l ( Z ) o u- ( u ) o > t l
'eroJereql
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(0s)
au)(u)o
leql os
-*' = o .. _u dhs (u)o
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'l
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swuol J.{o)uvhl cNV sussl InN llo)uvw
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2. MARKOFF NUMBERS AND I\4ARKOFF FORMS
Tsronru 5. SupposeA is a sequenceof form (41) with ),i@) < 3 - 6 for some 6 > 0 and all integers i. Then A is periodic with period of the form 2, S(lt,u),l,l,2where2p is thenumberof I's in theperiodand2u is thenumber of 2's. Conversely,if p and u are any relatively prime positive integers,then theperiodicsequence A with period2, S(p,u),1,1,2 satisfies Ai(A) < 3-6 for some 6 > 0 and all integers i. Pnoor. The first part of the theorem follows from the conversepart of Lemma 6 and the description of S(p, z) in Theorem 4. For the secondpart of the theorem, suppose(a 1) is the periodic sequence I with period2, S(lt,u),1,1,2. If u: I the desiredresultis immediate.If u ) l, the correspondingsequence{r(i)} is balancedby Lemma 6, and hence is Markoffbalanced by Lemma 5. Now by the sufficiencypart of Theorem 3 in Chapter l, we have tri@) < 3 for all integersi. Since,4 is purely periodic this meansthere is a d > 0 such that tri@) <3 - d for all i, and this proves the theorem. Theorems4 and 5 can be usedto checkwhether a given periodic sequence of l's and 2's actually correspondsto a Markoff number. For example,if we are given the sequencewith period
2tt22lt22tt22tttr2, then by Theorem 5 we must have p: 5 and u : 4 if there is a corresponding Markoffnumber. Theorem4 with F:5 andu:4 showsthat the above period givesthe correct sequence,S(5,4),so the sequence,4 doescorrespond to the Markoff number m(5,4) : 43,261. We concludethe chapterby discussingthe asymptoticbehaviorof the function // (x) : number of triples (m, mr , m2) satisfying( I ) and max(mr, mz) 1 m 1 x osx ---+ m. If the conjectureon the uniqueness of the triple (m,m1,m2) for each Markoff number la is true, then "{(x) is simply the number of Markoff numbers not exceedingx. It is easy to compute many Markoff numbers using the recursionsin (3) and this leadsto the conjecturethat Z(x) growslike a constanttimes log" x. In fact, Zagier [1982] showed
(53)
.6(x) - Clog' x
as x---+oo, with C : .180717...
As pointed out by Zagier ll982,pp. 712-7131one can obtain a first approximation to this result as follows: lf m> m1 ) m2, it follows from (l) that m ) mrfttz. If m2 is large (which will occur in all but a negligiblepart of the Markofftree in Figure l), then z is much largerthan m1 and (l) gives m 2 < 3 m m 1 / f i 2 Im 2 + o ( m z ) o r m - 3 m t m z . Multiplying by 3 and taking logarithms gives log(3m1)+loelmz) : log(3la)+ o(l)
for m large.
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2. MARKOFF NUMBERS AND MARKOFF FORMS
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v 1 0 r
\
6
r
/
-
L
'{ecuenbescrpoued,(1arnde s\ :@)W) = V a {q srequnu 1uoJJo d los eql ougep ei6 '[0S0t] ,roperEourl pue euoloq pw 'euoleq 'rroperEourn [8S6I] s{nC pue ur punoJ oq uec llnser srql go sgoordJeqlo 'esecyercedse se I ruoJoer{I Jno sur?luocqcrg^\ euo lereue?ororu B sr llnsoJslg:[€62'd'61 ureroeql 'SS6I] ruroqurol fq uo,rf ssrr ruoroeql goord lsrg oql ,{ltuereddy I Jo '@)I/tf : (V).1'ecue11
'(s)w: (s)oypue 1r1,t,ryT,fi{ : (s)oy leql qcns Br ecuenbesB urelqo em 1 reldeqJ ul 9 Brutuo.1o1,fuu11oro3 eqt,(q'(!)tt KqV elelsuerl oql oq o1 ly ecuenbeselrugur ,(1qnopeqt Surugeq ecuenboseqr Jo 'popunoqsr y 'e1rugsl --full : Q)l l?ql qcns {({)tt} V)l ueq1y V)()tU slstxe eJeql y ecuenboselruuur ,(1qnopfue rog 'uorlrugep ,(g 'roou4
'tr^tr ) T 'l rrsuoaHl 'urupeds eEuerEel eql ur eq ol lou urulceds JoIJBIAI eql lueuole ue JoJ suorlrpuoc luerclgns pu? suorlrpuoc fuessecouetuos Jo qlrm reldeqc sql epnlcuoc oiil 'urulcods oql JoJ lpser snoSoleue Jo{Jer\l u? sr € rueJoeql isecuenbescrpoued ,(1e1eldruoc Jo senl?^ e8uatel Jo les eql Jo oJnsolcoql sr turulceds e8uer8u'I eqr leqr e,rord em Z ruoJoeql uJ 'reldeqc srrll Jo t ureJoeqJur ue,trt se ' "'
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sl f ul lou lnq tr41ul JeqrunuuA\oDI lseSreyoql 'Z speacxetz ur serJlueeql Jo euo lseel 1e;r ,{po : pu? JI gJ| < v)W prlt,(uedord eql seq EliL enqn ' "'otrgr'e Efl e^oq? eprcuroctr tr pu? T l?qr ^le{rl sI rI ' "'gLzg'n ueql re}ee:Erequnu fueaa sureluoo (trAios pue) T rsqr ^\oqs IIBr{sen 7 reldeq3 u1 'urrulceds .goIJ?I/{ eqlJo lesqnsredord e sr uru1cods eEuertel oql l?qr e,rord ol eurquoc reldeqc srql Jo t pue I srueroeql '€ ^\oleq oprcurocsnql erlcedsJo{rer\l pue sr y acuenbeseqt g > (ilW sessrlesqclq,r\ a?uet?el eql icrpouad,{yelelduroc rurupads Jo{JeI I orll Jo luotuele qceo JoJ l?q} ur\oqs s?/r\lI 1 reldeq3 u1
perBdruoJBrpadg eEuurEBTpuu So{ruIN eql g urrrdvHJ
3. THE MARKOFF AND LAGRANGE SPECTRA COMPARED
36
and the closureof P is denotedby cl(P). The proof of the following theorem was first given in Cusick U9751. Davis and Kinney ll973l gave an indication of some of the ideas in the proofs of the right-to-left containment of both this theorem and our later Theorem 4. Tneoneu 2. L:
cl(P).
Pnoor. In order to prove L c cl(P), we take any sequenceA for which L(A) is finite. For any € > 0, we shall construct a purely periodic sequence C suchthat -e < L(C) - L(A) < e. By Lemma 2 of Chapter I there exists z ) I such that (l)
l[0,4i+r,...,ai+m,x]-[0,Cti+r,ai+2,"'11<el4
for any choice of integral i and any x limsup; tri(A), and so there exists{frU)} such that L(A) : limT-- ka@). Clearly we may assumethat k(j + l)- k(j) - oo as / - oo. Defining the Aiby olj) : a*1iy+iandusing {A} in the Corollary to Lemma 6 sequences of Chapter l, we obtain the sequenceB and choose,/ so that (i) k(J + l) - k(J) > 2m; (ii) for any j > J, ar(j)+i : bi for all i, -2m < i < 2m; (iii) li/) < L(A) + el2 for att i > J. From (ii) and the choice of m we have that llrrit@) - L(A)l < el2 for all j > J. For u : k(J + l) - k(J), we define the purely periodic sequenceC by : ci aku)+i for all i,0 < i < u. We shall now show that for all i there exist an integer n and some x,/ > I such that (2)
(3)
L i ( C ): l a n , a n + t , . . . , a n + m ,+x ][ 0 ,e n - t , . . . , Q r - ^ , ! 1 .
For any i > 0 with min{i, a - i\ > m,we immediatelyobtain that (3) holds f o r n : k ( J ) + i . S i n c eu ) 2 m , e i t h e ri ) m o r u - i ) m. If 0 < i 4 ffi, then ( c i , c i - r , . .). : ( a * e ) * i , ' . . , a k e ) , a k u + t )- -. .t , a k ( J + t ) - ^ ,-.). = ( 4 * ( t ) + i , . ., a k ( J \ - ^ , . - ) ,
bY(ii)'
and so (3) holds for n : k(J) + i. lf j : ts- i I m,then ( c i c, i * t , . . . ) : ( c r -j , . ' . ) : ( a * r*t t ) -j , . . . , a k ( +J t ) , '. . , a k ( +J t ) + ^ , ..' ) , and ( c i - t , c i - 2 , . .) .: ( a * u * t t - r - t , . ' . , a k ( J + t l - i - , n , ' . ' ) , againby (ii); for this case,(3) thus holds for n = k(J + l) -7. Therefore,by the periodicity of C, (3) is satisfiedfor all i. Combining(l) and (3), we havethat lliG) - 1,(A)l < el2, and ,ti(C) < L(A) + e then follows from (iii). Moreover,for i = 0 we have n: k(J) in (3) and so (2) implies that -e < ).0(C)- L(A) < t, as desired.
lsql qcnssreSelur enrlrsod;o{(+ 'l)ry} acuenbssp asooqJe^\ (V)\rt\ o?f6nrrull Jo uorleJoprsuoo e tuo4 'Z r.ueJoaqlJoyoo:d eql ur sV '1 a t 'rgo ocroqcfuerog s > l ( V ) ! v- f t ' u t - ! o ' ' ' ' ' t - ! 0 ' 0 1+ [ x ' u t + ! D' (' ' ' t + ! o ' l D l l (g) I IIB qcns roJ Lu slsrxoaJeql fue:o; a raldeqJ < (, euruel ,{g luql < I 1 0 Jo 's > (y)W 6)W > s- qrrq^\ JoJ J ocuenbescrpouad ,{.1un1ue,re ue lJnJlsuoJ llpqs o/'\ 0 < a ,(ue .rogpue '(V)oy : (V)W qcns Jeprsuooo^\ 'uorsnlcurosJOAoJ aql e.tord o1 lurll X '(1 relduq3 ur g ueroor{a eos) los pesolr e sl tr{ leql pue IAI ) g l?qr stceJeql ruor3 fllcerrp s^\olloJI I f (g)lc uorsnlrur eql 'Jooud .(fl)lc = hI uaqJ '{sap1sqpq uo trpouad (t11oruuar.a) q V :(V)W} : B p7 't. wauoaHl 'runrlceds JoITUIAIor{}roJ llnsar SumolloJ eqt o^eq 016 'Z ruoroeql go goord eql seteldruoopuu (C)l :7 ll-rlt sa-rordslql '0 < 3 lle rog a j ll - O)ll e^?q snql e,r\ (/) uorg :(y)/ < u Ue roJ a + 7 > O)'V uretqo e,r\ '(S) qll^\ slql Euturquo3 'Z,l?+ I j (V)!V'(g) ,{q os 'r 11en1 (VSoy > (tV)!V 'X < [ IIe roJ reql sorldu! .f llu to1 (y)oy = ({V)W'pusq roqlo oql uO
' 3-
7 < Z l s - ( t + x V ) o7y 6 l t x t I y
e^eq e^\ 'l + X:
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'zls>t-(v)oy>T.ls-
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roJ leql qcns t1 < X slsrxeeroql'(fz)0f --Irull : (V)l oo*ftull: Z ocurs '0+ (S) {)l > i > 0'1lero3 Zls>(t+tV)ty-13y+{t)IytZl?tf ? [ IIB roJ luqt qcns Ir 'ut>r>wslsueeJeqlg ,(q'ecua11 e , { u e r o g ' y r e l d e q 3 u l ? t u t u e ' I < Z ';lero; q l l l \ l , ( u e pue f < t,(uuro; tr'*Iio:t+t+()Ic'(t+!)I > 1> 0 '(t) pue uorlrugep eqt ,tg lsql qcns ,l/.slsue oreq] I Q ru KueroJ J Jo 'l : (C)l pue A\oqs o^\'t+[d ) l"ql > { /.{ue I II€r{s 0 1 - t+()12 qllt\ '0, lnoqe crJloruru,(s'ecuenbesaql oq ol roJ turugeq J rr*rlo '0 ' . f q c u er o J , ( u er o 3 { d + . . . + z d + t d : ([)l pue (g)/ a u g e p e A 1 < . f 0 : 'tV {4 vrl eunss? requnJ ,{eru errr aqt sr Id;y leql 7 Jo porred eql Jo Q18uay 'ut '.1 ) ! > 111- 1e rog tq: ,ri,
\r)
leql selldur t < [ wql qcns 1 slsrxe eJeq] | 7 tu ,tue rog l?ql os ',(r?ssoceu Jl'{tV} elecuul a1yI retdeq3 ur 9 eruue'I o1,ire11oro3eql ruo4 ecuenbes e sl g leql pue { 1e rog ({V)oV: (V)W : ({V)l teql eunsss fieu em tV Jo ,$rcrpoFredaqr ,(g 'l : (V)l --.rrull leqt qcns secuenbescrpoued ,(1ernd 'uorsnlJur esJeAeJeql e,rord o1 ;o acuenbese sr {{7} teql esoddnsem LT,
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3. THE MARKOFF AND LAGRANGE SPECTRA COMPARED
38
( i ) / c ( 1-,r )> m;k(i+ l ,+) - k(i ,+)) 2m, and a sequenceB+ and an integer J+ 2 | with the property that (ii) for any ,/ > J*, dk1i,*y*r: bl for all lil < m. We obtain an analogoussequence{k(t, -)} of negativeintegers,a sequence B- and an integer,I-. For J : max(J+,"/-) we define the (singly infinite) purely periodic sequencesC+ with respectiveperiods dke,+)+t,ate,+)+z'. .', ak(t+t,+) and period lengthsut : lk(J + l,+) - k(J,+)1. Using these,we definethe eventuallyperiodic C by C : (C-)*,ak(t,-),...,Cty,...,ap11,15,C+ (where * denotesthe reversedsequence). We shall prove that IM(C) - M(A)l < e. Since k(,r, *) ) m, (8) implies that lo!)
