Gaps, Frequencies and Spacial Limits of Continued Fraction Expansions
de Jonge, Jaap
DOI
10.4233/uuid:e0b37188-c8b6-4c96-9d04-93ac1f6899e3
Publication date
2020
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Final published version
Citation (APA)
de Jonge, J. (2020). Gaps, Frequencies and Spacial Limits of Continued Fraction Expansions.
https://doi.org/10.4233/uuid:e0b37188-c8b6-4c96-9d04-93ac1f6899e3
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G
APS
,
FREQUENCIES AND SPACIAL LIMITS OF
CONTINUED FRACTION EXPANSIONS
G
APS
,
FREQUENCIES AND SPACIAL LIMITS OF
CONTINUED FRACTION EXPANSIONS
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. dr. ir. T.H.J.J. van der Hagen, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op vrijdag 24 januari 2020 om 12:30 uur
door
Cornelis Jacobus
DE
J
ONGE
Doctorandus in de Wiskunde Universiteit van Amsterdam, Nederland,
promotor: dr. C. Kraaikamp promotor: prof. dr. F. H. J. Redig
Samenstelling promotiecommissie: Rector Magnificus, voorzitter
Dr. C. Kraaikamp, Technische Universiteit Delft Prof. dr. F. H. J. Redig, Technische Universiteit Delft
Onafhankelijke leden:
Dr. K. Dajani, Universiteit Utrecht
Prof. dr. ir. G. Jongbloed, Technische Universiteit Delft
Prof. dr. S. Marmi, Scuola Normale Superiore di Pisa, Italië Prof. dr. H. Nakada, Keio University, Japan
Overig lid:
Prof. dr. H. Jager, Universiteit van Amsterdam
Reservelid:
v
Printed by: GVO drukkers & vormgevers B. V.
Copyright © 2019 by C. J. de Jonge ISBN 978-94-6384-087-3
An electronic version of this dissertation is available at http://repository.tudelft.nl/.
C
ONTENTS
Summary ix Samenvatting xi Preface xiii 1 Introduction 1 1.1 General introduction . . . 11.2 Semi-regular continued fractions. . . 2
1.3 The natural extension. . . 3
References. . . 4
2 Three consecutive approximation coefficients 5 2.1 The regular continued fraction . . . 5
2.2 The mean value of |ϑn+1− ϑn−1| . . . 9
2.3 Six patterns. . . 11
2.4 The optimal continued fraction. . . 13
2.5 The Nearest Integer Continued Fraction . . . 26
2.6 The measures of the six patterns of the NICF . . . 29
2.7 Discussion - Connecting the results with our understanding of the contin-ued fractions involved . . . 39
References. . . 42
3 The natural extension of Nakada’sα-expansions 45 3.1 Nakada’sα-expansions . . . 45
3.2 Applying singularisations and insertions . . . 49
3.3 The caseα ∈ (g,1]. . . 50 vii
3.4 The caseα ∈ (12, g ]. . . 52 3.5 The caseα ∈ (p2 − 1,12] . . . 57 3.6 The caseα ∈ ( p 10−2 3 , p 2 − 1]. . . 59
3.7 The long way down to g2 . . . 67
3.8 The ergodic systems (Ωα,B,µα,Tα) . . . 74
3.9 Theα-Legendre constant. . . 76
References. . . 79
4 Orbits of N -expansions 81 4.1 Introduction . . . 81
4.2 Sufficient conditions for gaplessness . . . 88
4.3 Gaplessness in case Iαcontains two or three full cylinders . . . 107
4.4 A world of gaps . . . 116 References. . . 123 5 Conclusion 125 References. . . 127 Acknowledgements 129 Curriculum Vitæ 131 List of Publications 133
S
UMMARY
In this thesis continued fractions are studied in three directions: semi-regular continued fractions, Nakada’sα-expansions and N-expansions. Whereas the first two had been studied quite thoroughly already, the third was still young and hardly explored.In Chapter1the general concept of a continued fraction is given, involving an oper-ator that yields the partial quotients or digits of a continued fraction expansion. The ap-proximation coefficientsϑn(x) := qn2|x−pn/qn| are introduced, where pn/qn, n = 0,1,2,... are the convergents of the continued fraction. Some well-known results on semi-regular continued fractions are given. Finally, the concept of ‘natural extension’ is explained.
Chapter2is about orders (called patterns) of triplets of three consecutive approx-imation coefficientsϑn−1(x),ϑn(x) andϑn+1(x). The asymptotic frequency of pattern X (n) is defined by
AF (X (n)) := lim
N →∞ 1
N#{n ∈ N|2 ≤ n ≤ N ,X (n)}.
Starting with the regular continued fraction (RCF), it is shown that, for instance, the asymptotic frequency as n → ∞ of the pattern ϑn−1(x) < ϑn(x) < ϑn+1(x) is smaller than the asymptotic frequency of the patternϑn(x) < ϑn+1(x) < ϑn−1(x). The asymptotic fre-quencies in the case of the RCF are explicitly given: two of them are 0.1210 ···, the others are 0.1894 ···. After this, these patterns are studied of two other semi-regular contin-ued fractions: the optimal contincontin-ued fraction (OCF) and the nearest integer contincontin-ued fraction (NICF). The asymptotic frequencies of the OCF prove to be more equally dis-tributed: the two less frequent patterns of the RCF now have the asymptotic frequency 0.1603 ···, where this is 0.1698··· for the other patterns. The asymptotic frequencies of the NICF prove to be different for all six patterns. However, summation of specific pairs yield once 2 · 0.1603··· and two times 2 · 0.1698···, thus showing a great correspondence with the OCF.
Chapter3is dedicated to the natural extension of Nakada’sα-expansions. By means of singularisations and insertions in these continued fraction expansions, involving the removal or addition of partial quotients 1 in exchange with partial quotients with a mi-nus sign, the interval on which the natural extension of Nakada’s continued fraction map
Tαis given is extended from [p2 − 1,1) to [p10 − 2)/3,1). From our construction it fol-lows thatΩα, the domain of the natural extension of Tα, is metrically isomorphic toΩg forα ∈ [g2, g ), where g is the small golden mean. Finally, althoughΩαproves to be very intricate and unmanageable forα ∈ [g2, (p10 − 2)/3), the α-Legendre constant L(α) on this interval is explicitly given.
In Chapter4N -expansions are introduced for natural numbers N larger than 1. These
expansions, like semi-regular continued fraction expansions, are also sequences of par-tial quotients, called orbits, existing in the interval Iα= [α, α+1] for some α ∈ (0,pN −1].
Depending on N andα, there is a finite number of consecutive digits that occur as par-tial quotient. It appears that there are conditions (that is, combinations of N andα) such that these orbits eventually do not land in certain parts of the interval Iα, called gaps. It is proved that if the number of digits is at least five, no gaps exist. If the number of digits is four, there do not exist gaps for most N , but in the cases that there areα such that Iα contains a gap, there is only one and it covers the lion’s part of Iα. When the number of digits is two or three, the number of gaps varies, but it is possible to give very clear conditions under which there are no gaps.
S
AMENVAT TING
In dit proefschrift worden de resultaten gepresenteerd van onderzoek op het gebied van kettingbreuken in drie richtingen: de klassieke semi-reguliere kettingbreuken, Nakada’sα-ontwikkelingen en N-ontwikkelingen. De eerste twee gebieden hadden vóór dit
on-derzoek al een lange geschiedenis, terwijl de N -ontwikkelingen nog in de kinderschoe-nen staan.
