The Markoff and Lagrange Spectra

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Author: Thomas W. Cusick

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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 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

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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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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

'/ r.ueJoeqlur ^\olaq 'r,unJlceds JooJd rurequeddo aql o^r3 elA JoIJSI I eql ur sJeqrunulsoll?tuseerql eql Jo uorle^rrepaldurs p se^rEqclq^\ yoo:d slesspJaqt l[1yralduq3 [Zg6t] urequeddg ;o reded e ,(q perrdsurse,nr, 'tS6Il pue slasseJ.{q ue.,rr8sen goo:d oql uorsrel raldurs [6t6I] {etue1 Jo '[gt6t] snrueqoJC seepr 'lVZ6tl V Jo Jo esn epetu oq1( {eruo; ,(q uanrE sBir\suorlJu{ ponurluoc ,{o1due lou seop qJlq,t\ s |'ueJooql 3o ;ooJd y 'l elqel ur uanr8 ate'V socuenbes elrugur ,{1qnopSurpuodserrocoql pue 'tr4.lur sJeqr,unu ual aql lsallptus ' o o* e S E ( l i l W p u e ( \ V ) W o ^ e q e ^ \ ( " ''(,'l : l) > f Z f 'rea1c 'trAI tzl 1V : JoJ ecurs sr srql ur ueql ssal sJequnu aql go lurod lng Z T, € 'g eqt ataldruoco1 lFurl B sr leql ,{Jue,t,{1uopaau arrr rueloaql € So 3oord 'popunoq oJp '''';l 's 'l sroqrunuEurpuodserror aql IIe pue'sraqrunulenbe11e go dn opetusr "''(t)t'(l)s'(.l)"r secusnbes oql Jo ouo sdels Joqrunu pepunoq e JaUBleql eôrd e,tr ',{e,nslql ul Surpaacor4 ;o 'y > pue 'papunoqqloq eJ? 'y > gs,{1r1unba S S'pup s os -ur oql ol speol s.(l).r go srred yenbeeseql Surluno3 'peceld ,{1pculoturu.{s ere s.(r).rgo srrud lenbe qcrqm punoJe 'pa1e1osr eq uec | | Z Z uraUed u os

' " ' ' ( 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 ) " '

ruroJor{l sr rnJro (t + puu (ry)seraq,{\ Jo ry)s 's.(r).1 y ecuenbes oq] uorgod eql asneceq go pnba go srred Sulpuodsarroc Jo etuolsrxoeql serldrurs.(l)spnbago srrudeseqlJoecuolsrxa eql 'l *S:.f roJlou lnq,s . J o J( . r - r y ) s= ( t + r + ? ) s ' s r l B r { l: ( 1 + 7 ) s ' ( r 1 ) s r r e d > i { t oql ruou s.(l)s lenbago srredJo Jeqr,unurunrurxetueql elouep S luelsrprnbe '(qder8eredsnor.rrerd eqt ur euo eql se etueseql eq lou paou Z srql esJnoc 1e1 yo) s: (t + Z)s I t + s: (Z)s.{us'lnccos,(l)slenbeunl?ql esoddnseA\oS 'peqsrugore e^\ pue pepunoqsr s ueql's urlogeqtJo sl (/)s ecuanbas eqlgl 'pepunoq oJ? y pu? / qloq os

'(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

'I,\l ur raqr,unu lsolletus 'hl eqt roJ sruroJ crle.rpenber,r goo:d poqs fuer e ene8 IwAil qleeqJul I '[glOl] ll€d puu Jo requnu tselleruspJnll oql urplqo osle fug pue uosIrIC [glOl] ^\uqsuorello'[gSOt]KeJ'l616tl uos{rlq'[OSO1]sIA?C,{que,rt8uaeq eêq IAI ur sJoqrunulsoll?rusoû eql Jo suoIlBAIJepalduts Jaqlo 'I{ uI sJoq -unu lsellutuseeJqlaql sent8llnse: SutmolloJoql leql JBoIJst 1t 'seueutrutlerd eseqlralJv 'Jeqtunu[BJr p sr o pue rf go lueutulJJslp oq] sl 0 < (l)p anqll.