2 A o @ )- e : M ( A ) - e .
We shall show that (9)
liQ)3M(A)+e
foralli >0;
an analogous argument yields (9) for i < 0. We defined C+ so that c[ : ak(t,+\+ifor all i, 0 < i 1 u+. Also, property (ii) above implies that c! : f o r a l li , k ( J , - ) < , S k ( J + 1 , + ) + c t k ( t + r , + ) + i f o ir,aOl l< i < m , s o c i : a i z. Therefore, (8) implies that A i G ) a 1 i @ ) + e f o r a l li , 0 S i S k ( J + l , + ) : k ( J , + ) + u + . For i > k(J,+) + u+, from u+ ) m we obtain that
L i ( C ): l c l , . . . , c l * ^ , x+l [ 0 ,c l - r , . . . , c ! - ^ , ! l for somex,! ) l. Therefore, the argumentwhich establishedidentity (3) in the proof of Theorem 2 yields L i ( C ) : l a n , c l n + t , . .Q, ,n + *x, f * f 0 ,a n - 1 , . . . ,a r - ^ , ! l for someintegern and somex,y > l. Hence,(8) impliesIiQ) S )"r(A)+e S M(A)+e. This completesthe proof of (9) as well as the proof of the theorem. The setsL and M coincide below 3, and for a long time after the work of Markoff they were believedto be the same set. In fact, Vinogradov, Delone and Fuks tl958l claimed that the two spectradid coincide. However,Freiman t1968] gives a real number which is in M but not in L. His example is o : M ( S ) : , 1 0 ( S )r y 3 . 11 8 1 2 0 1 7 8 ' with
(10)
S:Tz1T1A
tzTzlzTzTZ,
'secuenbesqns JEIIIUIS Joqloer{lJo oJ?ceI?} suollelncl?c ' s - 0 I- o o p) I tlg| :lt z I t I 0lz+z> @){y oql Sr'eldruexeJoJ 'Jooud eABqe/rr'sJncco(91 ecuanbesqns arytq paprrputturuaaqts1{q ataqm's-01- aa s (g)ly uaqt ZSua$D 'z7zll.?,1(,27,2 (gt
z t t * z t z 7l, lzll*zlzz
(rt ot ful
*77
( S I
I I - Z I I
(et *I 'Z vnwa'I aqtlo tuo sutotuol Wltli^ aruanbaso sl g II sacuanbasqns 'sesecJeqlo eql Jo 0J3c0I?l suollslncFc JulIuIs ' s - o l+ * o < E l o t: l z I g l + [ z I z l < @ ) w 'Joodd ueql 's:ncco (1 ecuenbesqns JI '4sua$oatq tq paporput xapw aLils2 {q ataqu'e-0I + aa < (g){y uaqt
' l z l l z l z z z l | * ( , 1 7 , 7 7 1l zl z ( , 1l T l z z z l | . ( , 1 2 7 , l7z, l1 I l7,l(,zzll.T,lzzellzz
Qt (ll (ot
' i i | i r i i : i i | i i i i il :
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3. THE MARKOFF AND LAGRANGE SPECTRA COMPARED
Tnronrrra 4. For any n > 4, an and ae are elementsof M which are not in L and limn-- M(Ar) : ooo. Moreover,M(A*) : |i(A*) only for i : 0; for all n> 4 the equalityM(A,): ).i(A,) holdsonlyfor i:0, -17 - n. Pnoor. We first supposethat B is any sequencefor which there exists an rn such that
(13)
A^(B))a--10-8
and (14)
Ii@) S a- + l0-8
for all lil:_ m.
By Lemma I and (14), none of the sequencesl) to 12) is permitted for ljl 2 m; likewise (13) and Lemma 2 imply that the sequences13) to 18) are prohibited for j : m. The prohibitions of the sequences13) and 14) yield b^ : 2 and bmlt : L Moreover, using the prohibition of the sequence l5), and reversing.B,if necessary, we have that bm-z:2 and bm+2: l. The determination of additional values of b; by means of Lemmas I and 2 is given in Table l. The secondline of the table gives the number of the subsequencein Lemma I or 2 which is used for the specifiedvalue of i. Tnnre l. Determinationof b^*1, for -10 < i < -3, 3 < i < ll
,
I
3 - 3 4 - 4 5 - 5 - 6 6 7 - 7 8 * 8 9 1 0- 9 - l 0 l l
Subsequence
N u m b e r|
2
316 17418
567
89 l0I 2 ll
312
bm+i |
2
2 2
ll2
2t
2 2
22
I
ll
2
2
Hence, B - . . . 2 2 | 2 1 2 \ | 2 * t 2 2 J| 2 1 2 2 2 . . . , where b- is indicated by the asterisk. An analysisof Table I revealsthat only the subsequences l) to 12) were used in the determinationof b^ai for < < (14) I l. Therefore, 5 i and Lemma I imply that ^Bhas the form (15)
B -...22 t*2 tfl3 t,
wherethis asteriskindicatesb--a. By way of contradiction we assumethat for some n ) 4 or n : @, o, is an element of L, and that B is a sequencefor which an = L(B) limi*- luit@). By restrictingthe sequence{k(i )}, if necessary, we assume that it is a monotonicsequence. Sincelon o-l < l0-8, thereexists1> I suchthat for all i > I (13) and (14) hold for any m: k(i). Hence,B has the form given in (15), where the asteriskwill denoteany bp1i1-sfor i > 1. Since{k(i)} is monotonic,we thus have that a, = L(B) : t(2 tA l) < cr- - l0-8, contraryto assumption.Therefore,no o, is an elementof L. To completeour proof of the theorem,we observethat if M(Ar) = Ai(An), then lA1(Ar)- n-l < l0-8. From the aboveargumentwe concludethat An
'splor{ uorsnltuoceql 'snq1 '(73) lcrperluoJplno^\ srql 'Surlrrurlsfenrlesr ecuenbes crpoued ,{1arnde eJurs 'D : (V)h{ luql qJns y ecuenbescrporred ,(lernd euo ls?el le sr ereql uor{l f ur sr ll JI '(lJ) pue Z tuoJooql ,(g 'uoou4 '1u| tou st D uallt 'ploll qpq SutlrutrluousI V uary 'o: (flW It ftC) pun l141loTutodpayolos!uD s! o (lJ) puD lN ut sr o 11 'S w:ruoaHl 'uorlrpuoJfuessacau e lou sr (tJ) '1ruJo sluoruoloJo{'^r} ocuanbesoqlJo lurod 1rrur1 eqt sr ((lt) ot ra;ar) *o acurs 'pueq reqlo oql uO 'l ur ur sr qJrqa Jequnu e;o eldruexoueureJC lou lnq 14l 'o o1 pnbe onle^Jo{rBI qlp\ I [gqOt] yeu€uo eqt sreloo S tuerooql'ocuoH ecuanbesalrugur flqnop ,(1uoaqt sr g acuanbesSurlrurrluouaql leql pue 14tr ur potslosrsr (enoqu(Ot)otreye:) o leql s^\oqs'[7 ueroeql'896I] ueturerC pue v tueJooql ur lsql ol Jelrurs 'uorlBlnclBJ ''I ur lou lnq tr{trur eq ol o Jequnu e JoJ suorlrpuooluercgns eldurs les u sar'rEruoJoorlllxou eql Jo 'vll (g)At > > sP uorJ fllcerrp s1r\olloJ .trou (91) ,(tlpnboul '@)!V : @)W qclq^\ "rogtq aqt roJ erul sr srql'relncrped uI 's_01 > l-o - (A)!Vl r{clq1(roJ lq futue,{ue selecrpur IsrJels?oql eJer.l^\'(St)ul ue,rrEu:o; aql seq g reql pue'unrlcods stuertel oql Jo lueuele ue lou sl @)I/{ leql ur?lqo o^\ t tuerooql ;o goord eql ruor; ' g - 0 1* @ n > t n > ( A ) W > s o > - -^) s_01
ecurs'Jooud ' l z 17, t Z I Zl> x > ItZzt Z t)
(st) aDqttr
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ler4l'.8-!Dfutue sqt selucrpur{srJelseoql oJeql\'(St) ul us,rr8urog eql s?rl CISUVdI^IOJVUIJEdS EONVUCV1 CNV JIO)UVW
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3. THE MARKOFF AND LAGRANGE SPECTRACOMPARED
Berstein [1970] developedthe following terminology: For any e > 0, a doubly infinite sequenceI is said to have the right (or, respectively,left) eproperty if there exists an integer i6 such that for any different l' which has d'i : ei for all i < i6 (respectively,; > io) we have M(A') > M(A) + e. Lnuua 3. If A has the right (or left) e-propertyfor some e > 0, then A is periodic on the right (or left). Pnoor. Let e > 0 be given such that A has the right s-property. We may assumethat M(A) is finite, so that A is a bounded sequence.Hence, for any /, > I there exists a block of integersof length n which occurs at least twice to the right of an in A; that is, there exist i(l),i(2) which dependon n such that i(2) > ,(l) + n > 2n and ( a r o \ * t , . . .a, i o \ + ) : ( Q i e ) + t , . . a. ,i e ) + ) ' W e d e f i n et h e s e q u e n c eAsn : { o ' , " } A y r e m o v i n gc t i t r ) + t , . . . , o i eftr o m A ' . namely, we define ol') : ai for all i < i(11; a(,n): ai+i\z)-i(t1,otherwise. M(A). If I is not periodic on the right, then I is Then limn**M(A,): not equal to A, for all n. Therefore,A doesnot have the right e-propertyfor any s > 0, contrary to hypothesis. The converseof Lemma 3 is not true: the sequencel- given in (l I ) is periodic on the left, but the existenceof {o,} shows that A* does not have the left s-propertyfor any e. Bersteinll973a, $11, pp. 4l-461 has shown that if 7" has the right s-property,then the length of the period of 7" is at least 7, which is the length of the right-hand period of the sequence,4-. The following theoremof Berstein[1973t, $8,Theorem 1, pp. 30-33] gives another set of sufrcient conditions for o to be in M but not in L. In an earlier version of this theorem (see Theorem 3 in Berstein U9701), the condition (C5) is stated incorrectly. In both versions, Berstein adds the unnecessary condition (C3) M(A):
o holds only for finitely many sequences,4
to the hypotheses.It is not known whether or not (Cl) and (C2) together imply (C3); perhapseven (Cl) or (C2) alone implies (C3). Each of the other implications involving these three conditions is falbe: From Lemmas 7 and 9 of Chapter I we obtain that o : ,ftt is an isolated point of M and,M(A) : ,,,8i holds only for the nonlimiting sequenceA : 3, and so (Cl) and (C3) togetherdo not imply (C2). Also, by Lemma 9 of Chapter I, CI= (65 +9r/r122 is a nonisolatedpoint of M and M(A): o holds only for A:T-1.I l2T; hence,conditions(C2) and (C3) the nonlimitingsequence do not togetherimply (Cl). THroneu 6. If a is in M and the three conditions (C2); (C4) if M(A) : a, then A is periodic on at leastone side; and (C5) if M(A): a and A is periodicwith periodat,...,ap on the right (or leJt),then A' : at,...,ak has tne rtgnt \or rclt) *propertyJor Some
' g Z- o s+ ( . g ) W < g - ( ; ) ( e ) t y< ( V ) \ e ) l + t t ) t Y '(OZ)pue'(6t)'(t1),{g '(eroqeroJ pa^\ollee^eqe,trse) g o1 lcadsarqtrm a8rel .{lluarcgns 1 o{Bl o.,rr'1 rsldeqJ Jo Z etutue1 Eutsn
'9 - os+ (.g )w < ( J) ( e ) ' r
( o z)
rsEelurue sr aJeql '0 < g .{uu:og 'ecue11'03 leql qons 0t < Q)t '(8 Kq + (.9)W < O)W leqt 0aJo otloqt rno luo{ s^\olloJtl .g * J oculs t) 'est/rjaqlo t+(1\:1p I _ , ."