In Hoofdstuk1worden algemene eigenschappen van kettingbreuken genoemd. Elk type heeft een eigen afbeelding die de wijzergetallen van de kettingbreuk voortbrengt. Ook de benaderingscoëfficiëntenϑn(x) := q2n|x −pn/qn| worden geïntroduceerd, waarin
pn/qn, n = 0,1,2,... de convergenten van de bijbehorende kettingbreuk zijn. Verder wor-den een paar klassieke resultaten op het gebied van kettingbreuken genoemd. Ten slotte wordt het begrip ‘natuurlijke uitbreiding’ uitgelegd.
Hoofdstuk2gaat over volgorden (die ‘patronen’ worden genoemd) van drietallen opeenvolgende benaderingscoëfficiëntenϑn−1(x),ϑn(x) enϑn+1(x). De asymptotische frequentie van patroonX (n) wordt gedefinieerd als
AF (X (n)) := lim
N →∞ 1
N#{n ∈ N|2 ≤ n ≤ N ,X (n)}.
Eerst wordt in het geval van de reguliere kettingbreuk (RCF) aangetoond dat, bijvoor-beeld, de asymptotische frequentie voor n → ∞ van het patroon ϑn−1(x) < ϑn(x) < ϑn+1(x) kleiner is dan die van het patroonϑn(x) < ϑn+1(x) < ϑn−1(x). De asymptotische frequen-ties in het geval van de RCF worden expliciet gegeven: twee ervan zijn 0.1210 ··· en de andere zijn 0.1894 ···. Daarna worden deze patronen van twee andere semi-reguliere kettingbreuken onderzocht: de optimale kettingbreuk (OCF) en de kettingbreuk van het dichtstbijzijnde gehele getal. De asymptotische frequenties van de OCF blijken meer gelijkelijk te zijn verdeeld: de twee minder vaak voorkomende patronen van de RCF hebben asymptotische frequentie 0.1603 ···, terwijl die 0.1698··· is van de andere pa-tronen. De asymptotische frequenties van de NICF zijn voor alle patronen verschillend, maar sommatie van specifieke paren geven een keer de somfrequentie 2 · 0.1603··· en twee keer 2 · 0.1698···, wat precies de waarden van de OCF zijn.
In hoofdstuk3wordt de natuurlijke uitbreiding van Nakada’sα-ontwikkelingen be-handeld. Een belangrijke rol is daarbij weggelegd voor singularisaties en invoegingen in deze kettingbreuken, waarbij wijzergetallen 1 worden verwijderd of juist toegevoegd in ruil voor wijzergetallen met een minteken. Daarmee wordt het interval waarop de natuurlijke uitbreiding van Nakada’s kettingbreukafbeelding Tαbekend is verlengd van [p2 − 1,1) tot [p10 − 2)/3,1). Uit de manier waarop dit gebeurt blijkt dat Ωα, het domein
van de natuurlijke uitbreiding van Tα, metrisch isomorf is metΩg voorα ∈ [g2, g ), waar
g de kleine gulden snede is. HoewelΩαuiterst ingewikkeld blijkt opα ∈ [g2, (p10 −2)/3) en niet meer goed te beschrijven, wordt deα-Legendre constante L(α) op dat interval expliciet gegeven.
In hoofdstuk4worden de N -ontwikkelingen geïntroduceerd, waarbij N een natu-urlijk getal groter dan 1 is. Net als bij de reguliere kettingbreuken hebben we hier te maken met rijen wijzergetallen, die we ‘banen’ noemen. Deze banen bestaan in het in-terval Iα= [α, α + 1], waarbij α ∈ (0,pN − 1]. Afhankelijk van N en α is er een eindig
aantal opeenvolgende getallen dat als wijzergetal in een baan kan voorkomen. Het bli-jkt dat er omstandigheden (combinations van N enα) zijn waaronder banen vanaf een bepaald moment niet meer in bepaalde delen van Iαterechtkomen, die we ‘gaten’ noe-men. Zulke gaten blijken niet voor te komen als het aantal verschillende wijzergetallen ten minste vijf is. Als dat aantal vier is, zijn er meestal geen gaten, maar als dat toch het geval is, dan betreft het precies één groot gat, dat het grootste deel van Iαbeslaat. Als het aantal verschillende wijzergetallen twee of drie is, dan varieert het aantal gaten, maar het is mogelijk zeer duidelijke voorwaarden te geven waaronder geen gaten bestaan.
P
REFACE
This thesis is the conclusion of my research as a PhD-student while being a profes-sional teacher of mathematics in secondary education. This research was supported by the Netherlands Organisation for Scientific Research (NWO) under project number: 023.003.036.Chapter2is a slightly adapted combination of two separately published papers: "On the approximation by three consecutive continued fraction convergents", published in Indagationes Mathematicae in 2014, and "Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases", published in Tohoku Mathematical Jour-nal in 2018. Chapter3, only superficially adapted, was published with the title "Natural Extensions for Nakada’s alpha-expansions: descending from 1 to g2" in Journal of Num-ber Theory in 2018. Chapter4is not yet published, but will be the first of two papers on gaps in orbits of N -expansions, both to be submitted for publication soon.
Cornelis Jacobus de Jonge Delft, September 2019
1
I
NTRODUCTION
1.1.
G
ENERAL INTRODUCTION
This thesis is about continued fractions. The first two chapters in which I present the results of my research are on semi-regular continued fractions; the last, third chapter is on N -continued fractions. Although both types of continued fractions are about ap-proximating irrational numbers by (rational) fractions, the main difference is that the semiregular ones involve an infinite alphabet of partial quotients or digits, while N -expansions make use of a finite set of digits.
Generally speaking, a semi-regular continued fraction expansion of an irrational num-ber x ∈ [α,α + 1], with α ∈ [−1,0], is associated with an operator
K (x) = ¯ ¯ ¯ ¯ 1 x ¯ ¯ ¯ ¯− c,
where digit c is a natural number such that K (x) ∈ [α,α + 1] \ Q. By means of such an operator any irrational number can be expanded as a continued fraction if, in the case of numbers with an absolute value larger than 1, the subtracting with a natural number is done first.
In the case of N -expansions, the operator involved is
L(x) =N x − d,
where x ∈ [α,α + 1] for some α ∈ (0,pN − 1] and digit d is an integer such that L(x) ∈
[α,α + 1).
Whereas the semi-regular continued fractions had already been profoundly researched when I started my own research, the N -expansions have a very recent origin.
1
fore, Chapterscontinued fractions, while Chapter2and3are quite specific, built on an extensive literature on semi-regular4is a true exploration of fundamental properties ofN -expansions, more specifically of orbits of N -expansions.
In the next section I will give an introduction to semi-regular continued fractions. In the subsequent section I will introduce the natural extension of an invertible measure-preserving dynamical system, which is important for both Chapters2and3. The intro-duction to N -continued fractions will be given in the related Chapter4itself.
1.2.
S
EMI
-
REGULAR CONTINUED FRACTIONS
A continued fraction of a real number x is defined as a finite or infinite fraction
a0+ ε1 a1+ ε2 a2+ ε3 a3+. .. (1.1)
In this expression one hasεn= ±1, n ≥ 1, a0∈ Z and an∈ N, n ≥ 1. In the following the
more convenient notation [a0;ε1a1,ε2a2,ε3a3, . . . ] will be used for a continued fraction.
A finite or infinite continued fraction is called a semi-regular continued fraction (SCRF) when an≥ 1, n ≥ 1; εn+1+ an≥ 1, n ≥ 1, and, in the infinite case, εn+1+ an≥ 2 infinitely often; see for instance [P] or [K]. In this thesis solely infinite continued fractions are investigated.