'

* r) : (,t'x)l rt(hpI - z(,tu

ulelqo OrrrOr?nbs 'l : (0'1)rf eurnsse,tetu a^\'lT = Jg - OY',tq + ,x3 eqt 8ur1e1druoc,{g : t ',[fl * ,xV : x uoIlBIUJoJsueJl Jslnpoulun e ,{q ueql 's.leEelut3o rted JoJ runrururur slr surelle I wrp erunsseo^\ JI 'l : ()tu tunlulullu eutrros oql leql os luelsuoc olqellns e Kq I ,{1dt11nuUEOe^\ 'tuntulullu oJezuoupu? sluerJgooc Ieer qll^\ (,('x)l tuJoJ JlleJpenb fueutq ellugopul uB UaAIC 'roldeqc eql ur 'sutloSctlerpunb JorlJeapesnsenbruqoaluolloe{ penulluoJ Jno ol }sPJluoJut Sursn sluaun8re ;o uorlerlsnlll pooE e sr yoord stql '[Zg6t] urtequeddg tur,nollog 'tr41ur sreqrunulselleruseoJqleql Jo uollellJap oslJuoce a,rt8lxau e16 'ernpefuoc eql rog goord snolcellu; e ane?lgt1il re3:equesog '*ut * w uo{l'E6101} Gut'ut)xert puB sploq (tt) qclq^ ro3 sa1dullJullslp o/tu ere pue zur'tut 'w Jr le\t pelord aq iernlceiuocslql JoJocuopllo lw'l*'*tu euos uenrSs?q [St6Il tlsoJog 'sploq (11)teUt qcns zu't'ttu'w IeJrrorunu 'spJo^\ Jeqlo sreSalurarrrlrsodgo aydul enbtun B sI oJeql ut uenr? oql JoJ ur lsu-rogcrlerpenb lueprrtnbs Jo slooJ eJesJoqlunuoseql ueql 'lr Jo senlel luoreJrp o^\l JoJsploq (t I ) pue uerrtt sl ut Jl leql ernlceluocol l?JnluusI lI 'l elqel ur uen6 (V)W lo sonl?Aeql ol pnba (o)z/oêqsnqt'rop:o ur uolel'sw:og eseqlJodrsloor eql:(€1)suro; '68'gtz

r 'r€'68

, 1 $ g - { x 6 6 1* 2 1 6 8

z'67.'691

I ' er

:[96]-,{x|tl+zxr6l : d t 8 t [ x L 9 € +z x 6 9 l

r

'tttZ-txl7g+zx€ez

,tp6 -

t{x97+ rxpg

'16r

re r 're (.'9'62

I'I'I

,"[-tx+rx

t 't'z

,[Z-[xr+zrz

t'z'9

,",(lE-[xE9tzx6., , [ 6 1- t x 6 7 + r x g 1 ,[c-{xtr+-xc

([' x)w1 rurogcrle;pen[

I'9'€l

7w'Iut'u uotlnlos

(91) surog crlerpenb Surpuodss:-toc eql pu? (I y) uotlenbeJo suollnlos 'Z irlsv1' IÎNUIJEdS IJO)UVW