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> l/l y1eroy tq : t+QDtP
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3. THE MARKOFF AND LAGRANGE SPECTRA COMPARED
Taking d : (eo- W@) - M(8.)))13, we thus have A t o + , o t ( A ) >M ( B ) + 6 . Becausesuch an inequality can be proved for infinitely many /, then L(A) >M(B) * d, which contradictsthe assumptionthat L(A): M(B).Therefore, ,B must be periodic on the left and has the left e-property. Letting ,8. be the purely periodic sequencewith the same period as the left-hand side of .8, in a manner similar to the above argument we obtain :nk(r)+j-m were to hold for all the analogousintegersIs and I. If ap11y,r1 j < 2Io, then for any d > 0 we could find an i such that A*rit@)> M(B) - 6
and hrtt@) < M(8.) + 6; theseinequalitiesare contradictory for any 6 < (M(B) - M(8.))12. Berstein$973a, $12,Theorem l, pp. 47-49)used Theorem 6 to determine the largestintervals containing the numbers o and a* (defined in (10) and (l I ), respectively),but not containing any elementsof L. The interval for o has length approximately equal to 1.7 x l0-r0; the one for a- has approximate length 2 x l0-7. Freiman U968, Theorem 3l showedthat the interval for o actually contains countably many isolated points of M (among them o) which are not in L. The interval containing a- also contains all of our other ar, as given in Theorem 4. We next consider the question of determining necessaryconditions for a number to be in only the Markoff spectrum. By a lengthyargument,Berstein |973a, $$2to 7, pp. 22-291 has shown that, when o is restricted to certain intervals, the set of sufficient conditions in Theorem 6 is also necessaryfor o to be in M but not in L. For example,Bersteinproved: THronEI{ 7. If a is inl.4 but not inL and 3.1666< o < 3.2656,then (C2), (C3), (C4) and (C5) hold. Although it seemsunlikely that the conclusionof Theorem 7 holds for all elementsof M which are not in L, no exampleto the contrary is known. In [973a, $10, pp. 39-4ll Bersteingavea list which contained22 otherintervals for which the conclusion of Theorem 7 remains true for any such a in the interval. The interval in Theorem 7 is by far the longestof these intervals. The method of proof does not ensurethat there actually exists an element of M which is not in L in any of theseintervals. One of the intervals given by Berstein does contain a (as defined in (10)), but none of his intervals contains any of the
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4. HALL'S RAY
Hall [1947], we shall show that F(4) belongs to the class of Cantor point sels, which are defined by the following procedure: Take a closed interval A - fx,x * a] of real numbers, and remove from it a middle open interval At2: (x + a1,x * Q * an\ two closedintervalsAr : lx,x * a1l and A2: [x + ar * dp, x * a] remain. The secondstageof the procedureis the removal of middle open intervals from A1 and A2. The processis continued, so the nth stageof the procedureis the removal of 2n-t middle open intervals. The set which results in the limit, when every closed interval which arises has been subdivided, is a Cantor point set. To begin the procedurefor obtaining F(4) as a Cantor point set, we take - D,2\/2 - 21. ln defining the subdivisionsof I which produce : A $6n .F(4),1hree types of intervals need to be considered. Intervals of the first type have the form
(2)
. ,a,,fll, .,a ,,a 1 ,f0,a1,.. 1 1 0 ,h ,..
where o and / denotethe infinitely repeatedpartial quotient periods 4, I and T7, respectively,if r is even; orTl and 4), respectively,if r is odd. Thus I itself is an interval of the first type with r : 0. Intervals of the second type have endpoints (3)
1 0 , a t , ..., a r , 4 J f
and
[ 0 ,a r , .. . , a , , 2 , 4 J 1 .
Intervals of the third type have endpoints (4)
10,ar,...,a,,4i1 and
l0,dr,.'.,a,,3,411.
In the subdivision process,the interval with endpoints (5)
10,ar,...,a,,2,4J]and
l},ar,'..,a,,131
is removed from each interval (2) of first type; the interval with endpoints (6)
[0,4r,...,a,,3,4J1 and
10,ar,..',a,,2,T]1
is removed from each interval with endpoints (3) of secondtype; and the interval with endpoints (7)
1 0 , a r , . . . , a , , 4 , 4 J 1a n d
[0,4r,...,a,,3,131
is removed from each interval with endpoints (4) of third type. Thus each interval (2) of first type is divided into an interval of second typeononesideandfirsttype(withar+t:l)ontheother;eachinterval of secondtype is divided into an interval of third type on one side and first type (with ar+t :2) on the other; and each interval of third type is divided into an interval of first type (with ctr+t:4) on one side and first type (with ctr+t:3) on the other. Thus the subdivision processcan be started with the interval ,4 and continued accordingto the aboverules. The resultingCantor point set is plainly F(4),
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4. HALL'S RAY
50
Pnoor. DefineAl : A,+t U A,+zandAi : Ai (2 < i < r). From (10)and Lemma2 we have (ll)
and Ar*Ai:Ai+Ai
Ar'rAt:Ai+Ai
Now
Q
r
B + B : l J { , a+r A j ) u U @ i + A ; ) 2 < i <j < r
j=I
:
U @i+Aj):B**B*,
l
wherethe secondequalityfollowsfrom (l l). closedsetsof realnumbers of bounded Leuun 4. If Ct,C2,.. . is a sequence Ci*rfor i : 1,2,..., then andCi contains
(r2)
frr*frc:frtt,*cl.
i=l
i:l
l=l
Pnoor. For any i, the set Ci * Ci contains the left side of (12), so the right side of (12) clearly contains the left. In order to prove the opposite containment, let I be any number in [lI' (Ci + C1). DefineV : {@r,x2): x1 andx2 arein C;} for eachi : 1,2,..., and define T : {(xr,x2): x1 * xz - /}. SinceVi is a boundedclosedset and 7" is closed, the set T nVi is bounded and closed;since I is in C; * Ci, T o tr{ is nonempty. Also ?"n 4 containsT iV;a1 for i : 1,2,..., so by the Cantorintersection theorem
irtn vi):rnfln, i=l
l=l
is a nonemptyset. If (tt,t) is any memberof this set,then 11and t2 are in C1,and this proves flL C and /1 t2: l. Hence/ belongsto |Er C +llr the lemma. l),2(rt - l)l and conW e c a n n o w p r o v e( l ) . W e t a k e S o : t i ( 2 struct a sequenceof closedand bounded sets,56,Sr, Sz,. . . with the following properties: (13)
(14)
S ; c o n t a i nSs ; a 1
(i:0, 1,2,.'.);
frr,: o1o1'
l=0
and (15)
, S *i S , = S i + t* S i + t
( i : 0 ,1 , 2 , ' . . ) .
We saw earlier that F(4) may be obtained from 56 by removing an infinite set of disjoint open intervals, namely the set of all open intervals of form (5), (6) or (7) with each a; 1 4. We can arrangethis set of open intervals
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52
where h < kz < . . . is any strictly increasing sequenceof integers. By defini tion, Z(o) : limsuplr(c); to find the limsup we need only considerthosen for which ctn+t: a ) 5. For such n ) . n ( a ) : a + f 0 , b r , b 2 , . . . ] +[ 0 ,c t , c z , . . . -] I
a sn + o o ,
so I(o) : /. A more careful use of Theorem I gives a better result. Tneonela 4. The Lagrange spectrum, and so the Markoff spectrum, cont a i n se v e r yn u m b e rg r e a t e trh a n4 + [ 0 , ] ; ? l + [ 0 . 1 , 5 , 1 3 - ]: 5 . 6 8 1 9 5 . . .. Pnoor. Let p :4+[0, Tl4]+[0, l, 5,T7] ana suppose), > p. By TheoremI we can write ,t in the form ( l7), where none of the b; and c, exceeds4. Since 1 > p > 4 + [ O , T V ] + [ 0 , T 1 4 :15 . 6 5 6 . . . , w € m u s th a v ea ) 5 . w e d e f i n ea b y ( 1 8 ) ,s o lim,supir(o) : tr, and limsup,ln(a)< 4 + 10,l, a, tt+l + [0,T7]. an,t+a
Thus if a ) 6 we certainly have limsup,tr(o) = )., and if a :5 we have the theorem. limsup,lr(o) :,1 provided,t > p. This establishes The bound of about 5.682 in Theorem 4 seemsto be about the best that can be obtained by using this method of Hall (and Cusick and Lee). A new idea was provided by Freiman and Judin [966]. They claim a bound of 5.118in their paper,but in fact their method givesthe following: Tnsonrrra 5. The Markof spectrumcontainseverynumbergreater than
4 + [ O , T J+] [ 0 , 3 , 4 , T 3 ]+: + l 1 t / 2 -t 3 ) +* r ' t x + I l ) : 5 . t 0 2 9 3 9.. . . The proof of Theorem 5 is basedon the following result, which is similar to Theorem 2 but usesa smaller set of continued fractions. Tneonrl{ 6. Any real numberin the intervat[5- tE, tE -3] : [,417..., 1 . 5 8 2 . . f c a n b e w r i t t e ni n t h ef o r m f 0 , b 1 , b 2 , . .].+ f 0 , c 1 , c 2 , . . . 1w, h e r et h e partial quotientsbi and ci Q:1,2,...) do not exceed4 and whereno pair is everequalto 1,4 or 2,4. bi,bi*t ot c1,c1a1 Pnoon. Let D denotethe set of o : f0,a1,a2,...] suchthat no partial quotienta; is greaterthan 4 and suchthat no pair ai,a;1t is everequalto 1,4 or 2,4. Define
: ]6/n - y, 62: lo,4,T3l:]1s- ,/n1, d|: [o,TJ1 , / : [ 0 , 3 J:][ 1 r / n - y . We first show that D is a Cantor point set obtained by dissectingthe interval tr :162,61).
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(equality could occur only for k : 2); and if k is odd and positive we must have
M*(A)< 4 + [0,ivl + [0,13]< ry. By symmetry,the sameinequalitieshold for negativek, so we have M(A) : )' provided ), > 4. This provesthat every number in the interval lq, | + JUl : [5.103,...,5.582...] belongsto the Markoff spectrum. To fill in the gap not coveredby this result and Theorem 4, we use Theorem 6 to write any ,t satisfying l0 - r/Tl < 1< 2 + \E in the form A:5+41]-).2, w h e r e , t r:1 [ 0 ,e 1 , a 2 , . . . ]a n d ) , 2 : 1 0 , a - t , a - 2 , . . . ] a r ei n D . W e d e f i n e A - . . , , C | - z , a - at ,o: 5 ,a l r a 2 r . . . , so Mo(A) : A. lt is easy to see that in this casewe always have M(A) : MoU) : tr, so the Markoff spectrum contains the interval ll0 - \8, the proof of Theorem5. Z + tql: [5.418...,6.582... ]. This completes Using a more detailed considerationof the set D of Freiman and Judin [f966], Hall [1971] made a tiny improvement in Theorem 5 by replacingthe constantby
4 + [ 0 , T , 3+]t o , 3 7:l 5 . l o o 6 8 9.. . .
