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ROLE OF THE RATIONALS IN THE MARKOV
AND LAGRANGE SPECTRA

Dissertation for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
YUAN « CHWEN YOU
1976

 
    
 

Lms‘g 113}!

Egg. ’13! 0’3 E2 (‘mm

 

This is to certify that the

thesis entitled

ROLE OF THE RATIONALS IN THE MARKOV
AND LAGRANGE SPECTRA

presented by

Yuan-Chwen You

has been accepted towards fulfillment
of the requirements for

Ph ADJ degree in Mathema Li CS

//r4 J? Law

Major professor

Date _Augu.s_L3.._1316_

0—7639

 

 

ABSTRACT

ROLE OF THE RATIONALS IN THE MARKOV
AND LAGRANGE SPECTRA

By
Yuan-Chwen You
For each infinite sequence of positive integers g = {xi},
we let
. = j_1 l l
[x0,xl,x2,...,xn] xO +lx + x +...+ x ,
l 2 n
[xo;xl,x2,x3,...] = lIm [x0;xl,x2,...,xn],
n-HXD
M(€:") = [xk;xk+laxk+29 '0] + [0;xk_l,xk_29--o]l
ME) = SUP “(ask)y LUZ) = lim M(€.k)
k k—mo
is known as the Markov spectrum and the

The range of M(£)

range of L(£) as the Lagrange spectrum.

is shown how to construct rationals in the

In section I it
difference set E2 9 E2 and the sum set E2 9 E2 by purely periodic

pairs in E2 3 E2. Non-denumerably many pairs in E2 3 52 are found
for each rational so obtained in the difference set. It is also shown
This shows that

that such rationals are dense in the difference set.
there are infinitely many rational values in the Lagrange spectrum.

Yuan-Chwen You

In sectionll, it is shown how to choose an interval of the
complement of the Lagrange spectrum containing uncountably many
points in the Markov spectrum. In fact a non-denumerable set P of
points in the Markov spectrum and not in the Lagrange spectrum is
found. Moreover, every point in P is a limit point of P.

In section III, the dimension of some level sets correspond-

ing to small values of the Lagrange spectrum are shown to be small.

ROLE OF THE RATIONALS IN THE MARKOV
AND LAGRANGE SPECTRA

By

Yuan-Chwen You

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Mathematics

I976

© Copyright By
YUAN-CHWEN YOU

1976

ACKNOWLEDGEMENTS

I would like to thank my parents for their patience and

support during the completion of this work.

I am deeply grateful to Professor John Kinney for his
guidance and invaluable assistance in the writing of this thesis. I
would also like to thank the Department of Mathematics at Michigan

State University for their financial support for my graduate studies.

Chapter
I.
II.
III.

IV.

TABLE OF CONTENTS

Page
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . l
RATIONAL APPROXIMATION . . . . . . . . . . . . . . . . A
NON-COINCIDENCE OF MARKOV AND LAGRANGE SPECTRUM . . . l9
FRACTIONAL DIMENSION OF LEVEL SETS . . . . . . . . . . 28

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . 37

 

 

 

all»! a ... n. I»... .I. II:
D IFJ .. ’ p i 'r «(a’.. .‘H..wn ., n

CHAPTER I

INTRODUCTION

The Markov and Lagrange spectra are defined as follows: For
x E A, the set of all sequences x0,xl,x2,...,xk,... of natural numbers,

we let

, I I I
[x0,x],x2,...,xn] x0 + 511+ genie LK—I

n

[x0;x],x2,x3,...] = lIm [x0;xl,x2,...,xn] .

n-NO

Let D be the set of doubly infinite sequences of positive integers

_ . . . j _
E - {xi}-m<i<w' Define the Shlft transformation L E - {xi+j ‘m<i<m,
j = :j,:?,... We let
I) M(g,k) = [xk;xk+l’xk+2""] + [0;xk_l,xk_2,...],

2) Me) = sup ”(€.k). Hg) = TENN)

k k+m

The range of M(E) is known as the Markov spectrum, the range
of L(g) is known as the Lagrange spectrum. The first function arises
in the study of the maxima of binary quadratic forms [2], [8], [IS],
the second in the approximation of real numbers by rationals. O. Perron
[2] noted the connection between the two and showed that the lowest
parts of the spectra are a discrete sequence of values approaching 3
from below, and corresponding to the numbers most poorly approximable

by rationals. He also noted that the range where the xi are

restricted to l and 2 is always less than the range where threes
appear in the expansions. Hall [l3] showed that the sum and dif-
ference sets of the Cantor sets formed by continued fractions allow-
ing l,2,3,h as partial quotients filled an interval, from which he
deduced [l9] that the ranges take on all values above 5.

The work of Hall was difficult to extend to more restricted
sets of continued fractions but [l8], [IA] and [4] showed that the
lower ranges of the spectra are relatively Sparse, in fact, the
portion below V75. is of measure zero. It was recently established
[6] that the Lagrange Spectrum is the closure of the range over
periodic sequences. The Markov Spectrum was known to include that of
Lagrange and also to be closed. It was only recently established
[ll], [5] that the Markov and Lagrange spectra do not coincide.
Kogonija [20] gave a sufficient condition for the two to coincide
above /TO, namely that the lengths of the repeated 12 blocks in the
continued fraction expansion be bounded. The sufficient conditions
for the two to not coincide was given in [6] recently. Some of these
conditions are shown [5] to be necessary in certain intervals. Never-
theless, this theorem [5] does not insure the existence of a non-
coincidence in these intervals. Hightower [l5] found countably many
gaps above 3.

In section I it is shown that the rationals are dense in the
difference set £2 9 E2 and sum set 52 e 52, where E2 is the Cantor
set of continued fractions with entries l,2. It is Shown how to con-

struct rationals in the sum and difference sets by purely periodic

pairs.