3HI NO SI'INSEU UA(I'IO

'I

t2


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

:@)oy:gyyy?ir{} (il,yTf,i : (il,v 'u 'leql : ({V)W pue Z eturue'I ueql 'e3:21,{lluetcgns r4cee .{1dur (ry)oy rog aqt ur ta e:11et en gy '{fy} oq o1fuu11oroJeql ur ue,rtt {Q)tV\ eq o1,fue11o.roJ ecuenbesqnseql e)1eluec e^\ 'ty e\l Eurlleqeler,(g 'f:e11oro3 '9 euuol r.uo{rJBr ecuanbosoql urelqo eA\ pue pepunoq eJe tV oqt uI sJoqlunu eql Us 'a1rugeq ol pounsse oq uec ({V)W qc?oecurs './ 11erog (V)oV : (y)yg wr4t erunsse,{eu e16 'I tr ul ecuenbeslueErerruoce eq {({V)W} ta1 'uoou4 aLtJ 'g r reuoaHl 'ps pasop o sr utntpads [o4tnq

'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 'ûv'I'IodoJ tlqnop D apuap "''(;D'dr't6o' ur poprulsuoc "''rq'uq't-Q'

.(g)oU : : (V)W ,(peo1c ,(em (g)W segsrles srqt " - g ecuenbeseql 'l fueu ,(1e1tugul JoJ

qll/r\ Isslluepl s! I InUIJAdS IJO)UVW

EHI NO SI'InSaU U3C1O

'l

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) 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^\

JorlJSe S.UOJJodJO

oJB/r\?unsB^\ or.I^\ 'lzt6tl uos>lcef ,(q uarrlE sB1| '1 rueJoeql 3o Soord aqt go anbruqcel oql 6ursn '6 er,urue'Iur deE eqt Jo uorlulrJep lueJeJrp V '6 Brrrtue.1 go goord aqt salalduoc uer$ ',Q, l) Z t tt(Z l) : tV Kq,tV slql 'oo e I sB 0^, * (V)W ,{y-reeyc eugop o,r\ 'tr^tr lurod polslosru? lou sl 0o lsql A\oqsol ,(1uosuteuer 11 JI 3o 'op (V)W > g_Jl Eur,{gsrles > s e r r o r d s l q 'lo o : ( I l ' Z ' t D l < @ ) W a c u e n b e s , ( l u o e q Il sZI € t Z l : V 'l ,{.1reelcueqlt - r1:o I : ?JIosn?coq'elqtssodsl7,:1,{1uo'lana,noH < Z tE ruroJ eql oq lsnru wrtrts/t\olloJlI'0o < (V)yg sar.t? oruosroJ V l I Z Z Jo 'E'Z 'sesec qceo g Burrrutrre.1 uI Jo Jo $ rno; osoqlJo I'l Jo euo eq Z'Z Z'l '!o 'sJncco '7 rted eql qltm t+!D '!p lsnru rted oql uoql 1 uorJ luoJe:Slpt+!D " ( l ' T ) ' g u r e l l e de u n 1 1 e d ?s n q l u o r u o sr o J lsnu 1 I ' 7 ' g | J I ' J n c c o '/ ur JnJOu o e c7 ' 7 ' E r o t ' Z ' e urouedou'uorlelnclecJ?lllulse,{g '0o : ([l'Z'tDl < (lto'l'tl)l < @)W '!, eluos JoJ eleq o^\ etutuol ,(q ueqr esngceq'7 ur lncco uuc g'1 ro 8 'E+v puer rr,{lrselJ 'oa} (v)w > > 1'g uralledo1q l l ? r o J e EJI "''ttrt'0o 'I-D'"' leql qJns ecugnbesollugul(1qnop e sl V esoddng 'uorpunJ ',-r - 0D ls'I : qpue (x)l ue rc turseercur * ZT,lG9+ EIO 'lceJJocsI eluluol eql ur uerrt8 'de8 aruosgo lutodpue lurodpue -reddneql leql fgrra,r ,{po peeu oa snql re^\ol oql sI €l/ l?ql L Btutuel;ogoord egl tuor; reelc ,(puarlesI lJ 'Joodd 'W to lulod Urull o st raqwnu sIW:l Z EE ?, | : V tol tluo : (v)w aôq al4 'utruucads [oqow aLfiut do8 lowlxoul o sI zzlGg + Eld