Bumby$9731,by carefullyshavingdownthesizeof thesetD, madea further small improvementto + + [ 0 , T , 3+ ] [ 0 , 3 ]: 5 . 0 9 4 0 6 . ..' Furtherprogress depended on provingthat the set.F(3)+ ,F(3)containsa longinterval.Sincethe smallestnumberin .F(3) is [0,3J] = (-3 + \/TD16 and the largestnumberis [0,T,3] : (-3 + \E) 12,we have
r ( 3 )+ r ( 3 )c 1 . s 2 7 s 21. . .5, 8 2 5 8 . . . I . However, it is easily seenthat ,F(3) + F(3) does not contain an interval of length I because
a :2 1 0 ,3 ,3 J1< .61279 and
> .62201, [0,2,T3] F =10,3J1+ so in the Cantor dissectionprocess,the interval (o, f), whoselength is greater than .09, is a gap in .F(3)+.F (3). The following theorem,proved by Freiman U973b,$9,pp. 108-l l3l, givesan intervalin .F(3)+.F (3) which is the longest possible. THeonrla 7. The set F(3) + -F(3) containsthe interval
lB, [0,T,3]+ [0,l, 2,T3]l x 1.62202,1.52'1531, wherep is givenabove. Further,Freimangavea precisedescriptionof .F(3)+ .F(3) as an infinite Theorem2, Corollary,p. l25l gave unionof closedintervals.Schecker 11977,
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urlJeds eql u! sd?c s uardvr{J
58
5. GAPS IN THE SPECTRA
the other hand, tfl2 is the largestM(A) for I containing only I's and 2's. Thesetwo assertionsboth indicate why this was the first gap obtained above 3 and also suggestPerron's method for proving the result. Continuing, he (see Lemma 9 in Chapter l) isolatesr/tf in M, by proving that M(A.) > ,/Tt, whereA* :rt 3 321, is the minimum of the valueswhich allow 3 2. It is also proved that M(A.) is a limit point of M. A generalizationof this method of Perron has becomethe standardtechnique for conjecturing and locating maximal gaps in the spectra. A partial bibliography is: Kogonija [1966], Kinney and Pitcher [1969], Hightower [1970], Hall [1971], Davis and Kinney [1973], Bumby [973], U976} Cusick ll974l, and Gbur 11976l. Severalother authors, among them Pall [1948], usetechniqueswhich involve the quadratic Davis [1950],and Jackson119721, form approachto the Markoff spectrum. They normalize to forms f (x,y) : (x -ry)(x -sy) and considerintegerpairs betweenthe curvesf (x,y): I *e. On the other hand, Ollerenshaw[19481considersarbitrary lattices for the form xy. Theorem I (seeLemma 4 in Cusick [19871)below showsthat the method of Perron will alwaysyield endpointswhich have eventuallyperiodic sequences. For a fixed integer / > I let S(t) denote the set of sequencesof length r whose entries are positive integers,none of which is greaterthan 4. For any T c S(t), we define C ( T ) : { [ 0 , c r c, z , . . . f : c i + t , . . . , c i +€ rT , f o r a l l i ] and for any positiveintegersat,. . . ,4n we set C ( T ; a 1 ,. .. , a n ) : { [ 0 ,a t , .. . , c 1 7 1 , c 1 ., .c] 2e, T . ]. Tnr,onru L Let t be a positiveinteger. If for some at,...,an and some max{f: f e C(T;ar,...,an)} or a : min{f:f e T c S ( t ) e i t h e ra : C(T; a1, . . . , an)), then the continuedfraction of a is eventuallyperiodic. r . : m a x { B :f e C ( T ; a t , . . . , a n ) } : 1 0 , a 1 , a 2 , . . . f . P n o o r . W e c o n s i d ee Since for i > n eachai S 4, then at least two of the 4t + I sequences A n + 2 k t + t , . . , , A n + 2 kf O t +f kr , : 0 , 1 , . . . , 4 ' , must be the same;that is, there exist i ) n and d ) t such that (l)
(ai*t,..',ai+t) = (Qi+zd+t,...,ai+zd+t)'
ai+t,ai+z,.. . ], from the e' : 10,a'r,a'2,.. . ] : 10,h, . . ., ai+2d, ConstructinE i n e q u a l i t yi 2 n w e h a v e t h a te ' j : a j f o r a l l I < i S n . A l s o , ( l ) i m p l i e s that every sequenceof length t in qt appearsin the expansionof a; thus, o' is an elementof C(T;at,...,an), and so the fact that a : max{f:f e C(T;a1,. . . , ar)| impliesthat o' ( o. This, combinedwith (l ) and a comparison of the continued fractions of o and a', yields (2)
"'f, ' . .] S fai*2d*,*t,ai+2d+t+2, [,ai*,nt,ai+t+2,
if i + t is odd,
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Extreme elementsof the spectra subject to the exclusion or inclusion of certain sequencesare needed in the Perron method of determining maximal gaps. To do this, the following result of Bumby 11976, Proposition 2l is useful. L e t \ , I M al . F o r A - . . . , a - t , a o , a t , . . . a n d f i x e d n > | w e d e f i n ex : f a o , a - t , a - 2 , . . . 1y, - f a n , a n + t , a n + 2 , . .a. n1 d, f ( t ) : f 0 , a t , . . . , a n - t , z l . I " f (ar,...,dn-r) is symmetric,thenx > y if and only if As(A) > 1,(A). Moreover, if (ar, . . . , an-t) is symmetric and x ) y, then
(i) ln/)
y impliesthat if n is odd,f(x) S f(V), so A,(A) l , s o z i s i n c r e a s i n gfunction + f(z) an lf'(z)l:icz*dl-'< on z > l. Therefora, x ) y holdsif and only if y + f(y) < x+ f(x) and 1^(A) < 1o(A),completingthe proof of the lemma. THsonrvr 3. The intemal G(TI-D),LFD-D is a maximal gap in both spectra. Pnoor. We firstnoretharLOln): r/+8011 1 2 , l , l , x l+ [ 0 ,l , l , 2 , l , l , x f . A l s o ,[ 2 ,l , l , x l ) y ] x h o l d s t; h a ti s , x - l 2 , l , l , x f < 0 ; x S t 2 T t l ;
M(A)> LFn).
We shallcompletethe proof of the theoremby showingthat L(T|D) : max{M(A): eachai < 2 and 121is excludedfrom l}. For anysuchsequence ,4,wehaveM(A) : 1o(A): 12,xl+l0,2,y7,forsomex andy with 0 < x,y < l. Sincel2l is excluded, Also,usingLemmaI with n :2, l2,xl < tm\ from M(A) : Ao(A)we have12,xl> y. Hence, M (A) S 12,xl + fO,2,2,xl S tTlDl + 10,22nl : L(TD). Therefore,the givenintervalis a maximalgapof both spectra. KogonijaU9661and Hall [197Uwereamongthe first to identifythe maximal gapgivenin Theorem3. This gapis alsothe first elementof an infinite sequence of gapsidentifiedby Hightower[1970]and Gbur 11976l:Defining : b^ L(TZ) for any m )- 2, Hightowerfound real numbersan for which is a gapin M. Gburextended eachof (azr,bzn)and(b2na1,azn+t) thesegaps
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Lnuua 2. Let s : min{n:22-n< (b - a)}. If (a,b) is a gapinM andthe A and B are suchthat tro@)> b and bi : ai for all lil < s, then sequences M(B) > b. Pnoor. From the choiceof s, Lemma2 in ChapterI impliesthat ) . 6 ( 8 ) : l a o ,a t , . . . , d s ,b r + t ,b r + 2 , . . ] + 1 0 ,a - t , . . . , d - s , b - s - t , b - r - 2 , . . . l
> t r o @ )- 2 . 2 t - n > t r o @ )- ( b - a ) > b - ( b - a ) : a . Since(a, D) is a gap in M, then 1o@) > a implies that M(B) > b. THronru 4. If (a,b) is a gap inM and A is any sequenceforwhichM(A) : trs(A) = a, then A is eventuallyperiodic on both sides. Pnoor. By Theorem 5 of Chapter 4, we may restrict our consideration to sequencesI in which each entry is at most 4. Using the notation which precedesthe statementof Theorem l, for.r : min{n:22-n < (b - a)) we defineU to be the set of all elements(r.t-r,...,1t0,...,11r)of S(2s+ l) for which there exists a sequenceC with Ci : ui for all i, -s ( i ( s, and ,10(C)2 b. We observethat if D is any sequencefor which there exists an index 7 suchthat di-r,. . . ,di+, € U, then thereexistsa sequenceC suchthat ci : di+ifor all lrl S s and ,te(C)> b:by Lemma 2, M(D) > b. DefiningT to be the complementof U in S(2s+ l), we thus havetirat for any sequence C e a c hc i - s , . . . , c ; 1 5i s i n T i f a n d o n l y i f M ( C ) < b . Let Abe any sequencefor which M(A) : A0(A): a. The aboveargument i m p l i e st h a t c r 1: 1 0 a , r , a 2 , . . . 1e C ( T ;d t , . . . , a r ) ; i n f a c t ,w e s h a l ls h o wt h a t a + : m a x { f : B : l 0 , a t , . . . , c t s , x tx,2 , . . . ]e C ( T ;a t , . . . , a r ) } . F o r a n y 1 0 , a t , . . . , a s , x t , x 2 , . . e. ] C ( T ;e t , . . . , d r ) ,w e d e f i n et h e s e q u e n cCe . . i.n. c eM ( A ) < b a n d f } , a t , . . . , a s , x t , b y C : . . . , a - 1 , a 0 , . . . , a s , x t , x 2 ,S g x2,. . . ] C(T;at,. . . ,ar), then eachc;-r, .. .,ci+sis in T, which impliesthat M(C) < b. From the fact that (a,b) is a gap in M we obtain that M(C) <
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crn+t,f n+t arelTml,f2-22|1(in this order if n is odd and in the reverseorder if n is even).We note that b, :2, bn-t: I would force bnal: 2. Thus, for bn : 2 we only consider bn-t : 2; such a subinterval will be said to be of (in this order if n is odd and Type II with cn11, Baa1 eeualto lTDl,tZlDl in the reverseorder if n is even). If 1r u 12 is the subdivision of a Type I interval and the original interval t h a se n d p o i n tas : [ 0 , b ,r . . . , b n - 1 , 1 , D 2 2 1B, : [ 0 , b r , . . . , b n - t , 1 , f f i \ t h e n1 1h a se n d p o i n t[s0 , b ,1. . . , b , - t , l , l , 2 2 T l , [ 0 . b , . . . . b n - 1 , 1 , 1 , 1 2 2 2 1 , a n d1 2 h a se n d p o i n t[s0 , b ,1. . . , b n - t , l , 2 , T l n l , [ 0 , b r , .. . , b r - r , 1 , 2 , D n 7 . A computation, using Lemma l, shows that the ratios of lengths for this subdivision are
n0 + | + 7r), lr,llvl: (13-,/nox'/no + 6+ 7r)176/ ItzlI It | : (r2 t/ nq (r/-no+ 6+ | 2r)I 30(,/n0 + 8+ 8r), w h e r er : [ 0 ,l , b , - t , . . . , b r l . S i n c el 1 1 l / l 1al n d l l r l l v l a r e ,r e s p e c t i v edl ye,functionsof r, from j S r < + we obtain creasingand increasing
l r l l l r l < ( 3 3- ' / n q I s t < . 3 8 7 , -szt/n011208s <.037. Itzllltls (1083 We similarly obtain the subdivision1rUIz of a Type II interval: If the origi n a l i n t e r v atl h a se n d p o i n tos: [ 0 ,b r , . . . , b , - 2 , 2 , 2 . T n 1 , f : l b o , b t , . . . , bn-2,2,2,Tn1, then11hasendpoints[0,b1, . . . . bn-2,2.2,1,ffi1. [0,br, . . . , a n d 1 2 h a s e n d p o i n t s [ 0 ,b r , . . . , b r - z , 2 , 2 , 2 , 2 t 2 2 1 , bn,2,2,1, TDI A computationshowsthat the ratiosof lengths . . f0,br,. ,br-2,2,2,2,TD| are for this subdivision
lr,lI vl : (55- st/ nox'/no + I + 7r)I 6/ 120+ | + 7r), 8 + 8r), Itzllltl: (12-,/Tryr'/Tn+ 6 + I 2r)16$/T?/J* w h e r er : [ 0 , 2 , 2 , b n - 2 , . . . , b t ] . S i nl 1c 1 e l / l 1aln d l l r l l l l l a r ea g a i nr,e s p e c functions of r, from J 1 r 1 I it follows increasing and tively, decreasing that
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'Zl7,lsutntunV 'Z vwwa'I '8le9l : II OZtZZtZt)l" (v)w uaw 'gl_gf qclqa\ eql aoleq seII rurupods Jo uorgod eql sI slr1lleql '[€t6I] uBIuIorCpue s1(oqs([Sg 'O 'EL6l],(qulng ees)ztutuol Eurrnollo;eql 'peltqtqord s\ ?^ol^ed ,(q peuretqo s?A\pue /t\oloqz tueJoeql sl slql zl7,l ecuenbeseql qcrqa ur urrupeds eql Jo uotuod oql uo ulolsJegJo poqlour eql 'secuenbespe^IJop esn Ipqs ern'11nse:otuesoql Jo goord srq o,rtt ueql Joqlsg '01.61'Joq sJo{JBI I ol Jelrrurssr qcrqA\?oprue ol peuJnlulelsJegJoord sq u1 -olro ul Is$qI ut,fuoeql reqlunN uo ocueroJuoCuolun{V eql ls llnsor ot{l pegodar oq,rr '[qgl6l] urelsreg ,(q ,(ltuepuedeput per'ord se,r I tuoJoaql 'reldeqc 'sI ulgoord,(ue ur posntou sr ((7)ryldns: (V)141ll -qt lsql) urnruerdns sg1.ql orll Jo pedun IInJ eql 'IZI ou pue s.Z pue s.l fpo urc1uocqclq^\ osoql ol t/ seouenbaseql pulser o1 ,(yuopesn sI rurulceds Jo{JBI I eql Jo luaruele ue s (V)1,y leql peJ eql rueroeql e^oq? aql Jo Soord eqt uI l?ql elou al6 'oJez '1eurs ,{praltqre eperu 0q uec peJoloc eJnseeruseq los runs eql 1eq1Sutrrord qfual lelot eql os pu? ,(Zt,O')Jo JolceJe ,(q qfusl Ielol eql Jo uollcnpeJ3 uI sllnser serurl ?./Surpr,rrpqng'gg7' seuft JnoJJo Jolc€Je ,{q pecnperst qltual JoIUI lqEte ue uI sllnsal sl?^Jolul uolslllpqns plol esoqa\les runs oql IB Jo 'ssecordJolueJ oql JnoJ olur sle^Jelul pueluluns ssoql Jo qcee tutpt,rrq Jo 's1ss uolun eql e^eq ern aEels sle^Jelur oA\l Jo runs eql sI qclq^\ Jo qc€e 3o qceelV 'n+n les runs oql uo ssacordJolueJ slqlJo pe,gooql Jeplsuoc0AA 'ltlttZ' u?ql sselst sqfuel eql Jo Iuns oql qcg,t\ roJ sle^ -JelurqnsJnoJolur pepr,rlpqnssr ssscordJolue3 oql Jo 1 Ie^JeluIf.ue 'ocue11 VUIJAdS AHI JO SUNSV:IhI EHI
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6. THE MEASURE OF THE SPECTRA
2 implies that the set under considerationis containedin the translateof the sum set of u + u by 2. we shall prove that 2 +lJ + u has Lebesguemeasure zefo. We fix e > 0. For any o : [0, d1,ct2,...] e U with convergents pt/Qt, we d e f i n en : n ( a ) b y
(r)
q;2<e
Then o lies in the intervalwith endpoints[0,et,...,ara1f and f0,at,...,ar) and the lengthof this interval is I f q,*1qr, which is boundedaboveby q;2 I e, bv (l). Defining N to be the number of pnlqn e U satisfying (l), we shall prove that
(2)
N = O(e-'4eee\.