A point w E £2 with continued fraction expansion

[a

O;al,a2,...] is said to be purely periodic if there exists a non-

negative integer k such that ai = ai+k for all integers i. Two
distinct pairs in E2 o‘Ez are found for each rational so obtained

in the difference set. An interleaving technique is developed Showing
that the set of pairs in E2 3 E2 whose difference is such a rational
is non-denumerable. A way of finding another rational In the difference
set if one rational in the difference set is known is also given.

Many rationals in the sum set can be obtained from the difference set.
The hope here is to extend the methods of [l8], [l4], [A], to restrict
the dimension of the spectra in the range [3, 2/3].

In section II, the non-coincidence of the Markov and Lagrange
spectra is considered. It is shown how to choose an interval with un-
countably many Markov points which does not contain any Lagrange
points. In fact a non-denumerable set of points in the Markov Spectrum
with no limit point in the Lagrange spectrum is found.

In section III level sets of the Lagrange spectrum are dis-

cussed. The dimension of one of those corresponding to small values

of the Lagrange Spectrum are Shown to be small.

CHAPTER II

RATIONAL APPROXIMATION

 

If w€[0,l) we define Tw=(%-[%] If wa‘O
0 If w = O
an(w) = [ nll J n = l,2,... where [x] is greatest integer < x.
T w

p_,(W) = 1. p0(w) = 0. q,l(w) = 0. q0(w) = l

pn(WI = anIWIpn_IIW) + t%_2(WI. anW) = anIW)qn_IIWI + qn_2(w)
i.e.
pnIWI
m = [0;31(W),32(W),ooo,an(W)]
n

w E 62 if and only if an(w) = l or 2 n = l,2,..

If a and b are the words alaz...an and blb2"'bm respectively, we

define a.1 to be the inverse word a a ... ,
n n-I 2 l

a ...a b b ...b ,

a o b the composntlon word al 2 n l 2 m

La the left shift word a a ...a a ,
2 3 n l

n n times
a = aoao...oa .

 

5 = [0;a ,a ,...,a J = . =
1 2 n [O’al’82""] where an+j 3j

and we define bxw = [0;b],b2,...,bm,al(w),a2(w),...],

If P',P2,... is a sequence of words, we define

.3 Pi = Pl a p2 a... = [0;PH,P]2,...,PIn ’PZI’P22’°°°’P2n ,...J
l-I l 2
where Pi = P lPi2'° PIn

Remark 0.0. al(w),a2(w),... are the partial quotients in the continued-

fraction expansion of w.

(0‘1) Pn_l(w)qn(WI - pnIWan_l(w) = (-l)n n >_

O

<o-2> = I": I)

pn_l(5) pn(5)‘

”’3’ "a= qn-,(5) qn(5)I
_ t _ ’pn_l<5) qn_,(5)i
(o-A) Ma_l — Ma - Epn(a) qn(5)

since (AB)t = BtAt for any matrices A and B

t
and Mk — Mk

 

Lemma I. Let a be the word alaz...an, x = 5 and y = La.I . Then
qn-I(y) = qn-l(x)’ pn-I(y) = qn(x) - anqn_'(x)
qniy) = pn-IIX) + anqn_l(XI and pn(y) = pn(XI - an(pn_IIXI - qn_2(X))

Proof. By preceding remarks,

_ - n I 2p _ < i q _ ( )
La-I a; 3-] an = I: oIIpZolox q:(l)x II? Ln)

3
II
3
3
Z

(-8” l)(qn_IIXI pn_IIX) + anqn_l(XI)
I 0 qn(X) pn(x) + anqn(x)

= (anX) ' anqn-l(x) PnIX) ' anpnqixl + an(qn(x) - anqn_l(x)))

q (X) pn_l(XI + anq (x)

n-I n-l

Theorem I. If x = 5 is purely periodic in E2 with period

 

-l . .
a = a a ...an, n :_2 and y = La , then x - y IS a rational and

l 2

x - y =[pn_](x) ‘ qn_2(xD/qn_l(x) = [0;al,...,an_l] - [0;an_l,...,a'].

Proof: Let y = :E—E—ZE . By the preceding lemma,
Q = an_l(XI. B = pn-IIXI - qn(X) + Zanqn_l(XI

o = t<pn_,<x) - qn<x)) + zanqn_,<x)12 + Aqn_l(X)[pnIXI - anIpn_,(x) -

qn(X) + anqn_l(XI)] = (qn(X) - pn_l(XI)2 + Aqn_l(x)pn(X).

But x =(pn_l(x) - anx) + /D)/2qn_ (x).

I

Hence x - y =[2(pn_l(x) - qn(x)) + Zanqn_](x)3/2qn_IIX)

=[pn_l(x) - qn_2(x)]/qn_l(xl,
Corollary I. If X = :E—E—ZE is purely periodic in E2 with period

a = a a ...an where P,Q and D are positive integers, then

II
N

La.1 and a

II
*
fl
0.!

II

-h
'h
m
II
II
—n

La.I and a
n

Proof: x =[pn_l(x) - qn(x) + /Dj/2qn_‘(x) for some positive integer D.

By preceding theorem,

if and only if pn_l(x) - qn_2(x) = O i.e. pn_l(x) - qn(x) = -anqn_'(X)

a
ifandonly if x=-—”+ '5
2 2q x)
n-l

Corollary 2. If x = 3 is purely periodic in E2 with period
a = a a ...a a , n > 2, then x' = 5' with a' = a ,...,a a' and

l 2 n-l n -— l n-l n
a; = 3 - an is purely periodic in 52 Such that

 

 

 

 

Proof: This result follows from Theorem I Since x'-La'-l and x-La-l

are independent of a; and an respectively.