( r r e q q, .ses s 0 9o. €( ()s q +rtd+,ELt) '6 vr{wa'I lo\ralul aqJ IÎNUJ]EdS JJO)UVW

EHI NO SI'INS3U UEC'IO

'I

t l

0: tu + lu t + xe w tu t7 ,x - ( x) l uorlenbo crlerpenb oqJ 'sooud ur sta&atutaaqt aqJ'(I'I'l)lo

'sttod ut awud {1an1lo1at ato uoqnlostfuo aoqqSou 3ut4o7rQamssaccns [q paulntqoatp

zutuwe : lw _llut + ,w nSaqt aa4tsod arylo py 'l rrauoirHJ lo (w'ttu'ut) suouryos '(Ttu'tut'az)uorlnloseqlJo uoqq?tau(7) suorlnlos eql IIEoel[ 'ey - ty111tg: lul

'lt,,u- |unilg - t,ut

,ut - cuttutt ,ut

(g)

Surugepfq (lw'tut'ut)'(7w't,ut'u.t)'(Tut'tut',ut)

G) suorlnlos

Jeqlo eeJqlurclqo eA\ueql'(I) Jo uorlnlosrelnEursuoue $ (es'tut'ut) JI 'suorlnlosrelnEursoql Jo euo twl : ut eleq o^t pue'Z to sr snql'twl1 : lsnutu I ZI I z4 sarr6(1) ueqt 'rut sepr,rrp '|ut: tw Wql '[es'esoddns'poopul 1nd emJI pue lw ueq1 'lcurlsrp lou eJ? zut pue'tLu'u,t qclq^\ ul (l)Jo suorlnlos,(Fo oql oJ?oseql 'suounpsnln8uts eql (I) Jo (I'I'T) pue (I'l'I) suorlnlosoql IIBcol[

''''

E E z ' v 66l '9 16' g ' r e6' z t' l ' s ' 2 ,I'

qltm surteq sJoqrunu 'saqunu Jo{rBr\I;o ecuenbesoql leql eos olr 1 roldeq3 Jo Z olq?I uroq [oryoq eql uorlenbe srql Jo suoqnlos ur Jncco uec qcrq^\ sro?e]uroql flec eA\ .ztttrutuE : (t ) lut * [ut + ,ut Jo suorlnlos re8etur errrlrsodaql lnoqe uorleuJoJur oruos errnbe: e^\ lsJrC 'roldeqc luese:d oql Jo t Bruruo'I puu g qEnorql Z srueJoeql tuo{ s/r\olloJ'e:otu pu? 'rueJooql slql 'I reldeq3 Jo 9 rueJoeql ,(1euuu '[Oggf]JoITBI I Jo rrorueru puocos eql q llnseJ ureru eql Jo goo-rde urelqo o1 sr reldeqc srql ur esofund rn6

srurodg0{TBIAIpuu srequnN go{rstrAl z u:rJdvHc

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.

aaur {Joxrvru aHI'I axn9lc I'l

c

!

l99L'16l'€l =n6l

_\

l

'l

l

c

't'l's

L6gz'r6t's

\ \,,.

Ltrglr'sllr'rl

,r,r(,,

\

/

g z t ' t' n E ' € t

w g r s ' s t , r ' e t' - ) 'LLo6'68 szgtzvT, LLo6'68'ot sgLsz6'LLo6'nE'---> \

I'l

/ - 68't€ 'r

'tzzg'E'z ,rr,r( 6WgZn'Org'EEZ\ -

oI9

/ ' € € z' I

/65I'0t9'l

shtdot clovuvw cNv suEslInN cJovuvw

'z

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

(,('x)"T : Q(@- tuE) a xut'[(0, - nt) + xn)u'l

(f t )

sorlrluoprOql e^Bq 0lA 'lo,z{l ? lozl pug 0 ) 020[

(q t )

