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Ergodi
Properties of Continued
Fra
tion Algorithms
Ergodi
Properties of Continued Fra
tion Algorithms
Proefs
hrift
ter verkrijging van de graad van do
tor
aan de Te
hnis
he Universiteit Delft,
op gezag van de Re
tor Magni
us prof. dr. ir. J. T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 3 februari 2003 om 13.30 uur
door
Yusuf HARTONO
Master of S
ien
e in Applied Mathemati
s,
University of Missouri-Rolla, USA
geboren te Kundur, Indonesie.
Dit proefs
hrift is goedgekeurd door de promotor:
Prof. dr. F. M. Dekking
Toegevoegd promotor: Dr. C. Kraaikamp
Samenstelling promotie
ommisie:
Re
tor Magni
us
Prof. dr. F. M. Dekking
Dr. C. Kraaikamp
Prof. dr. J. M. Aarts
Prof. dr. M. Iosifes
u
Prof. dr. F. S
hweiger
Prof. dr. R. K. Sembiring
Dr. W. Bosma
voorzitter
Te
hnis
he Universiteit Delft, promotor
Te
hnis
he Universiteit Delft, toegevoegd promotor
Te
hnis
he Universiteit Delft
Romanian A
ademy of S
ien
es, Roemenie
Universitat Salzburg, Oostenrijk
Institut Teknologi Bandung, Indonesie
Katholieke Universiteit Nijmegen
The resear
h in this thesis has been
arried out under the auspi
es of
the Thomas Stieltjes Institute for
Mathemati
s, at the University of
Te
hnology in Delft.
Published and distributed by: DUP S
ien
e
DUP S
ien
e is an imprint of
Delft University Press
P.O. Box 98
2600 MG Delft
The Netherlands
Telephone: +31 15 27 85 678
Telefax: +31 15 27 85 706
E-mail: InfoLibrary.TUDelft.NL
ISBN 90-407-2381-8
Keywords: metri
, arithmeti
,
ontinued fra
tions
Copyright
2002 by Y. Hartono
All rights reserved. No part of the material prote
ted by this
opyright noti
e
may be reprodu
ed or utilized in any form or by any means, ele
troni
or me
hani
al, in
luding photo
opying, re
ording or by any information storage and
retrieval system, without written permission from the publisher: Delft University Press.
Cover designed by Silvia Yulianti
Printed in The Netherlands
To the memory of my mother
As far as the laws of mathemati
s refer to reality,
they are not
ertain;
and as far as they are
ertain,
they do not refer to reality.
Albert Einstein (1879 - 1955)
The beginning of knowledge is
the dis
overy of something we do not understand.
Frank Herbert (1920 - 1986),
The fear of the
Ameri
an Writer
Lord is the beginning of knowledge, : : :.
For from him and through him
and to him are all things.
To him be the glory forever! Amen.
Proverbs 1:7; Romans 11:36
A
knowledgments
I am deeply indebted to many institutions and persons without whom this thesis would
never exist. At the rst pla
e I would like to express my gratitude and appre
iation to
my supervisor Cor Kraaikamp, who introdu
ed me to this ex
iting resear
h area, for
his many interesting ideas and
onstant help that kept me in the right dire
tion, for
his en
ouragement that kept me going, and for his patien
e that
omforted me during
the diÆ
ult time in the resear
h. It has been a great pleasure to meet Karma and
Rafael. Also, I would like to express my gratefulness to my promotor Mi
hel Dekking
for inviting me to the Netherlands and providing me with the opportunity to
arry
out my do
toral resear
h at Delft University of Te
hnology (TU Delft). I should also
thank my
ommittee for their valuable
omments and suggestions.
The resear
h leading to this thesis was a part of the s
ienti
ooperation between
Dut
h and Indonesian governments. I am very grateful to the program
oordinators
Prof. Dr. R. K. Sembiring and Dr. A. H. P. van der Burgh who four and a half
years ago organized a resear
h workshop in Bandung that gave me a
han
e to be
sele
ted for this do
toral program at TU Delft. I thank Dr. O. Simbolon and Dr. B.
Karyadi, former proje
t managers of Proyek Pengembangan Guru Sekolah Menengah
(PGSM) { (Se
ondary S
hool Tea
hers Development Proje
t), in Jakarta as well as
Drs. P. Althuis, dire
tor of Center for International Cooperation in Applied Te
hnology
(CICAT) at TU Delft, for nan
ial support and assistan
e during my resear
h and stay
in Holland.
I personally wish to thank all members of Afdeling CROSS, in parti
ular vakgroup
SSOR, for de gezelligheid and a very
ondu
ive atmosphere, espe
ially Cindy, Ellen,
and Diana for their both administrative and non-administrative assistan
e, and Carl
for his
omputer assistan
e. I also wish to thank Durk, Christel, Theda, Veronique,
and Rene for a very wonderful friendship and for providing me with almost everything
I needed during my stay in Holland. My thanks should also go to all PGSM fellows
(Abadi, Budi, Caswita, Darmawijoyo, Gede, Happy, Hartono, Kusnandi, Sahid, Siti,
and Suyono) with whom I started the whole proje
t together for everything we have
had together from fun to serious dis
ussions about mathemati
s, parti
ularly to Suyono
with whom I shared an oÆ
e room and from whom I learned a lot about measure
theory; as well as to Agus, Komo and Julius with whom I spent some time sharing an
apartment in Delft.