P + tP
. = n n-l
Lemma 2. [0,a',a2,...,an+t]

+
qn tqn-l

 

holds for all positive

‘

inte ers n and an number t, h —fl-= ; ... = = .
9 Y w ere qn [0 al.a2. .an]. qo I. Do 0

a + Db + a-Db _ 2(ac - Dzbd)

 

 

Remark l. E—:—Ba- 2:53 - 2 2 2 for any numbers a,b,c,d,D.
c - D d
Lemma 3. If x e E2 and (a](x),a2(x)) # (l,2) then I - x 6 52.
Proof: If x = -l-T- where t E E then
2
l + --
I+t
_ _l - -l+t.._'_
l+t

Theorem 2. If P is purely periodic in 52 with period

p = Ip2p3...pn_'2 and if p = Lp-', then there exist positive integers

A?

m and D such that P = -I +-- . Let x = [O-a ,a ,...,a , 2 + ELfl
m ’ I 2 n-l /D

and let y = [0;al,az,...,a 2 - E—J where ai = l or 2 for all i<n

n'l’ 1/5

and any positive integer n, then x + y is rational in E2 9 E2.
JD— . . .
Proof: By Corollary l, P = -l + -E- for some posutive Integens m and D.

——-= = for some t E E .
f5 I+p I+1 2

 

 

50 -fl-& E and by Lemma 3, I - .9.6 E and hence x + y 6 E e E .
/5 2 D 2 2 2
By Lemma 2
m m
pn + /5 pn-I(x) pn D n-I(x)
X = and y =
q + “"1 q (X) q - —m (X)
n /5 n-l n D n-l

where pn = an-l(x) + pn_2(x) and q = an-l(x) + qn_2(x). Also by

 

Remark l
m2
_ 2(pann ' B— pn_](x)qn_](x))
x + y - 2 .
2 - fl- 2 (X)
qn D qn-l

Therefore x + y is rational.

Theorem 3. If P is purely periodic in E2 with period

p = 2p2p3...pn_‘l and p = Lp-l, then there exist positive integers

D and m such that P = -§ + 1%.. Let x = [0;al,a ,a l, 2 + p]

2"" n-2’

and let y = [0;al,a2,...,an_2,l, l-p] for any positive integer n

where ai = l or 2 V i < n, then x + y is rational in E2 0 E2.
f6 . .
Proof: By Corollary l, P = -& + —E- for some posutlve integers D and
l 2+5 _ l _ l
m. Clearly x E E2 and ]_p - 1+5 — l + 1+5 where P - 21S' for some

l0

s E E2 so y = [0;al,...,an_2,2,l + s] 6 £2.

 

 

x = [0;al,a2,...,an_2,l, 1 + (%-+ 12)], y = [0;al,a2,...,an_2,l,l + (%---%g)].
By Lemma 2
x = D” + (%-+-£g pn_l(x) y = pn + (%-- ig)pn_l(x)
q + (a'tigiqn-lei ' qn + (%--i§§qn_,<x)
where p - p I(x) + p 2(x) = pn_l(y) + Pn_2(Y) ,
qn = qn_l(X) qn_2(X) = qn_l(y) + qn_2(y)

By Remark l,

(X)qn_l(X)]

zupn + :pn,l(x))(qn + eqn_,<x>> - ia'Pn-1

 

x + y =

(qn + iqn_'(x))2 - 2g-q:_,(x)
ITI

Therefore x + y is rational in E2 0 E2.

Theorem A. If y,y' E E2 and x is purely periodic in E2 with

 

 

 

period a = alaz...an and if y - y' = x - La.I then
4. -l .I, |_ -1
a n y - La n y - x - La .
-jf- p _ (X) q _ (x)
Proof: By Theorem I, x - La I = n I q (x; 2 = y - y' ... (A)

* Pn_ I(X)Y + pn (x)
a x y=n17})y + qn (X) ’

 

 

 

-I . -I
‘I * ' pn_](La ) Y + pn(La )

 

 

qn_](La")-y' + qn(La")

p

- (X) - q _ ( )
q_2(X)(Y- n I n2x

q (x) )+pn(x)-an(pn_](X)-qn-z(X))

 

n-l
Pn_l(X)'qn_2(X)
qn-l(X)(y - qn-l(x) ) + pn_](X) + anqn_,(X)

 

M(
qn_ _2(X)y + pn (X) - (————7—7-+ an)(pn_l(x) - q n_2(x))

 

qln- W(X)y+q MTX)+aq _,(x)

q (X)
qn-2(x)y + pn(x) - af::y;7(pn_l(x) - qn_2(x))
qn-I(x)y + qn(x) . by Lemmas I, 2.

 

r
O”
I

‘ q (X) .
- (pn_,(x) - qn_2(x))(y + afl-TT;7)4qn_l(x)y + qn(x)]
. n -

 

 

= pn_l(X) - an,2(X) = x _ L8,]
qn_l(X)T
Corollary 3. If x and y are purely periodic in E2
With periods a = alaz...an and b = blb2"'bm respectively and

 

x - La- y - Lb.l and if p],p2,...,p ,... is a sequence of words

II

with pi = a or b, i = l,2,... then

 

 

 

 

I I.