leql lsrg esoddng 'g : 070[ l?ql /r\oqsol seJ$ns tt (C) Ued arord ol 'e,IoJaraq1 'w : (ut'n)3 : (0,('ox)u'lentE(71) pue (g) uodnereqr'r'w : o[ pve eJew pue lt oculs ueql 'g : o7 n :0x eûq lsnu e^\ '(9) ,{q euud '(1olt1u1er - 016 J I p u u 1 1: o 1 , ( 1 u r e 1pdu e l x u t : ( 0 ' 0 x ) 3: ( 0 ' 0 x ) ' ' t ' ( l t ) , t q ' u o q l ' 0 'lutulultu sI '0 - ozot JI l0zl + l0/l JI JoJ tttt: b leql ooso1 fsee sI lI ueql 0r('0t6teqt esoddnsroqunJ,{eu a16 leql os uesoqt ueoqeêq (91)Bur,(3sr1es '\tn - 0yy1 - 07

'tn - XUl = z

ougoq 'ueroeql eql Jo (g) ued Jo osn?roq

(sr)

01b=(0,('0x)"'l

osoddns,{eu orrrpue'b : l(}t'0v)utl I tuqt qrns 0,{'0x sJeEeluroJe eroql 'sluercseoc re8slur seq (f 'x)f ecur5 'oJoz t{loq 'x)*ll lou t'x sreEelurII? JeAoua1e1l(rf lo runrulullu aqt atouap D 101 '(9) 'tuoroeqleqr (g) uua e,rord(1 I ) pue (6 y) suorrenbo Jo ',{pelturtsperrordsl (tt) i(61) sa,rord(gt)pue (71)tututqutoc Jo osneteg

'(,tu- xut)fg * (t' x)B : (t' x)wl

('r)

Ocurs '(t'x)t8zute+ (tn - xu)[ = ((ttn - xrut)n- (t(ra- xtn)w)(,(rn- xrw) (gt ) '(t('x)t7qtt *

(f ) pue (9) ,{q'os1y - xtn)B (Zt) Q('x)3- : (ttn - xtt'u'[.ttu

(s)pue(s)

,(q l?ql e^JesqolsJg e/rr'(91) o,rordoI '(t'x)z?t1111a- *Q('x)wt-: I)€

'l ?tutue'I uI peugopsl I+:

QGn- xzut'tza- xzn)*!

3 eJO{^t (lt) pu?

(t'x)t?(,ut(a+ I)€ * Q('x)wl- : (r(tn- xtut'tta - xtn)*l

(Ot)

leql errord llsqs ea ueql '(Z'l : !)

: zfia + txtnZ ,xltu (t'x)t8 'eta+ txnTcxw: (t'x)B

eugep s,n 31 'ssqdde 1 eruutel snqJ '(I)Jo uollnlos relntutsuou e sr (zv1'tut'ut) lerll eunss? ,(eur em os '7 pue | : Ltr ro; pe4ceqc,{1tsee sl^tuoJJco)uvw cNv sdSswnNlJo)uvN

'z

22


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.

' ( t - u 7 4 ' " ' ' r r l ) X: o

. ( t - u ( 4 , . . . , r t 1 ) :X n

. ( t - u 2 4 , . . . , r 4 ) X: * 'arcwlaqunr

'lu4l' ' ' ' 'bl'01 = *p tDS 'st!8!p Jaqwnu uaâuD soq pouad asoLliluot1cotl lo panuvuoJ o sDq 0: (1'x)wl {o wD toor amilsod aW uaqJ 'uttol[o4to1rg o lo (V) uotuu{ap aqt m sta&atutaW aq tu'n 'Z < w p7 'e r^rsuoaHl 'n+nz-ut <w pue l : 7 ( n - u ) - ( a + n 7 - w ) u / e s n B c e(qn - u t ) l u t n 1 uorlce{ ponurluoc eql JoJ sploq etues oqJ 'culeturu,{ssr n f ut tog uorlJe{ penulluoc oql lBql e,rordem ,(eansrqt ur Surnurluoc,{q pue 't-uzD : zn setq? (Ot);o puelsul(19)tutsn luetuntre otueseql 'uro : lr se,rlfl(gg) 'erogeroql . ( z - u z D , .. . , z O ) X- ( t - u z p ,. . ' , t O ) X