Since the elementsof the set 2 + u + u lie in N2 intervals of length 2e, the measureof 2 + U + u is at most 2N2e. Therefore,the theorem follows from (2). Sincec, ( 2, then Qn: anQn_t* er_z .--3Qn_r.From (l) we thus obtain (3)
q;2<e<9q,2.
Also, 4n : anQn-t* en-z and a, < 2 imply 111a1)@-t)12 1 qn 4 3r;by (3) we have tol(t/e)
(4)
W e d e f i n et h e i n t e g e ris( l ) , . . . , i ( 2 u ) b y [ 0 ,a t , . . . , a r f : [ 0 , e t , a 2 , . . . , e j , 1ig1,2ip1,...,2ie,)l.WedefineN(l) (respectively, N(2)) to be the number of ones (respectively,twos) in this subsequence {as,...,an} of the partial quotients. Also, M( I ) (respectively (2)) M will denotethe number of isolated , ones (respectively,twos) in this subsequence, and M wlll denotethe number of stringsof onesin {i (2),i(4),. . ., i(2u)}. We denoteby N(n, N(2),u, M, M(2)) the number of pnlqn corresponding to the specifiedvaluesof n,N(2),u,M,M(2). From their definitions,each of theseparametersis boundedby O(n), which bV (4) is O(log(l/e)). The sufficiencyof (2) thus implies that it sufficesto show that N(n, N(2),u, M, M(2)) : O(e-.4eee4), which bV (3) would follow from (5)
N ( n , N ( 2 ) , u ,M , M ( 2 ) ) : O ( q , ) .
N o t i n gt h a tq r : ( a r + Q n - z l Q n _ r ) q n :_ r[ , e n , a n _ r , . . . , a .1qf r _ t , b y i n d u c tion we obtain n
(6)
n
ei:l.ai,...,atf. Qn:f[fai,ai-t,...,ar|)17u,, wheru
'sr 'Q)W + /,{ lseel lB sr I < (s)l qtt,rr s ppo Jo reqr,unueq1 leql :ploq lsnu I < (l + sd! pue I < (t - sZ)l qtoq srncco t = (sz)l roleueq^\'rncco lou aql seop ZIZI ecurs'leq] elou lsJg am'(rr7)t'"''(l)l secuenbes..ponrJop,, "'(elt7'(t)t 'srsleureredpoleclpul eql ol lcafqnsfu.')!2" lunoc ol JepJouI 1 'a)1g 'Lo' 'rr'01 teqt IIBJoJolA Jo requnu eqt sl k)w'w'/t'(T)N '(11y1,,68/S'l' : 6 1 1 8 Z l t ' 7 ' , , S [ 8 6 6' ' a t Z \ ' l ) O k)ttt'W "t'(()N'u)N
Q)
leql errordot seruns ll (s) Jo Soord eqt roJ t3q1sA\olloJ1l slql urou 'rr),r,'6815' |' ( z t w E ( .I0' ' S r V 6 '< " 5 e 8 6 " 2 1 p 8 Z l t ' Z ' ub ' ' ' trl,,,'68192 ' l 1 1 1 8 7 1 t ' ^ZZ' l 0 ' l 2 y a t t 6 6 ' f t ) w - z + n 6 l L 6<' l?ql seopZIZI ecurs ol?q eA\pue 'ploq * rt> (dW +(1)1tg'ncco lou lsnu Z ' 1 1 y 1 , , 6 8 1' 9 ' l ub g7p87lt'7'n7l0'l ' 6yatt66 ' Ow6l L6' < ol?q errrsrurel Sururqtuoc ' ne-0)n+fi)1,'68rS'l ' i nz-Q)w+k)xgzlV'Z 1 7 1 1 a - ^ 9t l' EO y 4 r - , J 6 tE' 6 7 a 6 L t ' l ' O r y E \ L ' Z1 u b eleq eA\elqu1srql pue (9) uorg 'elqet eql Jo urunlor {Unog eqt oleldruocsourl (7)Lr srncco z pu? saull (1)Lr s:ncco I lsqt slJru eql 'so/tuJo pue seuoJo sdnort r eJ? eJeql l?ql lceJ eql qlr^\ peurquoc suorllugap oseql tuo{ ^tolloJ t '(Z)W pue ( l )W lo suolllugep oql s/(or qlxrs qtnorqt pJrql eql l,{1a,rtlcedso uorg ,{11cerrp/rolloJ sa\or oaU lsrg oql : I elq?I Jo utunloc qgnoJ aql roC 'I elqel spled 7+!D't+!D'rr Jo senp,r elqrssodlq8re oql Jo uorl?JoprsuocV
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70
6. THE MEASURE OF THE SPECTRA
Hence,N(l) : t(l)+ r(3)+ "'+ i(2u - l) is a sum of z positivesummands, at least M + M(2) of which are greater than one. Subtracting one from each of these summands which is greater than one, we then have a representation of an integer which is at most N(l) - M - M(2) into z positive summands. The number of thesesums is equal to the number of combinations of N(l) - M - M(2) things taken u at a time, which we shall denoteas C (N (l) - M - M (2),2). Thus,the numberof sequences i ( I ), i (3),. . ., i(2u- I) i s b o u n d e db y C ( M ( 2 ) , M ) - C ( N ( t ) - M - M ( 2 ) , u ) . For the number of sequences i (2), i(4),.. ., i(2u), we recall that M(2) elements of { i (2), i(4), . . . , i(2v)} equal one and theseM (2) elementsoccur in M strings. As above, there are C(N(2)- M(2) - (u - M(2)),u - M(2)): C(N(z) - u,u - M(2)) ways to partition N(2) - M(2) into u - M(2) summands,sinceeachof these summands is greater than one. Also, these v - M(2) summands occur in at most M groups. Hence,the numberof sequences i(2),i(4),...,i(2u) is boundedaboveby C(u - M(2),M).C(N(2) - u,u - M(2)), implying N (n, N (2),v, M, M (2)) : no(t). C (M (2),M) . C (u - M (2),M) . C ( N ( 2 )- u , u - M ( 2 ) ) . C ( N ( l ) - M - M ( 2 ) , u ) , and for the proof of (7) it is sufficientto show that C(M(2),M).C(v - M(2),M).C(N(2) - r,, - M(2)) C(N(l)-M-M(2),u)
(8)
: O ( 1 . 0 2 3 M (.2. )9 8 3 5 ,. 2 . 4 t 2 8 N ( 2 )1.. 5 7 8 9 N ( r ) ) . For arbitrary real numbersa ) B, we define C(", f) : C([al, [B]), and we introduce four auxiliary functions:
: C( x- M( 2) ,M ) ( r .936) - ' ; f i ( x ) : C (M(2 ),x)(r.4 7 4 9 )-x;" fz( x) ; fz@): C(x - u,u - M(2))Q.ar28)-' : fq (x) C(x M(2 ) M, u) ( 1.5789) - ". Considering rt6), wehavethat fi (x + t)l fi @) : (M(2 )- x)I t.a74e(+x l) - .5959.Hence, is greater thanonefor x < .404M(2) M(2)+'404 |. fr @) 3 c (M (2), .404M (2))I Q .4749)'404 From Stirling's Formula we obtain C ( k t , j t ) : ( k t ) t l U t ) l ( k t - i t ) t : 1 k t \ o { t ) 1 k ol i i U , - i l k - j ) ' and so y u (z) C (M (2),.404M (2)) : no(t)(.404'+oa. 59 5g'srss = 911.9636u(z)1.
'llnsoJo^rlrugopororu leql go goord e er'tElou Ipqs oA\ 'psllolur suorl€lncluc Jo lunorue orll Jo esnecoq:[SSet] {JoA\ Jelpl s,,{qurng ur posn senbruqceloql elecrpur sruorooqlosoqtJo sgoordaq1 '[gg-79 'dd 'd'016I] 't,L611,(qungJo llnsor flree ue errorden t Iueroeql uI pue'[€tt sreEog luoJoeql ul uo,rtElFseJ e /r\eIAeJa^\ f LueJoeqlut 'urlcodsoql 19 Jo rno roC 'tl€S' > 6,)p > Ueg Jo uorsueurpJropsneH eqt Jo uorssnJsrp l?qt Joord e poqclo{s[SS6f] ,(qutng :lZtS' > (r1Jp> 90€S'leqt suollslntl?r olsroq?le ,(q pe,rord ISZZ-tZ| 'dd 'l?6ll pooD :pereplsuooueeq seq A Jo uorsuorurpgJopsn?H eql 'purru ur srql qll1l 'orez oJnseerus?q S + S les runs oql uoql 'f > (s)p segsrtess les B Jo uorsuorurpJropsneH eqr JI 'le^Jelur u? sur?luoc €!A ^\oleq unrpeds oq1 'prr:e1ul ue ul€luoc ol eJo^\ 1eq1,(1drurlou plnoA\ uolsnlcuoc reEuorls stql sl EI/,l\oleq 3uil1 a +A + z Jr ue^o'A +A + z los eql ur peulBluoc,(11culs 'l?^Jelur uB sureluoc + A A + Z los oql lou Jo IAI Jo uorgod eql esnuceg Jeqleqtr ua\oul lou sl lI 'Z u?ql roleor8luetlonb ptged ou suleluoc uotsued -xe uorl3?{ penurluoc esogA\sJeqrunu l?uorl?JJr IIe Jo 10soql eq A r0'I 'p > d I1eroJ 0 < (I)s1n pue p < d tp roJ 0 @)sgl p reqlunu IBor eql sl E Jo uorcuautp {topsnog eq1 'leq1 qcns (g)p :
' tt( t ( ,t) n q reeql l^ \t g a : g :d( g ) 9-3 ) t"f ld n s: 1q g ' ; , o ,
3
sl Jo amsoaut[topsnoH louotsuaturp-d oql '0 < d fue rog ? '(g)g releu?rp qltlr XJo tesqnspopunoqB !r qlIA\ acedscrrlaru e 0q X l0'I 'z pue suoJooql ur uerrrSrun-rlceds3ql I Jo sued oqt Jo uolsueulp '[€16I] JropsneH orlt Jo uoll?1nrlet eql uo [gfe t] uslulerg pue ?^ol^?d pue Kqung '[OgOt] roqclrd puu ,{euury Jo {ro^\ eql replsuoc lxou il?qs el6 'uerooql eqr goord eqt Eurleldruocpw (8) Sutrurguoc Jo '(11;1,'681S'I : e l t $ T I V Z ' 6 q Z 9 l ' ' ^ 6 1 8 6 ' 'n ( 8 6 6 ' ) O (,t'(dltt - W - (l),nf);2.
(k) w - / t' /t- (du) C - (0 )w 'Q) w - ,1 )) . ( w' Q ) w)C e^eq snql eM (x)Y Jo uorlrugop eql tuory
' (^ztos' tsLe' ^ 21 azgt')o : (0)u )'l (ft)u )tI@)T(w )V
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'(rztw+w68 L 9|'' , 6 8 L SI' r ) O : ( (t ) r u ) t / '(,Lov't I6agztr't)o : (dt't)tf '(6ya9E6' | ' w 9 E 6I 'l ) g : ( r t ) z f . e c u o 1. (1" _ N . r _ ( t N))o: @)y - x)) : (x){ ro3 luerunErereltrutse ,(g e^?q o/h!\ I < N rlllr ,-",t'Y '( . v w ' t ) t zw t 9 t96 'l ) o = (w ) V a w 8L 9 l' ) o : ( e w v o-rG 'eJoJoJeqJ VUIJIIdS AHI JO EUNSVEhI :IHJ
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72
6. THE MEASURE OF THE SPECTRA
Kinney and Pitcher [1969, Theorem 3.1] prove that the part of the spectrum which was considered in Theorem I has Hausdorff dimension lessthan one. Unlike Hall, they do not use an exact Cantor construction, but they do use more precisebounds oD rn: On-tlQn. Becausetheir ideas have been used and extended by Bumby 119731,we suppressthe details of Kinney and Pitcher's work and end the chapter with a sketch of Bumby's proof of Theorem 2. lf {p*lqr} is the sequenceof partial quotientsof a : f0,ar,...,an,en+rl, w e r e c a l tl h a t o l i e si n t h e i n t e r v a l o ( a t , . . . , a n ) : { [ 0 ,a t , . . . , a n , x l : x i s a t least l), which has endpoints Pn Pn'f Pn-l Sr' S, + Sr-,
and lengthequalto
L \ a t , . ., a n:) q ^ 6 + q ; J Then Lnuur 3. Let ctt,...,a1befixedpositiveintegers. L ( a r , . . . ,a i , l ) t 1 3* L ( a 1 , . . . c, t j , 2 ) t 1>3 L ( a t , . . . , a j ) ' 1 3 ,
and thereexistssuficientlysmalle > 0 suchthat 3 L ( a r , . . . , a i , l l z - e ) l t * L ( a 1 , . . . , a j , 2 ) Q - e I)3 < . L ( a 1 , . . . , a , ) Q - e I) .
Pnoor. For positives, we set R ( s' ):
L ( a t '" ' ' a i ' - l - .+) 'L ( a r '" ' ' a i ' 2 ) ' L(at,"',ai)t
.'