Proof: By the preceding theorem, pn * 5 - Lp; * La.-I

. .l. .1. - -‘ L
by the preceding theorem, pn-l a pn " a - Lpn-l %

o Lp2 o...o Lpn = x - La-

 

 

 

x - La-l. Again,

 

Continuing this procedure, we have pl * p2 *...* p * 5 - Lpl * Lp2

 

 

-I -I -I . -
7': = - :2 -
Lpn La x La I.e. (pI 0 p2 o...o pn) a

 

-I -I -I ... -1_ _ -1
(Lpl o Lp2 o...o Lpn ) La x La .

procedure one more time, we get

2...-_ 'I -I -12
(pl 0 p2 o...o pn) a (LpI o Lp2 o...o Lpn )

Using the same procedure, we get

n g - _ -l -l -| n g
(pl 0 p2 o...o pn) . a (Lpl o Lp2 o...o Lpn )

 

. . n ,
Since pI 0 p2 o...o pn = llm (p] 0 p2 o...o pn) «

n+°°

 

Lp;l o Lp;I o...o Lp;I = lim (Lp;l o Lp;l o...o

IT+°°

"=18

pi=l|mp‘np2 z..._:.p Ra,

i l n+w

.9
n

 

La-

 

I

If we go through the whole

 

= x - La- .

 

 

 

l3

 

. 'I L 'l
.. Lpn La

co
1\

and H Lp;l lim Lp;l * Lp2

i=I n+m

 

 

 

I -l

Hence pl 0 p2 o...o pn - Lp;l o Lp; o...o Lpn = x - La-‘

 

Theorem 5. If x and y satisfy the hypothesis of the preceding corollary

and a # La.l then there exists a non-denumerable set of pairs Q1,B)

 

in 62 e E2 such that a - B = x - La-l.

Proof: Without loss of generality, we may assume ak # bk for some k.

If w E (O,l) has the binary expansion w = 2 w. - 2 ', we let

8

aw = p(wl) * p(w2) * p(w3) *...= H p(wi) where p(0) = a. p(l) = b
i=I

on

_ ‘l - = _ ‘
and let 8w - 3:1 L(p(wi)) . Then aw 8w x La

 

I
If w, w' E (0,l) with w # w' i.e. there exists a first integer
2 such that w2 # w}. Then

(p(w£))k # (p(w£))k and thus aw # aw,.

Hence there exists a non-denumerable set of pairs (a.8) in £2 3 £2 with

 

a - B = x - La-

IA

Notation: Let Q be the set of all rationals obtained from Theorem l.

Corollary A. If q E Q, then there exists a non-denumerablezset of pairs
(a.B) in 52 952 such that a - 8 = q.

Proof: Immediate result from Corollary 2 and Theorem 5.

Theorem 6. If x and y are purely periodic in E2 with periods

a = a]...an_ll and b = a]...an_]2 respectively and k = l or 2 then

 

 

...... ___ (X) - kq (X) - p (X)
-I -1 _ qn-l n-2 n-2 .
k 0 a - L(k o a) = k 0 b - L(k o b) — pn-](x) + kqn-](x)

 

Proof: By Theorem I,

 

 

k 0 a - L(k o a).1 = k 0 b - L(k o b)-1

pn(k o a) _ qn_l(k o a)

 

qn(k o a) qn(k o a)

 

 

I '& pn-2(x) T qn-2(X)
_ k + Pn-u(*7 -% pn_l(X) + qn_,<x)
q x
n-l
qn_l(x) pn_2(X) + kqn_2(X)

 

 

pn_](x) + kqn-I(x) - pn-IIX) + kqn-l(;)

_ qn_,(x) - kqn_,(x> - pn_,(x) .

pn_l(X) + kqn_l(X)

Theorem 7. E2 8 E2 = Q.

Proof: If x E E2 9 62, then there exist a,B 6 E2 such that

If e > 0, there exist positive integers n and m such that

Ia'folal(a):azia)w-uan(a)ll <§ and IB-£0:a,(s).a2(s).....am(e)3| <§

Let w be the word a](a)a2(a)...an(a)am(8)am_l(B)...al(8)I

 

Let a = w and Bx = Lw-l. Then ad - 8% E Q and

3': 7': :‘L‘ f: E
Ia-B-(a -B)I:Ia-a|+|8-Bl<§+ =e

E.
2
which implies the Statement of the theorem.

Lemma A. If x,y E E2 with rational x - y and either

(al(x),a2(x)) # (l,2) or (al(y),a2(y)) # (l,2) then either y - x + l
or x - y + l is rational in E2 9 E2.

Proof:

If (al(x),a2(x)) # (l,2) then by Lemma 3, I - x e E2. Thus

y - x + l is a rational in E2 $ E2. Similarly, (al(y),a2(y)) # (l,2)

implies x - y + l is a rational in E2 9 E2.

Examples

Example i. Let a and b be the words l2l and l22 respectively and let

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ - _ .—————- _ - _ .—————- _ l = 2x + 3
l
2 + l+x
- + ¢=—'
3x2 + 2x - 3 = 0, so x = _l__?;_fll,
-l 2 _
La 1 = I = E§———:;3L- , 3La I + ALa I - 2 = 0 so
2 + 'l L -l + 5
1+ - 3 a
I + La"I
- - - ' p (X) - q (X)

La I = -2—:¥xj§:. Thus x - La I is the rational 1-= 2 I =

3 3 q2(X)
2;“)-.. _ ' -2y+5 2 - - -M

3 - Y - I + I - 3y + 7. 3y + By S — 0. so y - 6 -
2 ._L_
2+y

_—:T' I Lb-l + 3 “TT2 T'TTT 'TTTT -7 + JOE
Lb = l = ———-—————, 3Lb + 7Lb - 3 = 0, so Lb = ‘”'ZT"“' .

2 + Lb

-:1- I p2(y) - ql(y) 2 _ I

Thus y - Lb is also equal to §-= q2(y) = 3 . Moreover,

p2(x) = p2(y) = 2. q2(x) = q2(y) = 3. By Theorem 6.