0z)

eêqlsnu e^\'snlnpou oql wql sselpuu e,rrlrsodore ecuenrtuoceql ur sJoqrunuqloq eJurspu? (t-u(.D''' ,'eO)Xpout

( e - u z q ,' . . , z o ) X - ( t - u 1 . p ., . . , € n ) X

snqr ' ( z - u z D" ' ' z O ) X

I (t-uzr'"''Try)X'ZD

( t - u e D.,. . , e n ) X+ ( t - u e 7 , . . . , l o ) X r o

(OZ) JO

( u z o ' " ' ' z p ) X- ( t - u z v ' " ' ' t n ) X

(Ot)

pue lt: os 'r) : eêq lsnu ea\ eruulel aqlJo SOseqtodfqeqf r(g fi tU '(t-uZp< "''Zp)y : p u ? ( t u | n ' " ' ' t D ) X , f t , l , u AI0rusu '/l ) - fl,M ,/t'ttt ) ,ut pus I tnw leql qcns /n pue pt sroEelure,rrlrsod pug uec eA\a\oN 'l < n esoddnsu?c e/r\os 'l - lt Jr Ier rJl sr eurtuol oql pue ('zp'"''ln)X : u t os '(uzo'"''|o)X: n

,lrzo,..',1o7:nlut sr sluerlonb lerped go Jeqrunu uele uE qlll. nf ut JoJ uorpe{ ponurluoc aqt esoddng 'g xrpueddy Eurlpsuocfq pegue,req il? usc eJeqpesns?lnruJoJluenurluoceql 'dooud

ut

. ( 1 o . . . ' r l - u ? r u or u ? r . . . c T Dr l o ) x :

A

'(Zn'' "'l-uD'ilO'uD t "' ttO'7n)X :

. ( l D ( . . . , l - u D r u Dr u v . . . . . t D , z 0 ) X :

( e O ,. . . , | - u 0 , u D , u D . . . . . z D , l O ) X :

n

:plot!solnwtoltuonurtuoc 3utuo11ol aqtrcql qJnsuo'' "'tD sa&alu!attytsod anbrun$ua alaqt 'Jaôanry SIÎUOJ JIO)){VW

'Z

CINV SUAShINN IIO)UVW


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 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


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)

eûqe^\0 t l

'eroJereql

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@ ) o 7 t @ ( ) o> ( u ) o Z senr8qcrqm'u pue 7 sretelutentltsodII3JoJ

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(0s)

au)(u)o

leql os

-*' = o .. _u dhs (u)o

= : \u)o ,

D ' UB (/ )r l

'l

Ougep e1l\1l\oN < 4 sroEolur il? pue oatsretalur IIU JoJ

swuol J.{o)uvhl cNV sussl InN llo)uvw

'z

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@) b,q,o)

(ss)

/' (c o ' ) ' D ) + uorlcrulsuoc Surqcue:q eql ol sdeu 1 ernErg Jo oort Jo{rBIl{ oql saugep qclql\ (g) ,+o

( zu w E 'ut'T ut)N ( tw - r,yy

t u,y) . t .)\' ,( zw, ( t u t u t t ' t a ' t u ) - ( a u t- t u t u tt'E,y' t' u ' ,r

r , t ,w)

(rs)

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L

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'(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 ' "'

E 6 T , ':El t 7

e 1 T ,I r Z I Z z t t Z I 0 l + l Z t t Z 2 1 7 ) : r n

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 eôq? 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 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 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 (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 oêq 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)/ 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

eêq e^\ 'l + X:

.f roJ (g) qtl,tt 0: !'X:

'zls>t-(v)oy>T.ls-

U) f roJ (E) Surutquro3

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\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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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 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,

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z t t * z t z 7l, lzll*zlzz

(rt ot ful

*77

( S I

I I - Z I I

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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 bâi 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

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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

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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 ltl a lttvl+ 7

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4. HALL'S RAY

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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(g) '(Z) sotltlenbaut ,(1durr((.t) '(*Z) ',{lerrtlcedsa:) ', eql pue go rred aq1 'esecJeqlro uI 'uolo sI / +.? JI '(.9) ltpnbeut osJeAeJ < 1 " ' ' z + l + p z + ! p ' r + l + p 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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'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\ ?ôlêd ,(q peuretqo s?A\pue /t\oloqz tueJoeql sl slql zl7,l ecuenbeseql qcrqa ur urrupeds eql Jo uotuod oql uo ulolsJegJo poqlour eql 'secuenbespeÎJop 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 eôq? 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 eêq 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

L9

'9

68

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 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 ueô'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 , 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'

" ''to)'I

('o'"''to\

3

t

('o""'tD]'

"' "'r', s11t-71(Z'tP'

3 ( ! D ' " ' ' 1D l

: tttr-zl\'rD'"''to)7 3 tle-e)(t+{D'"''lD)T 3 a^sq a^l sle^Joluroseql JOôSunuuns 'oslv 'g ueql ssel sr rlcee qf8uel aql oJeq^\ 3o ' (uD' ' ' ' 'to)o[ sle^Jalur eql Kq paJeloc snql sr uorl?JeprsuocJepun les oql rc 'e> (to'' '''o)7 "tl:1i:j;'i::;ii::*ns

'{l

'0< e pexs srslxe eroqr roc

: ([O'"''tn)7 I o1 lenbe q€uel pue t-[+ub1 {+u6 ,t+u6 t={+,'+rAT7W llU slurodpuessq qJrq^\ ' l g ' 0 ] }: ( { o ' " ' ' t 7 ) o f

< 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) IIBJeôuoruneql 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

I

,ffi:td-al

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

VUIJ:IdS

AHI gO SUNSV:IH gHI

'9

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 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^\'eôqB 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
.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 eêqe/{\(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.zlrûn11pe,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 ) } .

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,I,I((nt

- a)ut, zN ,(tut nd)

(e)

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'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:

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ol W'N suollnlosler8elutpoxg fue rog ' - 6) ot pnba en1e,ra8uerEel eêq 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'INCOIÎ AHI VIA VUJJ:IdS AHI

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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 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?Joêrl 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 Jeô ue{"I sr uorlcesJelur eql eJoq/h I8

dnouc uv.Incow:rHr vIA vuJJSdsaHr '/

82


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;

'secuenbesrllnouJegeql pue spoued srq ueol\loq uorlcouuoc srql azru?oceJtou plp Jo{r?tr\l ,{ltuueddy 'IIInouJogol uteql polnqlrlte pue 'spoued {(l)r} secuenbesosor{lqlt,r pe{ro^\ [Zggt]Jo{reI\l JoU?I I Jo uorl -rsodelxnfeqt uo ruoroeqllxou oql urelqo ot {(l)r} sluouodxeeseqlJo tuJoJ prcods eql sesn [ft6f] uqoC '0grJo srenod eql Jo runs oql ol erurrdoc sr 0t'Jo sJol\od eqf ;o tuns eql qclqa ur 0Brpu? otl Jo sJoaod;o uotltsodetxnf z sl V 'sr l?ql '.09 '0V ,{q pelereueEdno:E oor3or{l ur pJofiraultutttd e s! y 'sdnort eerg leql setunsseeq uelqoJd srql ol slFseJ srq ,{1ddeol repJo u1 uo slpser preuet go uorleclldde ue se uerooql srgl surelqo [Zt6f ] uqoC '1n)xJA"' 17)xJOfi)xJA: V aôqall

'rt'''''Z'l: !