Now L ( a t , . . . , a i , o i + t ) - t : q j * r ( q l + t* Q i ) : ( a 1 + r Q j- r Q t - t ) ( @ i t
+ l)qi + 4i-r)
:aj@1, +r)(a1a + rI + r ) , w h e r ey = Q i - r l e 1 ; a l s Lo (, a r , . . . , a j ) - t : q j ( r + l ) . T h e r e f o r feo, r a n y s> 0 , R ( s )= ( r + 2 ) - ' + ( ( r + t ) l Q + 2 ) ( r+ 3 ) ) ' i s a d e c r e a s i n g ' f u n cot inorn> 0 . T h u s , 0 < r < I i m p l i e s ( j ) ' + ( * ) ' < R ( s )< ( j ) ' + ( * ) ' , so
( + ) ' /:' l , n ( * )r ( l ) " ' * ( * ) ' "r ( + ) ' / ' + n ( 3 ). G ) ' t t * ( * ) " ' . ( # ) ' t ' + Q 7 l t 2 5 ) 2:t 31 , which completesthe proof of the lemma.
tltr-zl(0'
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('o'"''to\
3
t
('o""'tD]'
"' "'r', s11t-71(Z'tP'
3 ( ! D ' " ' ' 1D l
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'{l
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< r;fx'lo'""to'uq
I?^Joluroql ur serl l] '{'tblld} ''" "q)A ' " ' t + [ o ' { o '' " ' t o ' u q ' ' ' ' ' t q ' O l f r q ( ' q f l e q l I I B c o Jp u e [ '('q'"''rQ) pexg JoJ - lr Jo sluerlonblurged;o ecuenbeseql elouep o,r\ ' { Z r o : l D r 4 c e e : 1 " ' ' z n ' I o ' u Q ' " ' ' t Q ' 0 1 } :( u q ' " ' " 9 ) A t (,q' "''tQ) IIBJe^ouoruneql sr iA ueql'g ueql sleseql Jo releert sluerlonb lerged {uuru ,{.1o1rug lsoru le sur?luoJ uorsuedxeuorlcu{ psnurluoc esor.IAsJeqr,unuleuorlsrJr IIB Jo les oql solouep *A JI 'Jooud 'i > (t)r > sa{suosuolsualulq \ 'Z [topsnop aW uary UDWnloat8 luaqonb Tot1todou sutoluoJuotsuodxauo1 -cott panu4un asolltrl puortortt pn lo ns aW st LII '€ hrauoaHl saqunu '[OfOf] sreSog ur 19 rueroeql sr ]lnsor lxeu eql 'peJrseps3 (I-ub + ub)'bLZ - _< ( t - u b + u b e ) ( t - u b+ u b z ) t
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aABqe/t\f : tt 'Z'Zf < t+'d 1[l 'Z] : € pue 'Zt+un pu? Z) t+ua > I luoq I : t+uoleql esoddnserrr,(ltpnbaulJeqlg eql JoJ 'snorlqo sr,$qenbeur opls lqtF agl'('n'"''tD)1 o1 pnbe qfuel qtur ('r'"''to)01 l?^Jolureql ut e\ g 'a sl€uorleJJroql ocurs 'Jool{d '('n' "''tn)1 > "''to)7 ld - nl > tzl (o' uaql 't+uq * t*oo{I 'z tsot'ulD st luauonb pvnd q J o ao p l t l f rur ! l " ' c z + u q < t + u q ' u o ' ' t D ' 0 f : d p u o 1 , " ' ' z o ' t o ' O l : p s u o t s -uodxa uotlcotl panu!run annq ! puo D saqunu lDuotlorn p7 'V vnn:r'I tL
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74
6. THE MEASURE OF THE SPECTRA
by Lemma3. Repeatingthis processwe thusobtain D
L ( a r , . . . , a i * r ) Q - ' ) l.t7 1 1 1 t 2 - d l az 7 1 2 1 t 2 - e<) 123.
( a 1 , . . . ,1a+t )
Thus, p(2-')lt(Y(br,...,br)) ( 2, which implies that 1t(z-e)lz(D< 2 and so p 2 l 3 ( V ): 0 . We next prove that ptl3(Y) is positive. If {1k} is any set of countablymany open intervals which cover V, the compactnessof V implies that there exists a finite subcoverof V, say {I r, . . ., I 7a),in which eachI po V is nonempty and none of the intersectionsis contained in another. In fact, eachIp o V must thus containat leasttwo points: if o: f0,a1,a2,...1isin Ip [tV, the opennessof Ip implies that for sufficientlylarge n we have that [0, at,...,en-t, 3 - a r , d n + t , . . ] i s a l s oi n 1 po V . For fixed k, | < k < M, we choosethe largestintegerK(k) such that there e x i s ta 1 ,. . . , a t 1 6 w i t h I p n Y c { [ 0 ,c t t , . . . , a y 1 p 1 ,xx )>: 1 ] : I o ( a r , . . . , a x 1 , \ )J: t . Since .I7.fl V contains at least two points, by Lemma 4 the length of 1a is at leastL(ar,...,a)127 : lJkll27. We shall prove that tttl3(V) > ll3 by showingthatl{=rlJr,lt13 > t. SettingN : max{k(l),...,k(M)}, we may reorderthe intervalsof the covering, if necessary,to obtain that k(M) = N. From the definition of V we observethat Is(a1,...,aN-t,3 - aN) c V also holds. If an elementof Io(ar,...,aN-t,3- aN)belongedto ,I7.for someft with K(k) < N, then we would have that Jy c Jp, contrary to our assumptionon theseintersections. : M - l, such that Ju-r Hence, there exists k, say k : replacing We refine the cover . . Ju} by Jya-1 Io(at,. . . ,ctN-t,3 aN). , {L. and Ju by the one interval Is(a1,...,ay-t).Recalling from Lemma 3 that L ( a r , . . . , a 1 , 1 ) t* l 3L ( a 1 , . . . , a j , 2 ) r l tL, ( a r , . . . , a j ) ' l t holds, in M - 2 stepswe obtain that M
Dltof,t
> LQ)tt3+ L(2)tt3> t,
k=l
completing the proof of the theorem. Bumby |973, pp. 34-351 provesthe following strongertheorem. Turonrrra 4. The HausdorffdimensionofY is greaterthan |. Pnoor. We consider a Cantor-type construction of V which is similar to Hall's construction given in the proof of Theorem l: V is contained in the interval from [02T] to t0T2l. This is subdivided into the intervals ttO2Tl,l027Ill and [[0lIZ],[0T2]l; the first of these intervals is subdivided into [[07T],l02lTZll and [022T2],l02Tll. Continuing in this manner, we s u b d i v i d et h e i n t e r v a, l s a yA : f r o m [ b s ,b t , . . . , b , , T j l t o [ b s ,b t , . . . , b , , - l 2 l
'i ueql sselsI las aql Jo '[gfOl] ,(qung uorsueurpJropsneH oqt leqt saoqsaq fu,n slql uI ;o xtpued -dV aqt ur pelsll sruurSordaql Sursnpeuroyad eJesuollslnol€coql 'sqt8uel eql Jo sloor arenbsaql Jo Lunseql ol enle^ Jelleluse p1et,{qclq/'\ lI Jo suoIsIA -lpqns Jo Jeqr.unuelrug oruossKerrrpsI eJeql 'ssecoJdJoluBJ eql Jo le^Jelul pexg ,{ue rog 'luql sanord ,tqung 'stueJootllsnotneld eql uI se EutnS:y ( t g ' 0 9t]u o { 'z'l'l'uq ' t q ' \ q lo l 'Izztz'T,'l'l"q [zlI I "'tq'1ql I e ^ J e l ueI q l p u e h l I ( , 'l ' l ' l ' u q ' ' ' ' ' t Q ' 0 q o] t I l I Z t ' t ' l ' I ' u q ' ' 'ssecordoqt ruo{ Iu^JOlureql :ere e8elslxou oql ls sle^Jalul ueql 3o 1eruet -ur uB go slurodpue eql JoJ sluatlonb lerged uotu(uoc 3o ecuenbaseql sI 'eldruexe:og 'ssecorduorsr^rpqnsJolueJ oql " ' 'tq'0q Jo lu^Jel ll'rq' ;r -ur ,(uego qceo pouad eqt v ro [t t Z t ] reqllo qclq^\ uI lurodpue lZZlZl Jo ss sJnJco,{1uo717 't\pey uollcnJlsuocJoluBJ u sI aJeql 'oJoJoJeqI 'Z(,17,7, -urrslllZII oq lsnu eql'uepplqJoJsI ZIZI eculs eJuorrnoco,(yuo IZI Jo 'f ueqr sseluolsuotulp JJopsn?H s€q ' ' z o ' t o ' 0 1: o } { s r n c c oZ l ( , 1 o u p u u Z > l o : f " 'cooud ttt lBr{l ^\oqs e^\'e^oqB pel?clpul sv * ;.jrj;:"t seq los leqt snql pue '1 u?ql sselsl 8/689^ A\oloqurn:lceds eql Jo ued aqt '7, tIlI/Y\polslcossespuetulunsOqlJo uolsuolulpJJopsn?H oql leql se,rordeH ruoroeql Jo [rt-9€ 'dd 'tr6l] goord s.,{qufg go I{clo{s e e,rlEoA\/t\olog 'f u?ql raleerEsl AJo uolsuaulp 'ocuelq'sle^roluleqlJo sqfuel oqlJo JropsneH oql pue'elrugur sl (A)tzlrtl sloor erenbseql Jo runs eql sessoJculssacordlueuaugeJ slq] leql s^\oqssIqI
'r <
7 1 / ( l v l l l t ve) r+l l v l l l t v l ) '(0t) pue (6) Sutsn sp1or,{ uollelncl€c
'(s+ [zr] + [zr r])(tzrI - trzl)I (s+ lztl+ trzl)(tzrI - tzr t l) < lvlI levl (0t) 's lerll serldur qolq,r\ +l1ll
> / Jo uollcunJ Eutsearcepe st
- tzltl) : lvlllzvl (.r+ [zrr])(tzrl-lrzDlU +lrzl)(lzrl ',{lreyrutg
'(g- trzol+ [rzzD(zt]- LtzD I @- [t zo]+ lzrD(ltzzl- I t zl) < lv|I ltvl G) os pue '3 - [IZ0] < / Jo uollJunJ Sutseorcurue
'(t+ltzzl)(tztl-f,rzDlQ - ttzl)= lvllltvl +IzrD(ttzzl o^eqe^\ I slulue'IKq'zV'tV'V sle^relureql Sururelqo'?+[ZI0] > .t > s-ll7,gl segsrtes [.tq'''''uq'0f = t'uq ' ' ' ' 'tQ'oq Kuero; 'leql qcns ?,/e3re1,{yluetclgnsslslxeeJeql'0 < 3 pexu JoJ , r q , \ q lu o t s : z V p u e . I T , , I , u q , . ., t Q , O qolt ll'7, fZ,l,l,uq , Z , u q , . . . ' t Q' 0 g l r u o J J : l Z ' s F ^ J e l u r q nosA Ue q l O } u I , Z , u q , . - . , t q , \ q lo l fl SL
VUJ,JEdS EHJ CO ENNSVIhI AHI
'9
76
6. THE MEASUREOF THE SPECTRA
Bumby U9821further extendedthis methodby usingresultsin Bumby 11976land obtainedthe following theorem. A sketchof the extensivecomputer calculationsusedto obtain the result is given in the last paragraphof Bumby[1982]. Tneonru 5. Theportion of the Markof spectrumbelow3.33437... has measurezero.
LL
go eteurr-1fuego Jolor,uurp oql 'srxeI?eJoql o1puotoquo sr qcrq,vr(g'Is) reddneql olouop(ts'{t)g p-t 'lrs - ll-*fdns ot (0'f.r)ruo{ olcrrcrrues : (a)rl e^?qea\ I sIuIue'I Kq'(n){r1: ts'(x){/l : Ll Eulugeg 'Jooud 'S/ 'louottarr!s! o 't wauoaHl 7 @)n uayl lI :ruerooqls.zlra\rnHe,rordol [gror] s[oqcrN,(q pue [gsol] '[416t] pro.{ ,(q pesn ser\ qcrq^\ urulceds eEuerEeloql Jo uotlelerd:elut eql sl sHI '{tqot satuaruoc (t) l(a)tr1- (*){tl ruqtqcns araqt:ry}dns: (a)l aUu{u!uo stslr,r.a J) {2) acuanbas 'D
fuo tog 'I vnns'I louotlDJJr :po,ro:de^sq eA\pu?