I7

[O;I,l,2,l] - [0:2,l,l,h]==[0;l,l,2,2] - [O;2,l,l,2]

_ 3 - l - l
_ _—E_:—§—_.

l.
S
We observe that there are two distinct pairs (a,8) with a # B for

l

both a - B = 1- and a - B = By Theorem 5, there is a non-denumer-

3 5'
able set of pairs (a,8) for both a - B = %- and a - B = éu In
fact, by Lemma A, there iS a non-denumerable set of pairs (0.8) for
A 6 .
+ =— + =-— ,
both a 8 3 and a 8 5 In E2 9 E2

Example 2. Let p be the word I 2 and p = 5. Then p = Lp and

 

 

 

 

 

____ l l .
p = [0,l,2] = -I + /3. Let x = l and y = 1 . Since
2+7; 2‘7;
l l _ -——
:=-IT6-[0', I,I,2I, X652. and
/3
y: I I = I I = l I = I 1 = [O:I,2,2,I]
I+I'7-3- I+-7§—— I'I'TTT— l+l+l
3'I Y3'I P
Thus x + y = ~B—-= 12- iS a rational in E e E .
l l 2 2
4._
3
——- -l
Example 3. Let p' be the word 2 l and I” = p'. Then p' = Lp'
and I” = [O;2,l] = :1—1412-. Let x = [O;I,2 + P'] and y = [O;l,l-P']

 

l8

 

 

 

 

 

 

 

 

_ I _3+/3 _ I _3-/3
Then x-l+ I -5—1—75— and y-]+ 1 --5—-;—7§.
73-l 73-l
2+ l-
2 2
Clearly x E E2.
l l+p I ___
- .= - = = = ‘
I P I 2+P 2+P I where p [0,I, ] so
I + —-—-
l+P
l I ———-
Y = I = l = [0;2,l,l,2] €~E2.
1+]. I+I+TF—P

Thus there is another pair (x,y) with x + y = %Z- in E2 9 E2.

CHAPTER III

NON-COINCIDENCE OF THE MARKOV AND LAGRANGE SPECTRA

For each x = {xk} E A, we define M(x,k) = [ J +

xkixk+l"°°

[0;X ’XI’XO]. M(X) = sup M(X,I() and L(X) = m M(x,k).

k k

The ranges of M and L are known as the Markov spectrum and

k-l’xk-2""

the Lagrange spectrum respectively. It is known that the Markov spectrum
is closed and contains the Lagrange spectrum.

For each x = {xk} e A, we define

I(x,n) = {y = {yk} e A : yk = xk, k = l,2,...,n}.

Let 82(n) L-(I + m'z" and eu(n) = K-(l—g—fii‘z”.

We then state the following two lemmas without proof.
Lemma I. If y e I(x,n) n [IIX,n+l)]c, then 51(n) < |M(x,O) - M(y,0)| < eu(n)

Remark l. |M(x,0) - M(y,O)| < lO-7 if n = l9

and II

IM(X,0) — M(y,O)| < Io' if n = 29,

Lemma 2. For £,n E E2

[0;XI,X2,...,X2n+I,2 + g] : [0;xla--Oax2n+]9l + rI]

[0;xl,x2,.. l + n] :_[0;xl,...,x n,2 + g]

I9

"x2n’

20

Let g = {5,} e D be a sequence such that E, = €i-7’ I > ‘A;

m
I
V!
II
m
I
0“
l
m
I
\I
II

2’ 5—8 = I, 5.9 = 5-10 = 5“‘ = E’IZ = E_l3

E-IS = €fl6=2’€‘l7 " I, E-I8 = 2, €‘I9 = 5-20 =I ; £3 = gi+7, I 1‘2].

Let a be the word I l 2 2 2 l 2.

 

M(E,O) = 5 + 2 + [Otl,2,2,2,l,l,2,l,2,2,2,2,2,2,2,2,l,2,l,I] :_3.2930AA26SA

Let a = M(£,O). Clearly, M(E,k) < M(£,O) if 5k = I. (gk-I’Ek) =
(2,2) or (gk’€k+l) = (2,2). Without loss of generality, due to the

symmetry €-l2+i = €_13_i, i = O,l,2,... we may assume that M(g)

can only occur at Ek for k = 7n, n = -l,O,l,2,... . Since

(L-7E)i = 5,, i 3_-A, (L-7€)_5 = 2 and 6'5 - l we have, by Lemma 2,
n(t‘7g,o) < a i.e.
M(€9-7) < 0..

Since (L7nE)i = 5,, i :_-Il (L7n

l
N

9 €_]2 - and €)_12 = I,

3
II

l,2,... we have, by Lemma 2
M(E,7n)‘<<x n = l,2,...

Hence M(E) = a.

We define words W3 = W as follows:

I .
2... n., --°9

2l

w = 2 I 2 2 2 I 2 I I 2 2 2 I
W7 = I I 2 2 2 I 2 I I 2 2 2 I I
w = I I I 2 2 2 I 2 I I 2 2 2 I 2

v9 = 2 I I 2 2 2 I 2 I I 2 2 2 I 2 2

2
II
N
N

l l 2 2 2 l 2 I I 2 2 2 l 2

W = l l 2 l I 2 2 2 I 2 I I 2 2 2 I 2 l l 2

w'2 _ 2 2 2 I 2 I I 2 2 2 I 2 I I 2 2 2 I 2 I I 2 I
w'3 = I 2 2
w'“ = I I 2 I I

w'5 = 2 2 I 2 I I 2 I

W'6 = l 2 2 l 2 I l 2 2

W‘7 = 2 2 2 2 l 2 l l 2 2 2

Lemma 3. If a sequence x = {xk} contains the word Wi with W; = xj
for some i E l2, where j > IA for x e A; and k = 3 if i = l;

k = i + I if i = 2,3,A,S or 6; k = i if i = 7,8 or l2 and k = i - I

if i = 9,Io or II, then, M(x,j) > a + Io'7.