The SRCFs have been studied extensively (e.g. [B,K]), as have their approximation
coefficients, defined by ϑn(x) := qn2 ¯ ¯ ¯ ¯x − pn qn ¯ ¯ ¯ ¯ , n = 0,1,2,...,
where x is a real irrational with continued fraction expansion (1.1) and pn/qn, with
n = 0,1,2,..., is the corresponding sequence of convergents, obtained by truncation of
the infinite continued fraction (1.1). These approximation coefficients, giving an indica-tion of the quality of the approximaindica-tion of x by pn/qn, have been studied intensively; in Chapter2some important results will be presented. For convenience, in the rest of this thesis the suffix ‘(x)’ behindϑnwill often be omitted.
The sequence (pn, qn)n≥−1has also been studied much. Many well-known proper-ties can be found in, for instance, [DK], among which gcd(pn, qn) = 1, n ≥ −1, and
p−1:= 1, p0:= a0, pn= anpn−1+ εnpn−2, n ≥ 1;
q−1:= 0, q0:= 1, qn= anqn−1+ εnqn−2, n ≥ 1.
1.3.THE NATURAL EXTENSION
1
3
In the study of the sequenceϑnthe ‘future’ and ‘past’ of the continued fraction ex-pansion [a0;ε1a1,ε2a2,ε3a3, . . . ] of an irrational number x play an important role,
de-fined as tn:= [0;εn+1an+1,εn+2an+2, . . . ], n ≥ 0 respectively v0:= 0 and vn:= qn−1 qn = [0; an ,εnan−1, . . . ,ε2a1], n ≥ 1.
The following relations, which can be found in, for instance, [JK, p. 303], are essential for Chapter2: ϑn−1= vn 1 + tnvn, n ≥ 1, (1.3) ϑn= εn+1 tn 1 + tnvn, n ≥ 1, (1.4) and ϑn+1= εn+2(εn+1ϑn−1+ an+1 p 1 − 4εn+1ϑn−1ϑn− an+12 ϑn), n ≥ 1. (1.5)
1.3.
T
HE NATURAL EXTENSION
In Chapters2and3the continued fraction operators are embedded in dynamical sys-tems, defined as follows (see [DK], page 16):
Definition 1. A dynamical system is a quadruple (X ,F ,%,T ), where X is a non-empty set, F is a σ-algebra on X , % is a probability measure on (X ,F ) and T : X → X is a surjective%-measure preserving transformation.
In particular, these chapters are about invertible dynamical systems, requiring the notion of natural extension, for which the definition of a dynamical system as a factor is needed; both definitions are taken from [DK] (pages 98 and 99) as well:
Definition 2. Let (X ,F ,µ,T ) and (Y ,C ,ν,S) be two dynamical systems. Then (Y ,C ,ν,S) is said to be a factor of (X ,F ,µ,T ) if there exists a measurable and surjective map Ψ :
X → Y such that
(i ) Ψ−1C ⊂ F (Ψ preserves the measure structure);
(i i ) ΨT = SΨ (Ψ preserves the dynamics);
(i i i ) µ(Ψ−1E ) = ν(E),∀E ∈ C (Ψ preserves the measure).
The dynamical system (X ,F ,µ,T ) is called an extension of (Y ,C ,ν,S) and Ψ is called a factor map. In case (Y ,C ,ν,S) is a non-invertible measure-preserving dynamical sys-tem, the measure-preserving dynamical system (X ,F ,µ,T ) is called a natural extension of (Y ,C ,ν,S) if Y is a factor of X and the factor map Ψ satisfies W∞m=0TmΨ−1C = F . For
convenience it is common to speak of the natural extension of an operator S instead of the natural extension of the full dynamical system (Y ,C ,ν,S).
1
patterns such asIn Chapter2the natural extension is used to calculate the asymptotic frequency ofϑn−1< ϑn< ϑn+1; in Chapter3it is investigated itself, in the sense thata natural extension is constructed.
R
EFERENCES
[B] W. Bosma, Optimal continued fractions, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 4, 353–379.
[DK] K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs 29, Mathematical Association of America, Washington, DC, 2002. [JK] H. Jager and C. Kraaikamp, On the approximation by continued fractions,
Ned-erl. Akad. Wetensch. Indag. Math. 51 (1989), no. 3, 289–307.
[K] C. Kraaikamp, A new class of continued fraction expansions, Arith. 57 (1991), no. 1, 1–39.
2
O
N THE APPROXIMATION BY THREE
CONSECUTIVE APPROXIMATION
COEFFICIENTS OF CONTINUED
FRACTION EXPANSIONS
2.1.
T
HE REGULAR CONTINUED FRACTION
Let x ∈ Ω := [0,1] \ Q have the regular continued fraction (RCF) expansion
x = [0; a1, a2, . . .]
and let pn/qn, n = 1,2,3,..., be the corresponding sequence of convergents. Let the op-erator T :Ω → Ω be defined by T (x) :=1 x− ¹ 1 x º .
Since for x = [0; a1, a2, a3, . . .] we have T (x) = [0; a2, a3, . . .], T is called the one-sided shift operator connected with the continued fraction, also known as the Gauss operator.
In 1798, Legendre ([L]) showed that if p, q ∈ Z, q > 0, gcd(p, q) = 1, and ¯ ¯ ¯ ¯x − p q ¯ ¯ ¯ ¯< 1 2 1 q2,
then there exists an n ≥ 1 for which pn/qn= p/q, with pn/qnthe nth RCF-convergent of
x. In 1895, Vahlen ([V]) showed that for all irrational x and all n ≥ 2, min{ϑn−1,ϑn} <
1 2, 5
2
while Borel ([Bor]) showed in 1905 that
min{ϑn−1,ϑn,ϑn+1} < 1 p 5.
In the course of the 20th century several authors sharpened Borel’s result: min{ϑn−1,ϑn,ϑn+1} <
1 q
a2n+1+ 4
;
see e.g. [BM]. In fact, J. Tong ([T1]) showed in 1983 that one also has the converse
prop-erty: max{ϑn−1,ϑn,ϑn+1} > 1 q an+12 + 4 .
For the optimal continued fraction (OCF) expansion, which will be discussed in Section
2.4, one has even more impressive Diophantine properties: min{ϑn−1,ϑn} <
1 p 5.
Unfortunately, this is not the case for the nearest integer continued fraction (NICF) ex-pansion, which will be discussed in Section2.5.
In 1995, J. Tong showed in [T2] that for all irrational x, for all n ≥ 2 and for all k ≥ 1 one has that:
min{ϑn−1,ϑn, . . . ,ϑn+k} < 1 +³3− p 5 2 ´2k+3 p 5 . Note that lim k→∞ 1 +³3− p 5 2 ´2k+3 p 5 = 1 p 5.
In various papers the distribution for almost all x of the sequences (ϑn) and (ϑn,ϑn+1) for n ≥ 1 has been studied for the RCF, OCF, NICF, and several other continued fraction algorithms; see e.g. [BJW,BK1]. In this chapter, we will focus on the asymptotic frequency of triplets (ϑn−1,ϑn,ϑn+1); see Section2.2for a definition.
Now let ¯Ω := Ω × [0,1] and x = [0;a1, a2, . . .]. The natural extension of T is the two-sided shift operatorT : ¯Ω → ¯Ω, defined by
T (x, y) := µ T (x), 1 y + a1 ¶ =µ 1x− a1, 1 y + a1 ¶ . (2.1)
In particular,T is measure-preserving with regard to the measure m with density functionµ, where
µ(x, y) := 1
log 2· 1
2.1.THE REGULAR CONTINUED FRACTION
2
7
We remark that
Tn(x, 0) = (tn, vn), n ≥ 1,
with the ‘future’ tnand the ‘past’ vnas defined in Chapter1. In this section we will prove various arithmetical properties ofT , from which we will later deduce some metrical results on triplets of three consecutive approximation coefficients.