' V l n \ - 1 ) l- l r r l n t l : Q)x

( gt )

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iV t1: /tl-

rzpg

,(1durol ourquoc (Ot) pue (0) 'J > pue'gt-vt_g: €r oculs t-vx'lo|?lnruruoc eqlJo uorlrugopeqt,{g ' V r t : ( t t n + z z o ) z t q t z- q ( t t o a z z n ) z z q t t: q ( B ( O t) , _ V, _ A ) t t €8

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84


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 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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(l)

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l"''z-tO't-!O'01+ [''''t+t2'lDf = (V)!V

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(r)

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urpeds orll Jo suopluuec e^llsurellv I xraNaddv

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. 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 ,

'V)W pue urorC Jo senlel IIe Jo les oql sI IAIl?ql s/dolloJll € tuerooq-1, (9) 't sta&arut Dô uayzt s! wnut{ut aql ataqfli '!DlJtJl: (I)W ilo 'nrcatory'{r n&alut :to} Ps ary lo uatil 'uloqr ary lo utto! tuo s1 Q{'x)t t1 ruaulap uD sI vIDI4Jarfl u! utto! o r(q [padotd pa\uasatdatst rlttrlu Tlgf '0 qtlM p < luoulultfislp uoql ssalDqunu qcoalo aryDârupsqo aW ua$ 't hlauoaHl sutto! uuntponb pacnpatlo utDqr o uttot $) ut swtol aWtI 'U)ut pocnpor lo enle^ eql qtr,tr (l('r)/ ol luop,"tnbo sturoJ etue$e1Jo ueJoel{l Pc Jo ur?qJ e Jo sJoqrxoluJo sluelcseoc eql slceuuoc -rsselcturrrrolloSeql 'o : (ot'ov)t leql qcns of '0x sreSelutoruud ,(1e,ttle1er lsrxe ereql y [.padotd lr requnu IseJ eql ruasatdatol pIBs sI ( I ) IuroJ eql .1...,2-t3,r-t3,01*[...,r+r3.t'l:t+!ofpv:!1aln (g)

. 1 . . . , 2 _ t B( t _ ! g ( g ] _ ! 1 1

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t ^ , t L l -

VUIJAdS gHI CO SNOIIINIJSC AAIIVNUAI1V

68

u>ul>lroJ .(uO.....Z+ut7)X(t-ut7....,rO)X+(O(....t+ut7)X(utqc'..'tO)X=(un'"''tn)X

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( t O , . ' . , t - u O , u O ) X:

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(lp)y

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(l)

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(r)

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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).

€6

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94

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96

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68

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The Markoff and Lagrange Spectra

The Markoff and Lagrange spectra

Lagrange-Punkt

Lagrange Five

Lagrange multipliers and optimality

Spectra

Lagrange-Punkt

Purity, Spectra and Localisation

Reduction theory and the Lagrange-Routh equations

The Geometry of Hamilton and Lagrange Spaces

The Geometry of Hamilton and Lagrange Spaces

Vibrational Spectra and Structure

Lagrange- Punkt L5

Spectra of atoms and molecules

The spectra and dynamics of diatomic molecules

Spectra of atoms and molecules

Linear Operators and their Spectra

Atomic Spectra and Atomic Structure,

Spectra of Atoms and Molecules

Interpretation of the Hydrogen and Helium Spectra

Differential Systems & Euler-Lagrange

Inverse spectra

Exterior differential systems and Euler-Lagrange PDEs

Constrained optimization and Lagrange multiplier methods

The Problems of Lagrange and Mayer Under General End Conditions

The Overlap of Affective and Schizophrenic Spectra

Generalized Sturmians And Atomic Spectra

Random walks, boundaries and spectra

Generalized Sturmians and atomic spectra

Generalized Sturmians And Atomic Spectra

Atomic Spectra and Radiative Transitions

The Markoff and Lagrange Spectra