22

Proof: For i = I, for example, we have
M(x,y) > 2 + 2[0;I,2,2,I] = 3.h > a + l0“7 .
For the remaining i's, the inequality is obtained analogously.

Lemma A. If a sequence x = {xk} contains the word wi with w; = xj

for some i, I3 :_i §_I7, where j > 8 for x e A and k = i - II,

then M(x,j) < a - l0-h.
Proof: For i = l3, for example, we have
. , , _ 3 3_ _ -h
M(X,J) < 2 + [O,l,2,I] + [0,2,2,I] - 2 + E-+ 7 < 3.l8 < a I0 .
For remaining i's, the inequality is obtained analogoust.
i-l
Remark 2. Lemma 3 and Lemma 4 are also true if x contains W in a

similar way.
Lemma 5. If x = {xk} is a sequence such that M(X.J)€: (a - l0-7, a + I0-7)

for some j > Ih, then for any integer n > Ih;

I) xj+2=l and n(x,2)5_a+Io'7 for j-Io_<_2_<_j+n implies
xj+£ = 52’ 2 = 0,I,2, ,n

2) xj+2 = 2 and M(x,k) :.a + 10-7 for max(0,j-n) §_£ :_j + l0 implies
xj_£ = £2, 2 = O,l,2,...,mIn(n,J),

23

Proof: The proof is by contradiction.

Clearly, xj = 2 and (xj-I’xj’xj+l) # (2,2,2).

-I
wl3

x = x W13 or

j-l j+l l otherwIse xj-Ixjxj+l =

-h . .
By Lemma 4, M(x,j) < x-IO . This is a contradiction. For the remaInIng

2's, contradictions are obtained analogously according to the following

table.
Case I. xj+2 = l
. . i
xj+£ - £2 for _lO §_£ < n, otherwIse x contaIns w as

described in Lemma 3 or Lemma h which leads to a contradiction.

2 2 -2 3 -3 h -h 5 -5 6 -6 7 -7 8 :8 9 I0 -9 -I0 II I2 I3 lh
x. I222222III22III22222I2
J+£
i Ih23I5I6hI756789I0I2II3121I57

If n 3_l8, by repeating the last seven columns, we have

x = g 2 = I8,I9,...,n.

j+2 2’

Case 2. xj+2 = 2.
We repeat the same argument as above, after replacing 2 by -2,

and interchanging column I and column 2 in the table. The rest of

the columns remain unchanged, i.e.

2h

xj+£ l 2 2
i I 2 3
Then xJ._2 = 52’ £=O,I,2,...,mIn(n,J).

Let p = {pi} be a doubly infinite sequence such that pi = 5;, i 3_-II,

p_12 = p_l3 = I, p_]h = 2, pi — I :_-IS. Then

Oi+29

M(p,0) = E2;I,I,2,2,2,I] + [0;I,2,2,2,I,I,2,l,2,2,2,l,I,2]

 

Let e = a - n(p,ox Then Io"I < e < Io'7 .

Lemma 6. If x = {xk} Is a sequence wIth xj+2 = Eg for some J,

2 = O,l,2,... then M(x,n) : M(p,0) for all n > j + Ih.

Proof: Clearly M(x,n) :_M(p,0) if xn = I or (xn-I’xn) = (2,2) or

(xn’xn+l) = (2,2). It remains to prove M(x,j + 7i) :_M(p,0), i 2,3,...

M(x.j + 7i) = 5 + 2 + [0:I,2,2,2,I,I,2,I,2,2,2,I,I,2,...] , i = 2,3,.

(Lj+7ix) = p, k :_-l6, (Lj+7ix)_l7 = 2 and p_ = I by

Since k

Lemma 2, we have M(x,j + 7i) :_M(p,0) for all i 3’2. Hence
M(x,n) :_M(p,0) for all n > j + Ih.

Theorem I. The interval (a - e, a + s) does not contain a point of the

Lagrange spectrum.

25
Proof: Suppose there is a sequence n = {ni} such that L(n) 6 (a-e,a+e).
Then there exists an integer k > Ih such that M(n,i) < a + e for all
i 3_k, and there exist infinitely many integers j > k such that
M(n,j) E (a - e, a + 5). By Lemma I, there exists a smallest n0 such
that eu(no) §_e£(l6). Let jI be the smallest one of such j's greater

than k + n . Let be the smallest one of such j's greater than

0 J2

. + . = = . . .
J] 2I. By Lemma 5, eIther nj2+2 fig, 2 O,l,2,... thch ImplIes,

by Lemma 6, M(n,n) :_M(p,0) = a - c for all n > jz + I4 or

n. _2 — £2, 2 O,l,2,...,J2-k, thch ImplIes

J2
= = ' = + +
nj‘_£ §£,£ O,l,2,...,no and M(n,J]) 2 s t where
= ;i,l,2,2,2,l,2,l,l,2,2,2,l,2,...,. J: .
s [0 nJl_n0 ”0 [50.51.52, 1

t = [0;],2’2,2’l’l’2’1,2,2’2’l,]’2’1’2’2’2,I’I’2’...’nj "‘°] = [t .
2

Let v = [o:I.2.2,2,I.I.2.I.2.2.2,I.I.2.I.2J = [v0;v,.v2,...3

T e I(V .16) n [III/,.ImC and s 6 1(5, no)

where T = {tk}, S = {5k} and V = {vk} .
By Lemma l, v - t > 62(I6)
s - 3 < eu(n0) §_e£(l6) < v - t and

s + t < v + a.

26
Since M(p,0) = 2 + 5 + v,
M(n9jl) < "(090) = C1 ‘ 5

Since both cases lead to aacontradiction,(a - e, a + a) does not contain

a point of the Lagrange spectrum.