To make the description of these properties easier, we introduce a slightly different operator,S : ¯Ω → ¯Ω, defined by
S := RT , R being the reflection
R(t,v) = (v,t).
One easily sees thatS is an involution. Moreover, S is measure-preserving with respect to the measure m.
We will study the effect ofS on vertical strips
Ra:= µ 1 a + 1, 1 a ¶ × [0, 1], a = 1,2,....
Observe that (tn, vn) ∈ Raif and only if an+1= a. It follows at once that S is a one-to-one mapping of Raonto itself. Furthermore,S maps horizontal line segments in Raonto vertical segments in Raand vice versa.
Now consider the curve given by t /(1+ t v) = c (see for instance [BJW]). Provided that
c < 1/2, c ∉ 1/n, n = 2,3,..., this curve intersects both Ra and Ra−1, a = 2,3,.... Under S , both curve segments are flipped, each in its own vertical strip, resulting in an overall invariance of the curve underS .
While the operatorT has but one fixed point per strip Ra, i.e.
Ga=¡ga, ga¢ := ([0; ¯a],[0; ¯a]) = µ 1 2(−a + p a2+ 4 ),1 2(−a + p a2+ 4 ) ¶ , the fixed points ofS on Raare the points of the form
µ
t ,1 t − a
¶ ,
i.e. all the points of the graph of T . Figure2.1shows the strips R2and R3with the
associ-ated curves t /(1 + t v) = c and the fixed points of S . Since the equations
v =1
t− a and v = (1 − at )(a + v)
are equivalent, we derive from (1.3) and (1.5) that the graph of the Gauss operator T divides Rainto two regions: one, to the right, corresponding with
2
v t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... O 1 1 1 4 1 3 1 2 t /(1 + t v) = 3/11 t /(1 + t v) = 3/10the fixed points ofS on R3
the fixed points ofS on R2 ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ... ...... ...... ...... ...... ...... ...... ...... ... ...... ...... ... ...... ... ...... ...... ...... ...... ... ... ...... ... ...... ...... ... ...... ...... ...... ...... ...... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
Figure 2.1: The fixed points ofS as well as two curves t/(1 + tv) = c on R2and R3
the other one, to the left, with
ϑn+1> ϑn−1.
These two regions are mapped byS onto each other, so they have the same measure m.
This can be generalised in the following way. On Ra, it follows from (1.3) and (1.5) that instead of
ϑn+1− ϑn−1= λ we can write (omitting indices)
(1 − at)(a + v) 1 + t v −
v
1 + t v = λ, (2.3)
which is equivalent with
v =a − λ a + λ 1 t− a2 a + λ. (2.4)
The corresponding hyperbola, hλ, has a non-empty intersection with Raif and only if − a
a + 1< λ < a a + 1.
Again we see from (1.3) and (1.5) that hλdivides Rainto two regions, the one to the right of hλcorresponding with
ϑn+1− ϑn−1< λ, and the one to the left with
2.2.THE MEAN VALUE OF|ϑn+1− ϑn−1|
2
9
A simple calculation shows that
S hλ= h−λ, − a a + 1< λ < a a + 1 . (2.5)
In the next section we will use this to determine the mean value, for almost all x, of the sequence |ϑn+1− ϑn−1|, n = 2, 3, .. ..
2.2.
T
HE MEAN VALUE OF
|
ϑ
n+1− ϑ
n−1|
The sequence |ϑn− ϑn−1| , n = 2, 3, .. ., is for almost all x distributed over the unit interval with density function f , where
f (λ) = 1 log 2 ³π 2− 2 arctan λ ´ (2.6) and as a consequence, for almost all x,
lim N →∞ 1 N N X n=2 |ϑn− ϑn−1| = 4 − π 4 log 2= 0.3096 · · · , (2.7) see ([J1]). A similar result for the sequence |ϑn+1− ϑn−1| , n = 2, 3, .. ., is:
lim N →∞ 1 N N X n=2 |ϑn+1− ϑn−1| = 2γ + 1 − log(2π) 2 log 2 = 0.2283 · · · , (2.8) for almost all x, whereγ is Euler’s constant; see ([J2]). In this section we will show how to use the operatorS on the strip Rato deduce a crucial result for the proof of (2.8), in a much simpler way than in ([J2]). We need the following theorem; see [DK], Lemma 5.3.11 for a proof:
Theorem 1. The two-dimensional sequence (tn, vn), n = 1,2,..., is for almost all irrational
x distributed over the unit square in the (t , v)-plane according to the density functionµ defined in (2.2).
Now let 0 ≤ λ < a
a + 1 and let La,λbe the region whereϑn+1− ϑn−1> λ and La,−λbe
the one whereϑn+1− ϑn−1< −λ. From (2.5) it follows that
m(La,λ) = m(La,−λ), 0 ≤ λ <a + 1a , an important ingredient in the proof of (2.8).
Straightforward calculation shows that the boundaries of La,λare the line segment given by v = 0, t ∈ µ 1 a + 1, a − λ a2 ¶
, the line segment given by t = 1
a + 1, v ∈
µ
0,a − aλ − λ
a + λ
¶
2
hλ h−λ h0 v t a−aλ−λ a+λ a−λ a2 ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... O 1 1 1 a+1 1 a La,λ La,−λ ...... ...... ...... ...... ... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ... ...... ...... ...... ...... ... ...... ...... ...... ...... ... ...... ...... ...... ...... ...... ... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...Figure 2.2: The regions La,λand La,−λin Ra
We define the asymptotic frequency ofϑn+1− ϑn−1> λ as
P (ϑn+1− ϑn−1> λ) = lim
N →∞ 1
N# {n; 2 ≤ n ≤ N ,ϑn+1− ϑn−1> λ} (2.9)
and that ofϑn+1− ϑn−1> λ under the condition an+1= a as
Pa(ϑn+1− ϑn−1> λ) = lim N →∞
1
N# {n; 2 ≤ n ≤ N ,ϑn+1− ϑn−1> λ, an+1= a} .
These limits exist for almost all x and
P (ϑn+1− ϑn−1> λ) = ∞
X
a=1
Pa(ϑn+1− ϑn−1> λ).
From Theorem 1 it follows that
Pa(ϑn+1− ϑn−1> λ) = 1 log 2 Z Z La,λ dt dv (1 + t v)2.
For our computation it is convenient to rewrite (2.4) as t = a − λ
2.3.SIX PATTERNS
2
11 We find Pa(ϑn+1− ϑn−1> λ) = 1 log 2 a−aλ−λ a+λ Z 0 a−λ a2+(a+λ)v Z 1 a+1 dt dv (1 + t v)2 = 1 log 2 a−aλ−λ a+λ Z 0 t 1 + t v ¯ ¯ ¯ ¯ a−λ a2+(a+λ)v 1 a+1 dv (2.10) = 1 2 log 2 µ log(a + 1) 2 a(a + 2)+ λ alog a a + 2+ log µ 1 −λ 2 a2 ¶ +λ alog a + λ a − λ ¶ .This result, crucial for the proof of (2.8), is also obtained in ([J2]), but with consid-erably more and inconvenient calculations. The remainder of the proof of (2.8) runs similar as the one in ([J2]).
2.3.