'{J> roJ A fueu ,(1o1rugul ' r _ U) l @- a b ) b l : , _ l ( a ) t - @ ) l | : r 7 ) d n:s ( a ) r l 'l:
'ecue11 'J) A e^eqe/{\(d + zb)l(,d + z p): (z)A roJos 'd)JI lsql olouoA\ b,d- dp wrtrlqcns,D',dsro8elur lsrxeeJoql'l: (b '{l : (b 'd) ,(ueru,(1e1rugur roJ.,_U> l(d - ab)bl :t7}dns: (a)l
eq ol l?Jequnu leuorlBJJroqrJo (o)/ en1e,r eSuurEzleql peugope,rr1 reldeqJul'(zzaa 7t211)l(r111ztta) : (z)l ,{q uorleru.roJsueJt pelsrcosseeql eugop o,r\ 'J ) / Kueuo^lg '[fSOl]uDIu?Upu€'[X uqoCJo s?epleql ,(q pocuonguloq? .reldeq3'Vg6l1'lZS6tlreuqe'I'[SSO1] sea\{roA\ slql leqr s,(ess110qctN'[8t6I] sIIotIclN uI popuelxepu? pouger s?a\qcrqa {Jo^\ s.pJogJo uotlelufualu oql e,rrt e,r saolloJ l?LI^\uI 'tueJo 'IUAil pro.{ 'U '1 dnor3relnpotueql Euts61 -eql s.zlr^un11pe,rord [SgOt] 'ue,rrE eq Ulrrr;oordeqt 'e,rrlcedsted slql Jo JoA?Ueqt e,rrEol sI luelul Jo qrle{s e fpo ,{leuorsecco eql osneceg'euo ol lenbelueuruJolepqllr\ soclJletuprtelut Z x Z ile Jo les eql 'sr $rlt:(Z 'Z)'IS: 1 'dnort relnporueql esnqclq^\s;oordlueregtpe,rtt e^\retdeqcsql uI pue Z pue 1 sreldeq3Jo slFser eql Jo fueru raprsuoooJ
dnorg rulnpotr{laql uF Br}reds eql , uaIdvHJ
78
7. THE SPECTRA VIA THE MODULAR GROUP
G(r1,s1)is at most p(o), sincethe set over which the supremumis taken is f-invariant. For any integer n, we denoteby C(n) the upper circle which is tangentto the real axis at (n,0) and has radius equal to llt/5, andlm(z) : -r/5lZ is the image of C(n) under the transformationT(z): llQ - n). Hence,the radius of any semicircleC which is orthogonalto the real axis and intersects some C(n) must be at leastt/512. Moreover, for any semicircle C there exists Z e f such that Z(C) intersectsthe fundamental domain
o : { l z l >l : - i s R e ( zs) i } . Computations given in Ford and in Nicholls, which we shall outline below, show that either the radius of V(C) is greater than t/512 or V(C) intersects some C(n). From the remarks above it follows that in either casethe radius of V(C) is at least\/512. In particular,thereexistsa f-image of C : G(r1,s1) whoseradius is at least,/312. implying that l(o) > r/5. For the computations referred to above: We assumethat the radius of V(C) is at most ,/S12.By translatingby elementsof f, we may assumethat the center of V(C) lies in the real interval [0, ]1. From the geometryit thus sufficesto consideronly the semicirclescenteredat (0,0) and (j, 0). lf V(C) has center (0,0) and does not intersect C(0), then its radius satisfiesp > 2l'f5. The inequalitv2ltfs < p < t/5lz impliesthat A: (l - rlJS,ll'6) is inside V (C) and (l + | lJ5, I lr/3) is outside V (C); Z(C) intersectsC(l ). In the casethat Z(C) has center (+,0), the condition that Z(C) intersects the fundamental domain D yields p > J312. Hence, the point I is inside V (C) andp S '/5 l2 impliesthat the point (4/ 3. \/5 I 3) is on or outsideV (C). Therefore,C(l) and Z(C) intersect. This completesFord's proof of Hurwitz's Theorem. Nicholls [978] obtains the Markoff chain by continuing the argument given in Theorem l. A sketch of this follows: In the computations given in Theorem l, the semicirclewith center(j,0) anA p: t/SlZ intersectsany C(n) in at most one point; also,eachof C(0) and C(l ) is internallytangentto this semicircle.For this exceptionalsemicircle,o is equivalentto (l + JS) 12. Continuing in this manner,any semicirclewith centerin the interval [0, j] and radiusJ5l2 < p S J2 can be shownto intersectC(-l), and C(-l) is internally tangentto the semicirclewith center (0,0) and p : {2. Nicholls U9781then generalizesC(n) to the upper circle C(p,4;s), which is tangent to the real axis at plq and hasradiusequalto (2q2s)-r. Prescribinga certain infinite sequence{(pi,qi;s)}, Nicholls usesthe above type of argumentto identify the elementsof the Lagrangespectrumbelow 3. The next characterizationof Markoff numbersthat we shall consideruses the following definition. For any integer N > I, the principal congruence subgroupof levelN in the modular group is defined to be f ( N ) : { A e f : a l l = * 1 , Q t 2 : a 2 t= 0 , a 2 2 : L l
( m o dN ) } .
'srxe IeeJeql uo Jeluecslr seq qJIq^\ (g's) ot (6'"r)ruor; olcJlcluesreddn oql ',{youeuly e;o slutodpexg uoql 'J> V ctloqrsd,{q Jo srxueql olouopIII^\ (s't)g:,', 'e,roqe ueJoeqJ go goord eql uI sV '(t)J/*H I Ieuorlerrreql eJus pue ./ gr ot +H uory deu uottcefordeql oq z lol pue {O < (z)tut :z) : +H l:'-I .?tuluel slql Jo uorsnlcuoJeql sogsrleszg'& Jo euo'erogoreql '(i)J ) zg splel,{uorl€lnJluc e pue t ,(q elqrsr.rrpsl N 'sl teqt i(g pour) t : I I zN luql serydrur(Z) ueql '(€)J I 'w epl^lp lou soop € leqr lue^e oqt uI €r urplqo o^ (z) tuor; uoql 'g ,{q elqrsr.rrlp sl W JI'oslv 'J: (l)J f g'sr leql:1 : (g)tep pue uolo sI qree leql serldrur(€) uoql '(Z pour) tuonr8uocan 14(utg-nO+N esnprag 't, :
,I,I((nt
- a)ut, zN ,(tut nd)
(e)
eJUrS eq uec (7) ,(1t1uept'l-: was? uollIJA\eJ zn 'ztlewln '(e > 6) )l zg ro (e)J ) Brreqllo luql erord Ipqs 016 o1 pnbe enler eEuurEele,reqsnql pue 0 : (l 'x)*l Jo slooJ eql eJs qJIq^{
'utZl QtJt zu6)+ wt nT) en (rg Jo acueqpue) g yo slurodpaxgaqt leql olou elA 'Zl(WQat.: zzq 'wty'[- IT.q nT) 77) '(nE- a)yg- : ztq 'Zl(WQat- nZ)+ : ttq 1ur) solJluo0^3t{ ol Br xIJlsIu aql augep 0/l\ 'l:
,W(V
-
tu6)
-
zN
Q)
ol W'N suollnlosler8elutpoxg fue rog ' - 6) ot pnba en1e,ra8uerEel e^eq 0 : (l 'x)"'l Jo sloor ztJzw lt oql luql 7 ntdeq3 Jo (A)Z tueroeql tuor; Ilecor eM 'u/(L: I * zlt oroq/tr ' = r[(n;- o) + tx(n7 ut) + zxw (t'x)*l oq ol pougopsl *l ruJoJJoITBI I or{t '7 raldeq3 go (7) uorlenbe ur sy '(a.rpout) tur : xzta+ Jo uollnlos roEolul errtltsod1seel oql eq ol n eugepem 'uotlsnbeslql ol suollnlosJo (zu.t'tw'1a)eldu1,tuerog -f lu,r lw + ,ut .(uttulult :
uorlenbe eql ol suorlnloslerEelute,ttlrsodII? Jo los eql sI sJequnu 'dooud Jo{r?IAI Jo los oql leqt g reldeq3 Jo I tuaroeql tuor; Ilecor 0A[ 'ztt(zult 6) o7 pnba anlnt a?uotSoT sDq qcplu lo t4coa's7urodpax{ louot1DJtlolxt qltul 'Z vlllwil'I D s! ut (t)llo ruaulap uo stswa arary uaqt 'nqwnu [o4tory tt 'Br,uI'uel 8uulo1lo; oql esn o/v\{Jo^\ slql Jo uolssncslprno uI '(e )J Jo lesqnsuleuec B 'reuqel 'uopreeg s? srequnu Jo{r?t\I Jo los oql ,(;ltuept [qgor] uroEuteq5 dNOUC UV'INCOI^I AHI VIA VUJJ:IdS AHI
'/
7. THE SPECTRA VIA THE MODULAR GROUP
For any hyperbolicA e f(3), n(Ge) is a closedgeodesicon the Riemann surfaceH+ llQ). Sincefor any .B e f(3) and C : BAB-\ we haveB(G): Gc, we shalldefinen(Gt) to be a simplegeodesic if for any .B e f(3)\(,1),B(Gi doesnot intersectG1. ln Beardon,Lehner,Sheingorn of this is proved. [1986,TheoremLl] the followingsharpening Lnuun 3. Let,a e f(3) with irrationalfixed points. Thenn(G1) is not simpleif and only if thereexistsa parabolic^Be f(3) suchthat G1 intersects B(G,q). A proof of Lemma3 will not be givenhere. Lemma3 will be usedin our proof of the followingrefinementof Lemma2 above,whichis Theorem2.1 in Beardon,Lehner,SheingornU9861. Tnronrla 2. The real numberm is a Markoff numberif and only if there existsA € F(3) suchthat n(Gi is simpleand the irrationalfi.xedpoints of A haveLagrangevalueequalto (9 - +1az1t1z. Pnoor. The sufficiencyfollowsfrom the coincidenceof L and M below3. We denoteby S the matrix associated with the (parabolic)transformation S(z) : z * 3. For anyprimitiveI e f(3) with irrationalfixedpointsr, r and anyV el, SV (GA) lV (r) - t'(s)l 2 3 + V (G1)intersects <+G,aintersects V-tSV1G;. SinceS is a parabolicelementof f(3), the normalityof f impliesthe following restatement of Lemma3: (4)
n(G.q)is not simple + lV(r) - Z(s)l 2 3, for someZ e f.
If m is a Markoff number,then from Lemma 2 we obtain a hyperbolic I e F ( 3 ) w h o s e f i x e d p o i n t s r , . i hpa( rv)e: p ( s ): ( 9 - 4 1 m 2 ) t / 2S.i n c e misa Markoffnumber,eachof r ands is a quadraticirrationalityand lr-sl S p(r). For any V el. Z(r) and Z(s) are the fixedpointsof VAV-\ e f(3), and V(r), V(s) are equivalentto r and s, respectively.Therefore,p(V(r)) : p(V(s)): (9 - 4lmz\rtzand,asabove,
l v ( r ) - z ( s )sl p ( v ( r ) ) < 3 . Becausethis holds for all V el, from (4) we obtain that n(G) is simple. Cohn [971] gives an analogousinterpretation of the Markoff spectrum: Let Q@,y) : a(x - ry)(x - sy) be a form with integral coefficientsand irrational roots, ,,,.r.Assuming that Q is reduced,that is, 0 < -s < I < r (see Appendix l), we denote the continued fractions of -s and r by -S = [ 0 ,A _ t , e _ 2 , . . , f ' ,
r = lAO,et,...l,
Instead of using the fundamental domain D, Cohn considersthe domain
D o o: n { l z - n l > l : I m ( z )> 0 } ,
'e- =ltlv ',JBJrr:lr_V,r_glrl (S) 'relncrged ut '[rjp,'rjg] = Ir_V'r_glteqlqcnsJ > g
slsrxo oJeql t ?rurueT pue uorlrugep ,{q 'xuleru Jo{JeIAI-uqoJ e sl t, JI '[g reldeq3 '(g) '(Z)] ur ue,rrt suorl?leJelrsJncereql ,{q (t't'1) uorlnlos l?rlrur eql uro{ pourclqo ere suorlnlos esaql i[g reldeq3 '(l)l uortenbo sgolreyq ol suorlnlos IIB Jo tes eql sr srequnu Jo{reIAI Jo les aql wql Z reldeq3 Jo I rueJoeql uro{ IpceJ eA\ 'o,roqe sV 'cooud '{xlttout [otttDw-utloJ D s! V :tllv ttl] ps aqt st 7 nqdoq3 ur parapwuocuaqutnu[otltorylo ps aqJ '€ r rauoaHl
'lr!v' ' v u a r f' lv r l d : h - v ' r - . gpl u o 'r - ( g v ): ) ' J > J ' g nt|nut[oTtow-uqoJar,lttltll paruloossD a1d1t7 o s! (),g,v)lt .v vnwir-I B pellec oq il1,lr ()'g'V)
sp1eil uorlelnduroc ,{seouV 'V tltlm papaosso a1d1t7 e1drr1qcns Kue'y xuleru Jo{JeIAI-uqo3 ,tue.rog
f g I l : o , ' Ll zt - Il - rI : o " 'Ll r! rl rf : o n Lt- €J
eJoq^\'S pue g suorletedo eq1,(q pelereuet dnorE eql Jo '09'0V) uo Eurlerodoruo{ peur?lqoe1drr1ouos oleurpJo lueuele uB ^q (0J Jo -oc lsrg oqt su srncco ty 1t xrnew[o4tory-utloJ e peil?c eq illh\ U xrrletu eql
'(v{'{vv'v)
: (J'g'v)s
i ( r - B 't - v ' v g ) : ( J ' g ' v ) d ^q s puB
e1dlr1,{ue ro{ 'J ) g 'V A suorleredoeql eugep ea (r'g'V) s$Vteru Jo {q pelereue8dnorS oor3oql torcnwuln etqls! : pue V Jo &V lg,Vl r_{ r_V xulsru erpJo aJDQeql elouep (F,)Jl'J> y,{ue ro3'srrro11o; leq^\ uI ill,rr 'Q'dls: J Jo sluouelo3o seldr-r1;o les B ol sraqunu JIo{rBI I Jo les aql sel?ler [ssot] uqoJ ^\oq a\oqslsrg lleqs elil 'sruroJ Jo{rel I roJ s?lnruroJ uorsrncer eql pue ,filtuepl Jo{r?IAl eql ol Surproccepe^eqeqpecrlou ISSOI]uqoJ qcrqA\secrrleruuortnlrlsqnsgosdnort ur?uer peroprsuoc[q63t] e{orrC ':e1e1leqmeuos '[6rgl]go1:e6 ur pegode.r lsJU sea\ srruoJ Jo{r?I/{ Jo los eql l?ql pe^Josqoe^\ Z pue 1 sreldeq3 u1
' 1' l o o - ( z ) e u I
Iu^relur eql ur ooq seluel pue f > l(z)eUl 1e,r:elureqt ur ooq sretue (s'.r), 'sruro; qcns JoJ ',(lerueu!(s'.r)9 crsepoo8pelercoss?eql Jo .,serpedoJd1rxo,, eql segqueprosp [gg1 'd'IL6t] uqoC 'Z pue 1 ,(1uosur?luocecuanbespolerc -osseosor1,r\ surroJcrlerpenb pecnpeJJo ss?lcpreueE eJotu eql turreprsuo3 'f: (z)ut1ou{ FluozrJoqoql A\oleqsurerrrer€C ul (s'.1)g;o qled eqt gr r(po pu? € lsoru w s! O Jo onp^ Jo{rel I eql 'snq1 '(O)ur l(O)il JI j (z)utt 'G't)g > z roJ leql urclqo pue o : (O)ut leql eunsse ,(eru lp en'o7l(O)pf : Zlls -/l sl (s'.r)g crsepoeE oqlJo snrperoql ecurs :sluerc -geoc IBuorl?Jo^erl rIJrqA\(,('x)O go seldrllnru esoql ol luep,rrnbo sruJoJ crlerpenb oJezuouJo les eql pus oC ul (s';)9r crsepoeteql Jo se8eurr-1go les eql uee/y\loqecuepuodserrocB surulqouqoc '[0zz 'd '06gt] urou Jo {roa\ crsselc oql ol Euu.te;eg 'a s:etelur IIe Je^o ue{"I sr uorlcesJelur eql eJoq/h I8
dnouc uv.Incow:rHr vIA vuJJSdsaHr '/
82
7. THE SPECTRA VIA THE MODULAR GROUP
In Fricke U8961it is proved that for any unimodular matricesA, B (tr.l12 + (tr B)2 + (tr AB)z : tr A tr B tr AB + trlA, Bl + 2. (This can be verified by direct computation.) Combining this identity of Fricke with (5) above and the fact that tr(D-t ) : trD for all D e f, it follows that for g : (AB)-l (t A)2 + (r n)z + (tr C)2 : tr A tr B tr C.