>’
X'

For each A E [O,I] with its binary expansion A =
k

"MS
I

l

N

we define 5A = {5?} to be a doubly infinite sequence as follows:

A

gag-161%? - I

-I7 — E-I8 = 2 and

A = A =
g-(|7+2k) E-(I8+2k) k
Let P = {M(EA)|A is an irrational number in [O,IJ}.
Theorem 2. P is an uncountable set consisting of points of the Markov

spectrum which are not in the Lagrange spectrum. Every point of P
is a limit point of P.

Proof: For E: with k < -8, 2'5 occur in paris, so M(gx) can
only occur at a: = 2 with k.: -8. By an argument exactly the same
as that leading to M(g) = M(£,O), we have M(EA) = M(gA,0). Since

- C - A = —
g )i - pi’ I :_ ll, §_]2 2 and 912 - I we have by Lemma 2,

who) > M(p,0). Thus ml) > a - 2:. Since (5})i = 5,, i _>_ -l6,

5517 = 2 and €_]7 = I by Lemma 2, we have M(gx,0) < a and hence
a - e < M(EA) < a. By Theorem I, M(EA) is not in the Lagrange spectrum.

Hence P does not contain a point of the Lagrange spectrum. Let A(k) =

27

I-xk+.
—————i ' = 1,2,... then A(k)

”—'+ - . J
I 2k+J

I 2 j

M7?

A.
| O O O O
i Is IrratIonaI In

H P18

[O,I] since A is irrational in [O,I], so

M(aw‘)

) 6 P and
M(§A(k)) = M(EMIQO) and A(k) + A as k + w

M(EA) = Iim M(£A(k)) since M(gx,0) = Iim M(5A(k),0).

k+m k+m
- Also gA(k) ¥ 52 for all k. Thus M(gx,0) # M(gx(k),0) since
5? = g?(k) for all i 2.0 i.e. M(gx) # M(gx(k)) for all

k a l,2,...
I

If A ¥ A' then M(gx,0) # M(gA ,0) since 5? = E? for

I
all i > 0 i.e. M(gx) # M(gx ). Hence every point of P is a limit

point of P and P is nondenumerable.

CHAPTER IV.

FRACTIONAL DIMENSION OF A LEVEL SET

This section provides an estimate of the Hausdorff-Besicovitch dimen-

 

<¢221+5J§+A}
- l0 °

sion of the level set H = {w E E2 0 [O,I) : L(w )
The Hausdorff‘BesicovittfiIdimension of a set S, which we will write
dim(S), is defined as follows: let (Ii) be a covering of S by in-

tervals, and let [III be the length of Ii; then 6 = I.u.b.IIi| is

called the norm of the covering;

r(a,S) = lim g.l.b. ZII.IU,
5+0 '

where the greatest lower bound is taken over all coverings of norm 6,
is the a-dimensional Hausdorff measure of S. dim(S) is the number
such that, for every positive 5,

T(dim($) - c, S) = m
and

F(dim(S) +6, 5) = 0

28

29
Notation
E = [0;2,2,I,I,...,l,l,...]

pn = pnIE). qn = anEI

p.(a)
. . I _ .
If a Is the word aI a2...an, we defIne E:TSY-- [O,al,a2,...,ai],
I = l,2,...,n and I8 = {w E [O,l) : al(w) = al,...,an(w) = an}.

Let an be the word 2 2 I I ... I I of length 2n.

ILn = I15“ 0 22"

i =II I.
n,k gnogk o 22

Remark: p2n+l(£n o 2) = 2p2n + p2n-I’

q2n+I(En 0 2) = 2"2n + an-I’

q2n+2(€n022) 2(2q2n + q2n_l) + q2n = Sq2n + 2q2n_l,

p2n+2(€n022) 5p2n + 2p2n-I’
92(n+k)+I(5n°5k°2) = (292k T ka-l)p2n-l + (2q2k + q2k-I)p2n’
q2(n+k)+l(€no€k02) = (2p2k + pzk-I)q2n-I + (2q2k + q2k-l)q2n’

p2(n+k)+2(€n°€k°22) = (592k + 2p2k-l)p2n-l + (5q2k T 2q2k-I)p2n’

30

q2(n+k+l)(gnogk°22) = (szk T 2p2k-ITq2n-l T (quk T 2q2k-I)q2n’

II. = l
n
(qun + 2q2n-IT(7q2n + 3Q

 

,
2n-I)

2n,k =

I
E(592k+292k_])q2n,]+(3q2k +2q2k_l)q2n1[(7p2k+392k_])q2n,l+(7q2k+3q2k_l)q2n]'

n-l

3\ [1-] )A +

Lemma 1- Pn = %[(I + 7§7A + (I ' 7%)6n-T] and qn = (I +37%

 

 

h n-I __ I-+ /5 __ I - /5 2 _
(I 7§96 where A — 2 and 6 — 2 are roots of x - x + I.
Proof: Let pn = AA".I + Ben'-l
p = I = A + B p = 2 = AA + 38 = A:§-+ (A-B)z§-
I ’ 2 2 2 ’
/5
2 = 2 + (A + A ' II‘jf. 3'+ /5 = 2/5 A,

so A=%(I+7§) and awn-7%).
H _ 3 n-l
ence pn — &[(l +-7§)A + (I - 7§)6

Similarly, qn = (l + 739A + (l - 7%)en-l

 

 

Lemma 2. If I = Z (3245)“ as n + m then .206A < a < .206hl.
k=2 n
p n-l n-l
Proof: n AA + 86 + A as n + w since 9 + O as n + w
p n-2 n 2

 

3I

 

q
where A = &(I + 7%) and B = £(I - 7%). Similarly, n + A as n + m.