S
IX PATTERNS
Now suppose that, for instance,ϑn−1< ϑn. Then, regarding (2.7) and (2.8),ϑn< ϑn+1 seems less probable thanϑn+1< ϑn. In the latter case one might ask whetherϑn+1<
ϑn−1is more likely thanϑn−1< ϑn+1< ϑn. In this section we determine, for almost all x, the asymptotic frequencies of the six possible patterns
A : ϑn−1< ϑn< ϑn+1, B : ϑn−1< ϑn+1< ϑn, C : ϑn< ϑn−1< ϑn+1,
D : ϑn< ϑn+1< ϑn−1, E : ϑn+1< ϑn−1< ϑn, F : ϑn+1< ϑn< ϑn−1 (2.11) where the asymptotic frequency of a pattern is defined in a similar way as in Section
2.2. As remarked, we may not expect the six patterns to all have the same asymptotic frequency, which would be 0.1666 ···.
Our investigation of the patterns is based on the division into six corresponding re-gions of the strip Ra. From the equations (1.3), (1.4) and (1.5) we deduce that g , the curve dividing the region whereϑn−1< ϑnfrom the region whereϑn< ϑn−1, is given by
v = t; that h, the curve dividing the region where ϑn−1< ϑn+1from the region where
ϑn+1< ϑn−1, is given by the graph of the Gauss operator T , v = 1
t − a on Ra. Further k,
the curve dividing the region whereϑn< ϑn+1from the region whereϑn+1< ϑn, is given by v = 11
t− a
− a, which is the graph of T2on the fundamental interval∆2(0; a, a), i.e.the
set of numbers with a continued fraction expansion of the form [0; a, a, . . .].
It is easily verified that the three curves all intersect in Ga, the fixed point ofT in Ra, and that the six regions actually correspond with the six patterns; see Figure2.3.
2
For the frequencies of the six patterns that we are investigating, we define
Qi:=
∞
[
a=1
Qi ,a, i = 1,...,6,
where the meaning of Qi ,acan be read from Figure2.3. As observed earlier,S h = h (by takingλ = 0 in (2.5)). In addition, straightforward computation shows that g and k are mapped onto each other byS . Even stronger, if in Rawe divide both g and k in a left part and a right part (with regard to the fixed point), we haveS gl = kr andS kl = gr (and vice versa). Also, underS the horizontal edge of Q3,a and the vertical edge of Q2,aare
mapped onto each other, as holds similarly for the horizontal edge of Q6,aand the
verti-cal edge of Q5,a; for the horizontal edge of Q3,a∪Q4,aand the vertical edge of Q1,a∪Q2,a;
and for the horizontal edge of Q1,a∪Q6,aand the vertical edge of Q4,a∪Q5,a.
1 1 a 1 a+1 Ga h kr gl kl gr Q1,a:ϑn+1< ϑn< ϑn−1 Q2,a:ϑn+1< ϑn−1< ϑn Q3,a:ϑn−1< ϑn+1< ϑn Q4,a:ϑn−1< ϑn< ϑn+1 Q5,a:ϑn< ϑn−1< ϑn+1 Q6,a:ϑn< ϑn+1< ϑn−1 r ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
Figure 2.3: The six regions of Ra
From this, applying Theorem 1, we derive
m(Q1) = m(Q4), m(Q2) = m(Q3) and m(Q5) = m(Q6).
Clearly
m(Q1) + m(Q5) + m(Q6) = m(Q2) + m(Q3) + m(Q4) =
1 2. and hence it follows that
2.4.THE OPTIMAL CONTINUED FRACTION
2
13
Since the six asymptotic frequencies add up to 1, we only have to compute one. It is convenient to choose Q2,a, and we find:
m(Q2,a) = 1 log 2 1 a Z ga t Z 1 t −a dv dt (1 + t v)2 = 1 log 2 1 a Z ga à v 1 + t v ¯ ¯ ¯ ¯ t 1 t −a ! dt = 1 2 log 2 à log µ 1 + 1 a2 ¶ − log à 1 2+ 1 2 r 1 + 4 a2 !! .
Summing over a, we come across two infinite products, the first of which is the well-known ∞ Y a=1 µ 1 + 1 a2 ¶ =sinhπ π = 3.6760 · · · .
The factors of the second product are 1 + a−2− a−4+ O(a−6), a → ∞. We find1
ρ := Y∞ a=1 Ã 1 2+ 1 2 r 1 + 4 a2 ! = 2.8269 · · · . Thus m(Q2) = 1 2 log 2· log sinhπ πρ = 0.1894 · · · .
Considering our earlier remarks, we have now proved Theorem 2. Define the constantρ by
ρ := Y∞ a=1 Ã 1 2+ 1 2 r 1 + 4 a2 ! = 2.8269 · · · .
Then, with the notation introduced in (2.9) and (2.11), one has for almost all x:
P (B) = P(C ) = P(D) = P(E ) = 1 2 log 2· log sinhπ πρ = 0.1894 · · · P (A ) = P(F ) =1 2− 1 log 2· log sinhπ πρ = 0.1210 · · · . (2.12)
2.4.
T
HE OPTIMAL CONTINUED FRACTION
In this section we will investigate the six asymptotic frequencies of the previous section for the optimal continued fraction (OCF). In general, we take the same approach as we
2
used for the regular RCF, based on the division of the (t , v)-plane in vertical strips in each of which the measures of all areas corresponding to the six patterns are computed, by applying Theorem1. In this section we will show how to adopt this approach to the OCF, where the situation is more complicated. For convenience we will identify a pattern P ∈ {A ,...,F }, with the region corresponding to this pattern. Throughout this thesis, we will use g :=12p5−12= 0.6180 · · · and G :=12
p
5+12= 1.6180 · · · as abbreviations of the two golden means. Note that G = g + 1 and that g =G1.
As remarked above, we obtain the convergents pn/qn, n = 1,2,3,..., by truncating the infinite continued fraction (1.1) expansion of a real irrational number x, so as to obtain good approximations of x. The approximation coefficientsϑn, n = 1,2,3,..., provide a way of measuring the quality of the approximants. In [B], Wieb Bosma introduced the optimal continued fraction as a continued fraction that is both fastest (i.e. having an expansion for which the growth rate of the denominators is maximal) and closest (i.e. having expansions for which sup {ϑk:ϑk= qk|qkx − pk|} is minimal).
Optimal as this fraction may be as to its approximating qualities, in [B,BK1] it is shown that both the subset ofR2, which we denote byΥO, and the two-sided shift op-eratorTO:ΥO→ ΥOof the ergodic system underlying the OCF are less accessible than those of the RCF: ΥO= ½ (t , v) ∈ (−1,1) × (−1,1) : v ≤ minµ 2t + 1 t + 1 , t + 1 t + 2 ¶ and v ≥ max µ 0,2t − 1 1 − t ¶¾ , see also Figure2.4, and
TO(t , v) := µ¯ ¯ ¯ ¯ 1 t ¯ ¯ ¯ ¯− a(t , v), 1 a(t , v) + sign(t)v ¶ where a(t , v) := Ì Ì Ì Ì Ì Ê ¯ ¯ ¯ ¯ 1 t ¯ ¯ ¯ ¯+ ¹¯ ¯ ¯ ¯ 1 t ¯ ¯ ¯ ¯ º + sign(t )v 2 µ¹¯ ¯ ¯ ¯ 1 t ¯ ¯ ¯ ¯ º + sign(t )v ¶ + 1 Í Í Í Í Í Ë . (2.13)
It is not hard to see thatTOworks onΥOin way similar toT on ¯Ω in the RCF case: TO(t , v) = µε 1 t − a1, 1 ε1v + a1 ¶ , and (tn, vn) = TOn(t , 0), n ≥ 0.