(6)
A considerationof congruenceclasses (mod 3) thus implies that each trace is a multiple of 3. Settingltr,11 : 3m,ltr Bl : 3rfit, I tr C1 : 3m2, from (6) we obtain Markoffs equation, m2+ml*m/:3mm1m2, as given in equation (l) of Chapter 2. Extending the operation R given above to the triple (trl, tr B,tr C) 13,we observe that (7)
R ( m , m 1 , m z ) :( m 2 , m , m 1 ) .
Since tr(l-t n1 : azzbrr- tnbzr - az:bn * a11b22 : ( a r r + a 2 2 ) ( b 1- tr b 2 ) - ( b z z a z*z b p a 2 1I a p b 2 1* b r f l r r ) : tr A tr B - tr(B-r tr-r) : gmmr - 3m2, we obtain the analogousextension (8)
S ( m , m 1 , m z ) (: m , 3 m m 1 -m z , m r ) .
The identities (7) and (8) combine to give the Markoff recursion relations - trB6: trCo:3, the conclusion of [ ( 2 ) , ( 3 ) , C h a p t e r2 ] . S i n c et r l s Theorem 3 follows from the previously statedwork of Chapter 2. A. Schmidt fl9761,119771extendsTheorem 3 by replacing SL(2,Z) by a generalFuchsiangroup, I'. He finds that the Markoffidentity holds whenever F is any extendedFricke group. Continuing with Cohn's work on Markoff forms, for any Cohn-Markoff matrix I we define the Cohn-Markof form as -fe6, y) : a2tx2* (azz- arr)xy - arz!2. Tnsonrr4 4. Let A be a Cohn-Markoff matrix for which the associated Cohn-Markoffform is reduced,and define 4: ltrAll3. Then there existsa Markof triple (m,mr,mz) such that the Cohn-Markof form f1 is equivalent to the Markoffform f^. Pnoor. In Theorem 2(B) of Chapter 2 it is shown thar If^ are equivalent. Therefore, without loss of generalitywe may supposethat Ir A > 0. For any with l, by Lemma4 lB-t, A-tf : tB;', A;') = K. triple(A,8, C) associated Computation yields (9)
tr(K A-t ) : 6az,- rr A;
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7. THE SPECTRA VIA THE MODULAR GROUP
Tnronnla 6. Let P(lt,r) be the period associatedwith the Markoff number (p,u). If plu : lao,ar,...,axf, and pN-2f un-2, m with Frobeniuscoordinates px-tluw-r are theprevioustwo convergents of plv, then (with a(N): ar1 P(tt,u) : (P(pN-r,up-t))arnP(pu-z,uN-z), if N is odd, P(p,u) : P(ltN-z,uu-)(P(pN-t,t/N-t))ctnl, if N is even. Pnoor. We consider the case when pf u > l; the casepf u < I follows in a similar way. Defining{rc(i)} as in (13) and also K(i) : -[-iplul+ t-(, - l)plul, we note that each of the sequences{rc(t)},{rK(r)} has p' tt-ulplul terms equaltolttlul* I and Ltt: u -p'terms equalto lplrl. We also define R t ( p , z ) : ( l ) 7 6 101( yl ) r r z l0 . . . ( l ) K ( , )0 , R - ( t t , z ) : 0 ( l ) , 1 10; ( l ) , . r z0l . . . ( 1 ) , . r " r . Each of R1 has / terms equal to zero and p'(lttlul+ l) + u'lttlul: (tt'+ u')[plu)* p' : ulttlu]* p' : lt terms equal to one. Denoting the terms of R1 by R + ( l t , u ) :r ( * , 1 ) ,r ( * , 2 ) , .. . , r ( L ,p + u ) , we define two polygonal paths, {Rf : 0 < i < 1t + u} by Rf, = (0,0) and for i > l
Rl-r+ (l' o) if r(a'i) : o' Rl' : I l . R i , + ( 0 1, ) i f r ( + , i:)1 .
Each of thesepaths beginsat (0,0) and ends at (u,1t). If P denotesthe polygon formed by thesepaths, we seethat there is no lattice point inside P. From the geometric construction of the continued fraction given by Smith in Koksma [1936,p. 38], QIP is a convergentofplu if and only if (P,Q) is a vertex of P. Moreover, the sequenceof convergentsis obtained from the vertices on alternating sidesof the line y : px lu . From the construction of P , s i n c ep l u : l a o , a r , . .,.a N f , . f ( R - ( p u - r , y , v - r ) ) a t r , '-r(Rt t u - z , u N - z ) i f N i s o d d , R - ( u . z') = { t R - ( p N - z , u N - ) ( R - ( t t N - t . u N - r ) ) a t ui tf N i s e v e n . Sincethe period P(p,r) has the form P ( t t , u ): 2 ( l l ) a r p 2 ( l l ) " r z r ' 2 2 ( l l ) * @ 2 , ( I )r1,i, the conclusionof the theoremfoland R- (tt, r) : 0 ( I )"t rt 0 ( I ),tzr0 lows from the above identities. Further development of the ideas in this chapter is given in the recent papersof Haas [986], Haasand SeriesU9861and Series[1985],U9861.
98
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ALTERNATIVE DEFINITIONS OF THE SPECTRA
For any doubly infinite sequenceA, either L(A) : limsup;*- 7;(A) or L(A) : lim sup;--- li7). In the former casewe definea : fao,at, a2,. . .f, and in the latter casewe definea=lay,a-t,e-2,...1. In eithercase(4) implies that p(q) : L(A), so L containsall valuesof L(A). We next supposewe are given a real number a which has the continued fraction expansiongiven in (3). If we define the symmetric doubly infinite sequenceI by A - ...,b2,bt,b0,bt,b2,..., then L(A) : p(a) follows from (4). Thus, the set of all values of L(A) contains L. Defining M(A): sup,t;(,4),where the supremumis taken over all integers i, Markoff U8791proved that the Markoff spectrumM is the set of all values of M(A). The following is a summary, without proofs, of what is used from the classicalreduction theory of indefinite binary quadratic forms. More detailed accountscan be found in various textbooks, for example Chapter VII in Dickson 11929,pp. 99-l l1l. An indefinite binary quadratic form (l ) is said to be reducedif
0 . r l a U )- b < 2 l a l< 1 fd ( f ) + b ,
a c< 0 .
Equivalently, if for d : d(f) we let
'=-J#
a n d' : - J #
denotethe roots of ax2 + bx + c:0, l r l< 1 ,
then the form (l) is reducedif
l s l> 1 ,
rs<0.
THronru |. There are only a finite number of reducedforms (l) with integer cofficients and given discriminant d > 0. The set of reduced quadratic forms of a fixed discriminant d can be partitioned into one or more chains,as follows: We define, for any integer i, (5)
f,(x,y) : (-l)ia;xz 'r bix)t - (-l)'ciy',
whereeachdiscriminant d(fi) : d and 4;, bi ?rQpositive real numbers. There is a unique integer c; such that the substitution x : Y, | : -X * c;I takes the reduced form I to the (equivalent) reduced form fa1. (We recall that two forms are said to be equivalentif we can obtain one from the other by a substitutionof the form x : rX ]-sY, ! : tX +uY with ra-sl : +1.) In this manner a chain . . . , f-r, fo, .fi,. . . of equivalentreducedforms is obtained. Tnronnpr 2. Any two equivalent reducedindefinite binary quadratic forms of the same discriminant belong to the same chain. Becauseof Theorem l, the chains are periodic if the forms have integer coefficients. Defining 8i : (-l)ic;, ui : 6/A * b)12a1a1,and u; : 6 / A - b ) l 2 a i + r , t h e n g ; > 0 , u i > l , 0 1 u i 1 l a n d w e h a v eu ; : g ; * u - ; 1 1 ,
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92
PELL EQUATIONS AND AUTOMORPHS OF INDEFINITE QUADRATIC FORMS
Also, since "f(x,y) is primitive, s must be an integer. Defining the integer r by (6)
r:ct*6
and combining(5) and (6), we obtain 12: (d - o)2 + 4a6 : b2s2+ 4(l - acs2): 4 + ds2. where the second equality follows from a6=fT+l:_acsz*1, and the last equality gives (4). This provesthe ..only if' part of the theorem. To prove the converse,we observethatif f(x,y) is transformedby (l) into A X2 + RX Y + C Y2, thenA : aa2+ bay + cyi,' B' : 2aa * b (a6 + f y) + 2cy6, 0 c : a f2 +b 0 6 +c62. If we substitutethe equationsin (3) into theseequations on A, B, C and then use (4), we obtain A: a, B : b, C : c. Thus, (l) and (3) define an automorph, and this completesthe proof of the theorem. If the form f(x,y) in Theorem I is reduced(see,for exampre,either Appendix I or Dickson 1lg29,pp. 100-1021),then the following theorem shows that we can obtain an automorph by looking at the continued fraction expansion of one of the rootsof f(x,l):0. THronru 2. suppose the indefinite binary quadratic form (2) has integer cofficientswitha> 0 andis primitiveandreduced.Let( and4'with0 < 6i r and q < -l denote the roots of f(x,l) : g. Let the completelyperiodic continued fraction expansion of € be given by ( : where [0, dt:d;.:6], n is taken to be evenand a1,az,...,an is the shortist perfid. Let pilqr: Lar,az,...,ajlfor all j = 1,2,... denotethe convergents to ll€. tf we define (7) e:Qn-t, f:Qr, T:Fn-t, 6:pr, then the determinant of the transformationin (l) ls ad _ I and is an fy: automorph of f(x,y). Moreover, this automorph satisfies(li), where r, s is the least positive solution of (4). Pnoor. Since( : 10,dt, ct2,. . . , an,€-t7, by the theory of continuedfrac_ tions we have {-r : @,(-' * pn-r)l@,{t * Qn_)) using(7), t€2+6 -rr)€- P : 0. Thus,the form yx2+(6 -a)xy - y2: 0 has f integer coefrcientswhich are multiples of the coefficientsof (x,y). since n f is even,we have a d - B y : P n Q n --t F n - t Q n : l , so the transformationin (l) is an automorphof f(x,y). By Theorem l, the equationsin (3) hold for some integer solution r, s of (4). since this integer solution is associatedwith the shortesteven period of (, it follows from the theory of Pell equations that r, s must be the leastpositive solution of (4).
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94
REFERENCES
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