 

 

n-I

From preceding remark,

2
l q (5A + 2)(7A + 3)
n,k + 2n-I
2 2 + + ]
" q2n-l[T(Sq2k+2q2k-l)+5p2k+2p2k-l][T(7q2k+3q2k-l) 7p2k 3p2k-I

as n + w.
2
_ n,k

Let 6k - 2 Then

6k+l + (5* T 2)(q2k-IT T ka-l)(7T T 3)(qzk-I" T ka-l) I
=7 as k+°°

k (5x + 2)A2(q2k_lx+ p2k_,)(7x + 3)A2(q2k-lx + ka-l) A

 

 

(SA + 2)(7A + 3) = 35A2 + 29A + 6 = 35(A + I) + 29A + 6 = 6AA + AI
p3 = 3. pa = 5. p5 = 8. p6 = 13. p7 = 2|. 98 = 3A .
Q3 = 7. Q“ = 12. as = I9. Q6 = 3|. Q7 = 50. Q8 = 8|-

Since the terms have nearly constant ratio, we approximate the tails of

the series by the geometric series

32

IIMB

k 2

I(I55+38)A+65+l6]-a[(2]7+57)I+9l+2A]-a+

I(hos+Ioo)x+I7o+h2]'“[(567+Iso)x+238+63]‘“

I _ A-Qa

 

}

= (64A+hl)a{[(7AA+3l)(IOSA+AA)]-a+ [(l93A+8l)(27hA+l]5)]-a +

'0.

[(505A+212)(7I7A+3OI)]

l _ A'Aa

= (6AA+AI)G[(lh28lx+9l3h)-a+(9727IA+62I97)'“ +

(666094A+425897)'“

- J-
] _ A ha

kg, 5: 3_I.OOOO6IO78 when a = .2064 ,
E 6: 5.0.9999675709 when a = .206Al,
k=2

Hence .2064 < a < .206AI.

Let e

0 be the empty word and let en be the word I I ...

of length 2n. We define

5i é (6AA + Al)a{[(60 + IA)A + 25 + 63'“[(8u + 2])A + 35 + 9]““+

I

33

> 2, k = I,2,...} u {i}.

:_O and nk._

: no
The following theorem follows immediately from a theorem in Schweiger 2h].
2

(£2459a as n + m then dim T = a.
2 n

Theorem I. If I =
k

IIM8

Corollary I. .206h < dim T < .206hl.

Proof: This is an immediate result of the preceding theorem and Lemma 2.
Lemma 3. T 5.”:

,/ + ___—_— -
Proof: Let t = 22' lg/§T+ A =[2;2,I,I] + [0;I]. Let w be an element

 

of T.

Case I. There are infinitely many pairs (ai(w),ai+](w)) = (2,2).
Clearly, M(w,k) < t when ak(w) = I. If ak(w) = 2 then either

ak_l(w) = 2 or ak+l(w) = 2. Without loss ofgenerality, we may assume

ak+‘(w) 2. Since there exists j < k such that aj(w) = 2 and k _ j

is even, [0;ak_l(w),ak_2(w),...,a](w)] < [O,I]
ak+2(w) = ak+3(w) = I, since w E T. If ak+h(w) = I then

[ak(w);a (w),...] < [2;2,I,I]. Otherwise

k+l

(ak+h(w),ak+5(w),ak+6(w),ak+7(w)) = (2,2,l,l). Continuing this argument,

34

unless [ak(w);ak+l(w),...] = [2;2,l,l], there must exist a first i such

that ak+hi(w) = I. This implies [ak(w);a (w),...] < [2;2,l,l]. Thus

k+l

H(w,k) < t and L(w I < t i.e. w 6 H.

Case 2. There exists a positive interger k such that ai(w) = l
for all i z'k. Then M(w,i) < t for all i :_k. Thus

L(w)<t and wE H. Hence TEH.

Theorem 2. dim H > .206A.

Proof: By preceding lemma, dim H :_dim T. By Lemma 3, dim H > .206“.

Alternative method of computing dim H.

We approximate the covering of T which is found to be a gen-

eralized Cantor set, i.e. the set constructed by removing the middle
interval. It is clear that max T = [0;I]. Let b = max T. Let s =

min T = [0;2,2,I,I]. That is T c [5,b]. Since

max {w E T: a](w) = a2(w) 2} 22ll * b

and

ll
‘
—

r
U!

l}

min {w E T: a](w) 32(w)

we obtain our first stage covering r] of T by removing

35

(22II * b, ll * s) from [5,b] i.e. F] = [5, 22II * b] U [Ils,b] D T.

Similarly, we obtain our second stage covering of T.

r2 = [s,(22ll)2*b] u [(22ll)*(ll)*s,22ll*b] u [ll*s,(ll)*(22ll)*b] 0

[(II)2 * s,b] : T.

Continuing this process, we obtain the nth stage covering Tn of T for
each positive integer n.
Let m be the length of the middle interval we removed, let A

be the length of the left hand interval and let r be the length of the

C . . . 2 r .
_.__.__——— remaIn
rIght hand Interval. SInce the ratIos Q + m + r and g + m + r

approximately the same each time we remove a middle interval from the re-
maining intervals, we denote the former by p and the latter by q. We

can easily see, by induction, that at nth stage we have (?) intervals

of size pn-JqJ(b-s) in [s,b]. Hence P(a,T) will be approximated by

n . .
rn(a.T) = z (3)[p”'JqJIb-s)1°‘ = Ib-s)“<p“ + ea)“.
j=0

So the a which makes Fn(a,T) = I, will be the dimension of T. Hence

109(pa + qa) =

A short computation shows:

.2064 < a < .206AA

Again we proved dim H > .2064.

36

 

-a Iog(b-s)
n ,
as n + m
since p 5 .00h48lh86542 and

q .IA63587922

BIBLIOGRAPHY

II.

l2.

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