In [BK1] it is shown that (ΥO,BΥO, ¯µΥO,TO) forms an ergodic system, withBΥOthe col-lection of Borel subsets ofΥOand ¯µΥOthe measure with density function
dO(t , v) := 1 logG·
1
(1 + t v)2, for (t , v) ∈ ΥO. (2.14)
In particular, we have that (apart from sets with Lebesgue measure 0)TO:ΥO→ ΥOis bijective and that ¯µΥOis invariant underTO. In [BK1] the following version of Theorem
2.4.THE OPTIMAL CONTINUED FRACTION
2
15
Theorem 3. The two-dimensional sequence (tn, vn), n = 1,2,..., is for almost all
irra-tional x distributed overΥOaccording to the density function dOin (2.14).
For more detailed information about the metric properties of the OCF, see for in-stance [BK1].
From Theorem3we derive that for every Borel measurable set A ⊆ ΥO
AFO(A) = Ï A 1 logG· 1 (1 + t v)2dt dv. (2.15) ΥO −12 g2 −g2 0 t 12 g g v =12 v =2t + 1 t + 1 v =t + 1 t + 2 v =2t − 1 1 − t v
Figure 2.4: The domain of the OCF
The most important obstacle to following the approach taken in the case of the RCF is that a ’stripwise’ computation (with 1/an+1< tn< 1/an, n = 1,2,...) is not possible, due to the curved boundary of the domain of the natural extension of the OCF; see Figure
2.4. In view of (2.13), an obvious solution of this problem is regarding curved regions for every an+1= 2, 3, . . . . For (t , v) ∈ ΥO, the sign of t is obvious. However, it is not easy to find the regions ofΥOwhere the values of the digit a(t , v) is fixed using (2.13). We first will show that, apart from sets of Lebesgue measure 0, for every (t , v) ∈ ΥOa unique integer
a ≥ 2 exists for which
TO(t , v) = µε t− a, 1 a + εv ¶ =: (T,V ) ∈ ΥO,
whereε = sign(t). We first consider the points (t,v) that are sent under TOto the bound-ary ofΥO. Let (T,V ) ∈ ∂(ΥO), then we have the following three cases:
2
1. (T,V ) satisfies V =2T +11+T . In this case we obviously have that (α,β) := µε t− a−1, 1 a+1 + εv ¶ 6∈ ΥO, since T −1 =εt− a−1 < − 1
2. Now consider the point (α,β), given by
(α,β) := µ ε t− a+1, 1 a−1 + εv ¶ . In this case we have
β = 1 a + εv − 1= 1 1 V− 1 =2α − 1 1 − α ;
i.e. (α,β) is on one of the other boundary curves of ΥO. We conclude that (t , v) ∈ ΥOwas on the boundary of the regions where the digit is either equal to a or to
a − 1.
2. (T,V ) satisfies V =2T −11−T . In this case we obviously have that (α,β) := µε t− a+1, 1 a−1 + εv ¶ 6∈ ΥO,
since T +1 =εt− a+1 >32> g . Now consider the point (α, β), given by
(α,β) := µ ε t− a−1, 1 a+1 + εv ¶ . In this case we have
β = 1 a + 1 + εv = 1 1 V + 1 =2T − 1 T = 2α + 1 1 + α ;
i.e. (α,β) is on one of the other boundary curves of ΥO. Again we conclude that (t , v) ∈ ΥOwas on the boundary of the regions where the digit is either equal to a or to a + 1.
3. (T,V ) satisfies V =1+T2+T. In this case we obviously have that (α,β) := µ ε t− a+1, 1 a−1 + εv ¶ 6∈ ΥO, since T + 1 ≥ g . Now consider the point (α,β), given by
(α,β) := µ ε t− a−1, 1 a+1 + εv ¶ . In this case we have
β = 1 a + 1 + εv = 1 1 V+ 1 = 2 + T 3 + 2T = α + 3 2α + 56∈ ΥO. In this case the digit a was unique.
2.4.THE OPTIMAL CONTINUED FRACTION
2
17
Now let (t , v) ∈ ΥObe such thatTO(t , v) ∈ Int(ΥO) (here Int(S) denotes the interior of the set S). Then from the above it follows that we must have that
(α,β) := µε t− a ± 1, 1 a ∓ 1 + εv ¶ 6∈ ΥO, so we must have that a = a(t, v), i.e.
a = Ì Ì Ì Ì Ì Ê ¯ ¯ ¯ ¯ 1 t ¯ ¯ ¯ ¯+ ¹¯ ¯ ¯ ¯ 1 t ¯ ¯ ¯ ¯ º + sign(t )v 2 µ¹¯ ¯ ¯ ¯ 1 t ¯ ¯ ¯ ¯ º + sign(t )v ¶ + 1 Í Í Í Í Í Ë .
In the regular case, the value of an+1depends on tn only, but in the optimal case it depends on both tn and vn. We want to know how to determine the curves between which an+1 is constant, given tn and vn. For convenience, we will generally omit the indices n for t and v and n + 1 for a in what follows. We start in the leftmost corner of ΥO, where a = 2, εn+1= −1 and εn+2= +1. So
TO(t , v) =µ −1 t − 2, 1 2 − v ¶ (a = 2,εn+1= −1, εn+2= +1). (2.16) −12 g2 −g2 t = −135 r−2(t ) =13t +55t +2 0 TO v =12 v =2t + 1 t + 1 v =t + 1 t + 2 0 12 image of v =2t +1t +1 image of r−2 g g v =12 v =2t − 1 1 − t
Figure 2.5: The map of the leftmost corner applyingTO
The left boundary is given by (t , (2t + 1)/(t + 1)), for t between −1/2 and −g2, which TOmaps to the curve (T,V ) = (−1/t − 2,1/(2 − (2t + 1)/(t + 1))) = (−1/t − 2, t + 1), which we can write as (T, (T + 1)/(T + 2)), for T between 0 and g . The horizontal line segment with v-coordinate 0 is mapped to the horizontal line segment with V -coordinate 1/2. We now determine the right boundary, denoted by r−2= r−2(t ), such that r−2is mapped to
2
the upper right boundary ofΥO. Applying (2.16), we want to write (−1/t − 2,1/(2 − r−2(t )))
as (T, (2T − 1)/(1 − T )). A straightforward calculation yields r−2(t ) = (13t +5)/(5t +2) (see
Figure2.5).
This procedure is easily copied to the rightmost side ofΥO. This time we have a = 2,
εn+1= εn+2= +1. Now TO(t , v) =µ 1 t− 2, 1 2 + v ¶ (a = 2,εn+1= εn+2= 1). (2.17) The right boundary is given by (t , (2t − 1)/(1 − t)), for t between 1/2 and g , which TO maps to the curve (T, (T + 1)/(T + 2)), for T between −g2 and 0. The upper bound-ary is part of (t , (t + 1)/(t + 2)), its rightmost point being (g , g ), which is mapped to the curve (T, (2T + 5)/(5T + 13)), with leftmost point on T = −g2. We now determine the left boundary, denoted by l2= l2(t ), such that l2is mapped to (T, (2T − 1)/(1 − T )).
Apply-ing (2.17), we want to write (1/t − 2,1/(2 + l2(t ))) as (T, (2T − 1)/(1 − T )). We find l2(t ) =
(13t − 5)/(2 − 5t); see Figure2.6. l2(t ) =13t − 5 2 − 5t t =135 TO 1 2 v =t + 1 t + 2 v = g g v =12 0 v = 2t + 5 5t + 13 1 2 v =t + 1 t + 2 g g v =12 t = −g2 v =2t − 1 1 − t
Figure 2.6: The map of the rightmost strip applyingTO
Proceeding similarly, we establish formulas for all combinations of a = 3,4,...; εn+1 andεn+2. We remark that the boundary between two regions with equal a andεn+1are separated by the line t =εn+1
a , where r−a(t ) =(2a 2+ 2a + 1)t + 2a + 1 (2a + 1)t + 2 and la(t ) =−(2a 2+ 2a + 1)t + 2a + 1 (2a + 1)t − 2 .
We conclude thatTOmaps vertical regions from the left and the right side ofΥO alter-nately to horizontal regions from the top ofΥOdownwards; see Figure2.7.
2.4.THE OPTIMAL CONTINUED FRACTION
2
19 TO l2(t ) =−13t + 5 5t − 2 l3(t ) =−25t + 7 7t − 2 r−2(t ) =13t + 5 5t + 2 r−3(t ) =25t + 7 7t + 2 −12 −135 −257 0 12 g g −2= a −3= a 3 = a 2 =aimage region for a = −2 image region for a = 2 image region for a = −3
image region for a = 3
−12 0 12 g g 5 13 7 25
Figure 2.7: The alternating character of the mapTO
In Figure2.7we have not yet processed the value ofεn+2, which is indispensable for determining the six patterns. In Figure2.8, confining ourselves to the leftmost part of ΥO, we show how the six patterns are spread out overΥO, for a = 2,3,.... We have filled the regions with different shades of grey, such that patternA has the lightest shade and F has the darkest.
(εn+1,εn+2) (-1,-1) (-1,1) (1,-1) (1,1) ϑn−1= ϑn v = −t v = −t v = t v = t ϑn−1= ϑn+1 v = a +1t v = a2t +a at +2 v = a2t −a −at +2 v = −a + 1 t ϑn= ϑn+1 v =(a 2−1)t +a at +1 v = (a2+1)t +a at +1 v = (−a2+1)t +a at −1 v = (a2+1)t −a −at +1 ϑn−1= ϑn= ϑn+1 t =−a+ p a2−4 2 t = −a2−2+pa4+4 2a t = 2−a2+pa4+4 2a t =−a+ p a2+4 2
Table 2.1: The curves and their intersection per sign tuple.
Now that we have established a way of dividingΥO in regions where a, εn+1and
εn+2are constant, we will show how we compute the measure of all regions. From the relations (1.3), (1.4) and (1.5) we derive for each of the four possible ordered sign tuples (εn+1,εn+2) the three curves that establish the six patterns. In each strip, that is, for every
a ≥ 2, we will now draw the curves that divide the strip in regions that correspond with
the patternsA through F , for which we will use Table2.1. Recall that for convenience we use t := tn, v := vnand a := an+1.
Finally, in Figure2.9we have a generic situation for the patterns: we know the values ofεn+1andεn+2 (which in Figure2.9is −1 for both of them) and all patterns actually occur, which is not the case in the leftmost and the rightmost regions. In Figure2.9we have indicated the formulas belonging to the curves drawn and some noteworthy values
2
sign(εn+1) sign(εn+2) an+1 − + 2 − − 3 − + 3 − − 4 ... ... ... v =2t +1t +1 v =t +1t +2 v = −t −135 −13 −257 −14 B A A B E E B A A B E F F D D F C CFigure 2.8: The six patterns forεn+1= −1
of t .
For convenience, we have omitted the coordinates of most intersection points, which are a bit lenghty in some cases. For instance, the t -coordinate of the intersection of
v =(2a2−2a+1)t +2a−1(2a−1)t+2 (which is actually ra−1) and v = −t is p
4a4− 8a3+ 4a + 5 − (2a2− 2a + 3)
4a − 2 · (2.18)
The calculation of the measure of areas such asCainvolves computing the sum of two double integrals, the limits of which are expressions such as (2.18). Computing the mea-sures of all pattern regions for all four cases would obviously be very tedious and de-manding, and therefore it is convenient that several areas prove to have the same mea-sure. As in [JJ], we use a composed operator, which in the case ofεn+1= −1 is
S−
O := R−TO, R−being the reflection
R−(t , v) = (−v,−t).
This operatorS−
O is an involution that is measure-preserving with respect to the mea-sure m that was introduced on page6. We will show howS−
O works on the regions shown in Figure2.9, whereεn+2= −1 holds as well. We have (leaving the computations to the reader) S− O{(t , v) : v = 1 t+ a} = {(t , v) : v = 1 t+ a}; S− Or−(a−1)= {(t , v) : v =t +1t +2}; S− O{(t , v) : v = 0} = {(t, v) : t = − 1 a}; S− O{(t , v) : v = −t} = {(t, v) : t = (a2−1)t +a at +1 }. (2.19)
2.4.THE OPTIMAL CONTINUED FRACTION
2
21 Aa:ϑn−1< ϑn< ϑn+1 Ba:ϑn−1< ϑn+1< ϑn Ea:ϑn+1< ϑn−1< ϑn Fa:ϑn+1< ϑn< ϑn−1 Da:ϑn< ϑn+1< ϑn−1 Ca:ϑn< ϑn−1< ϑn+1 v =t +1t +2 v =1t+ a v = −tr−(a−1): v =(2a2−2a+1)t+2a−1(2a−1)t+2 v =(a2−1)t+aat +1 t = −1a
−1a 1−2a 2a2−2a+1 −a+ p a2−4 2 a 1−a2
Figure 2.9: The six regions forεn+1= εn+2= −1
ForX ∈ {A ,...,F }, we set Xε1/ε2
a = {(x, y) ∈ X | a1(x) = a,ε1(x) = ε1,ε2(x) = ε2}. Now,
using (2.19), we easily derive the following (whileεn+1= εn+2= −1): m(Aa−/−) = m(Fa−/−); m(B−/− a ) = m(Ea−/−); m(Ca−/−) = m(D−/−a ); m(A−/− a ∪ B−/−a ∪ Ca−/−) = m(D−/−a ∪ Ea−/−∪ Fa−/−). (2.20)
This is exactly what was found in the case of the RCF. We note, however, that at this place we are only dealing with the situationεn+1= εn+2= −1, while in the case of the RCF one always hasεn+1= εn+2= 1. Still, we can confine ourselves to computing three relatively easy measures, say m(Ca−/−), m(Ea−/−) and m(D−/−a ∪ Ea−/−∪ Fa−/−).
We have (in the caseεn+1= εn+2= −1, a ≥ 4):
m(Ca−/−) = −a+pa2−2a+2 a−1 Z p 4a4−8a3+4a+5−(2a2−2a+3) 4a−2 (2a2−2a+1)t+2a−1 (2a−1)t+2 Z −t dt dv (1 + t v)2+ −a+pa2−4 2 Z −a+pa2−2a+2 a−1 1 t+a Z −t dt dv (1 + t v)2, which is 1 2 Ã log p 4a4− 8a3+ 4a + 5 + 2a2 − 2a − 1 2 + log a −pa2− 4 2 + log( p a2− 2a + 2 − a + 1) !
2
and can be written as 1 2log p (2a2− 2a − 1)2+ 4 + 2a2 − 2a − 1 (a +pa2− 4)(p (a − 1)2+ 1 + a − 1)· Then, m(Ea−/−) = −1 a Z −a+pa2−4 2 −t Z 1 t+a dt dv (1 + t v)2= 1 2log a −pa2− 4 2 + 1 2log a2− 1 a · Finally, m(Da−/−∪ Ea−/−∪ Fa−/−) = −1 a Z −a+pa2−2a+2 a−1 t +1 t +2 Z 1 t+a dt dv (1 + t v)2 =1 2log 2a2− 2a + 1 a + 1 2log( p (a − 1)2+ 1 − (a − 1)).
Applying (2.20), we find forεn+1= εn+2= −1 and a ≥ 4: m(A−/− a ) = m(Fa−/−) =12log
(2a2−2a+1)(apa2−4+a2−2) (a2−1)(p(2a2−2a−1)2+4+(2a2−2a−1));
m(B−/−a ) = m(Ea−/−) =12loga− p a2−4 2 + 1 2log a2−1 a ; m(C−/− a ) = m(Da−/−) =12log p
(2a2−2a−1)2+4+2a2−2a−1 (a+pa2−4)(p(a−1)2+1+a−1)·
(2.21)
We remark that although for a = 3 patterns C−/−
a andD−/−a do not occur, the formula in (2.21) still holds, for it gives m(C−/−
a ) = m(Da−/−) = 0.
In the case thatεn+1= εn+2= 1, the approach is completely analogous, including the use of
S+
O := R+TO, R+being the reflection
R+(t , v) = (v, t),
instead ofSO−. In this case we find, for a ≥ 3, m(Aa+/+) = m(Fa+/+) =12log
(a2+2+apa2+4)(2a2+2a+1) (a2+1)(2a2+2a+3+p(2a2+2a+3)2−4);
m(B+/+ a ) = m(Ea+/+) =12log p a2+4−a 2 + 1 2log a2+1 a ; m(C+/+ a ) = m(Da+/+) =12log
2a2+2a+3+p(2a2+2a+3)2−4 (pa2+4+a)(p(a+1)2+1+(a+1))·
2.4.THE OPTIMAL CONTINUED FRACTION
2
23
In the cases whereεn+1· εn+2= −1, we get hold of the six patterns with a mixture of S−
O andS+ O: S+ O(Aa−/+) = Fa+/−; SO−(Fa+/−) = Aa−/+; S+ O(Ba−/+) = Ea+/−; SO−(Ea+/−) = Ba−/+; S+ O(Ca−/+) = Da+/−; SO−(Da+/−) = Ca−/+; S+ O(Da−/+) = Ca+/−; SO−(Ca+/−) = Da−/+; S+ O(Ea−/+) = B+/−a ; SO−(B+/−a ) = Ea−/+; S+ O(Fa−/+) = Aa+/−; SO−(Aa+/−) = Fa−/+, (2.23)
and we find, for a ≥ 3, m(A−/+ a ) = m(Fa+/−) =12log (pa4+4−a2)(a2+1)
(2a2+2a+1)(p(2a2+2a−1)2+4−(2a2+2a−1);
m(B−/+a ) = m(Ea+/−) =12log
a2+2+pa4+4 2(a2+1) ;
m(Ca−/+) = m(Da+/−) =12log
(a2−2+pa4+4)(a2+a+1+(a+1)pa2+1)
a2(p(2a2+2a−1)2+4+(2a2+2a−1)) ;
m(D−/+a ) = m(Ca+/−) =12log
(a2+2+pa4+4)(a2−a+1+(a−1)pa2+1)
a2(2a2−2a+3+p(2a2−2a+3)2−4) ;
m(E−/+ a ) = m(Ba+/−) =12log a2−2+pa4+4 2(a2−1) ; m(Fa−/+) = m(Aa+/−) =12log (pa4+4−a2)(a2−1)
(2a2−2a+1)((2a2−2a+3−p(2a2−2a+3)2−4)·
(2.24)
We are almost able to give the total sum measure of all the six patterns. To actually do so, we have yet to compute the measures of the regions in the leftmost and the rightmost part ofΥO, where a = 2. In the case εn+1= εn+2= 1, we can apply the formulas of (2.22). In the caseεn+1·εn+2= −1, the patterns C and D do not occur. In fact, on the left side we only haveA and B and on the right side we only have E and F , which can be mutually mapped onto each other as in (2.23); see Figure2.102. To compute their measures, we apply the formulas in (2.24).
2For visual purposes, we used different scaling for the left part and the right ofΥ
O, as a result of which not
2
S+ O v =2t +1t +1 v =t +1t +2 −12 −25−135 12 g v = t v =2t −11−t B A F EFigure 2.10: The four deviant regions, where a = 2 and εn+1· εn+2= −1
Now we can compute the total sum measures of all regions: PatternsA and F :
m(A ) = m(F ) = m(A2−/+)+m(A2+/+)+∞P a=3
¡m(A−/−
a ) + m(Aa−/+) + m(Aa+/−) + m(Aa+/+) ¢ =12log 3+p5 2 + 1 2log 5 13 + 1 2log 13(6+4p2) 5(15+p221) + ∞ P a=3 1 2log
2a2+2a−1+p4a4+8a3−4a+5 2a2−2a−1+p4a4−8a3+4a+5+ ∞ P a=3 1 2log ³
2a2−2a+3+p4a4−8a3+16a2−12a+5´³apa2−4+a2−2´³apa2+4+a2+2´ 2³2a2+2a+3+p4a4+8a3+16a2+12a+5´³a4+2+a2pa4+4´ .
Applying the principle of telescoping series, we can reduce this to
1 2log ³ 3 + 2p2´+12log³p5 − 2´+ ∞ X a=3 1 2log ³ apa2− 4 + a2 − 2 ´ ³ apa2+ 4 + a2 + 2 ´ 2 ³ a4+ 2 + a2pa4+ 4´ , and finally to log³p2 + 1´−12logG 3 + ∞ X a=3 log ³ a +pa2− 4´ ³a +pa2+ 4´ 2³a2+pa4+ 4´ ,
2.4.THE OPTIMAL CONTINUED FRACTION
2
25 3 2logG + ∞ X a=2 log ³ a +pa2− 4´ ³a +pa2+ 4´ 2³a2+pa4+ 4´ .In order to facilitate numerical computations, we write this last expression as
3 2logG + ∞ X a=2 log ³ 1 + q 1 −a42 ´ ³ 1 + q 1 +a42 ´ 2³1 +q1 +a44 ´ . PatternsB and E : m(B) = m(E ) = m(B2−/+)+m(B+/+2 )+ ∞ P a=3 m(Ba−/−)+m(B−/+a )+m(Ba+/−)+m(Ba+/+) =12log³3+ p 5 5 ´ + 12log¡p2 − 1¢+1 2log 5 2 + ∞ P a=3 1 2log 2³a2+pa4+4´ ³ a+pa2−4´³(a+pa2+4´, shortly 1 2logG 2+1 2log¡p2 − 1¢+ ∞ P a=3 1 2log 2³a2+pa4+4´ ³ a+pa2−4´³
(a+pa2+4´, which can be simplified further
to −12logG + ∞ P a=2 1 2log 2³a2+pa4+4´ ³ a+pa2−4´³(a+pa2+4´or −1 2logG + ∞ X a=2 1 2log 2³1 +q1 +a44 ´ ³ 1 +q1 −a42 ´ ³ 1 +q1 +a42 ´. PatternsC and D: m(C ) = m(D) = m(C+/+ 2 ) + ∞ P a=3 m(C−/− a ) + m(Ca−/+) + m(Ca+/−) + m(Ca+/+) = ∞ X a=3 µ 1 2log ³ 2a2− 2a − 1 +p4a4− 8a3+ 4a + 5´ ³2a2 + 2a + 3 +p4a4+ 8a3+ 16a2+ 12a + 5´ ³ 2a2+ 2a − 1 +p4a4+ 8a3− 4a + 5´ ³2a2− 2a + 3 +p4a4− 8a3+ 16a2− 12a + 5´ +1 2log 2³a2+pa4+ 4´ ³2a2 + 1 + 2apa2+ 1´ ³ a +pa2− 4´ ³a +pa2+ 4´ ³a + 1 +pa2+ 2a + 2´ ³a − 1 +pa2− 2a + 2´ ¶ + 12log 15 +p221 2 + 1 2log ³p 10 − 3´+12log ³p 2 − 1´.