A lot of help and en
ouragement also
ame from my
olleagues at the Fa
ulty
of Tea
her Training and Edu
ation, parti
ularly at the Department of Mathemati
s
Edu
ation, Sriwijaya University { Pak Ismail, Pak Purwoko, Bu Tri, and Somakim,
just to name some, as well as from Kapten O. Tengke, Happy and other members of
Bala Keselamatan Korps Kundur, and Kak Ibrahim. I would like to a
knowledge them
here as well.
ix
x
I have appre
iated all the suggestions,
orre
tions, and
omments during the preparation of this thesis. I am the only one responsible for any mistake remaining in this
thesis.
I am also pleased to a
knowledge all members of Het Leger des Heils Korps Delft,
espe
ially Majoor en mevrouw Loef and Kapitein en mevrouw Jansen, and of International Student Chaplain
y, espe
ially Fr. Ben and Rev. Stroh, for their generosity and
warm hospitality making me really feel at home and letting me parti
ipate in their
many a
tivities; to mention a few of them: Daniela, Bernardette, Carla, Ri
ardo,
Yenory, Mar
o, Bibiana, Sandra, Sarah, Yadira, Susanne, Irek, Fabiana, Carmen,
Ralph, Paul, Poni, Duleep, Ni
ol
o, and Fabio. Many thanks to Mieke and Reini for
onsidering me as kind aan huis in their home and family. Ik vind het heel leuk om
met Sara, David, en Nathan kennis te maken. Ik wil Tilak, Mart, en Jose, met alle
medewerkers ook bedanken voor de samenwerking en een hele mooie vriends
hap.
Moreover, I
ertainly enjoyed wonderful times together and very warm
ompanionship with Indonesian students, espe
ially from TU Delft and IHE Delft, during my
study
Bernadeta, Yusuf, Dwi, Teresia, Sri, Helena, Arief, Elisa, Raymon, Silvia,
Dian, Dedy, Reiza, Evy, Joy
e, Diah, Hilda, Firdian, Susi, Theresia, Zenlin, and Susana, just to mention a few. Their presen
e in my life shows that strong bonds of
love in servi
e with ea
h other are essential for developing a pea
eful and harmonious
ommunity. They deserve an a
knowledgement too.
I would like to dedi
ate this thesis to the memory of my mother who passed away
on January 3, 2001 at the age of 65. She taught me how to thank and have faith in
God, how to pray, and how to love and serve others. My father, who has taught me
the meaning of hard-working life and, more importantly, how to do arithmeti
during
my early years at s
hool, also deserves my spe
ial thanks for without him I
ould never
be what I am now; and so does my aunt who took
are of my son while my wife was
working.
Saya ingin mempersembahkan tesis ini sebagai peringatan pada ibu saya yang
meninggal dunia pada tanggal 3 Januari 2001 pada usia 65. Dia telah mengajar
saya bagaimana bersyukur dan memiliki iman kepada Allah, bagaimana berdoa, dan
bagaimana mengasihi serta melayani orang lain. Ayah saya yang mengajar saya arti
kehidupan dan kerja keras dan, yang lebih penting lagi, mengajar saya berhitung pada
tahun-tahun awal saya di sekolah, juga pantas mendapat u
apan terima kasih khusus
karena tanpa dia saya tidak mungkin menjadi seperti saya sekarang, demikian juga
bibi saya yang mengurus anak saya saat istri saya bekerja.
Finally, a sin
ere expression of thanks should go to my dearest wife Elfarida and
our beloved son Gabriel Ekoputra for their love, patien
e, and understanding during
my absen
e in the family due to my study.
Terima kasih yang tulus saya sampaikan kepada istri saya dan anak kami ter
inta
untuk kasih, kesabaran, dan pengertian mereka selama saya tidak berada dalam keluarga.
Above all, I thank God for everything that has happened in my life.
Delft, O
tober 17, 2002
YH
:
:
Contents
1
Introdu
tion
1.1 Some histori
al ba
kground and basi
properties .
1.2 More re
ent developments . . . . . . . . . . . . . .
1.3 Some results in ergodi
theory . . . . . . . . . . . .
1.4 Approximation
oeÆ
ients . . . . . . . . . . . . . .
1.5 A brief des
ription of the thesis . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
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2
2.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . .
2.2 Insertions, singularizations and the OddCF . . . .
2.2.1 A singularization/insertion algorithm . . . .
2.2.2 Metri
al properties of the OddCF . . . . .
2.2.3 Approximation
oeÆ
ients . . . . . . . . . .
2.3 Grotesque
ontinued fra
tions . . . . . . . . . . . .
2.4 Other odd
ontinued fra
tions . . . . . . . . . . . .
2.4.1 Maximal OddCF's . . . . . . . . . . . . . .
2.4.2 Non-maximal expansions with odd digits .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
2
6
8
10
12
15
Odd Continued Fra
tions
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17
3
3.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Basi
properties . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Ergodi
properties . . . . . . . . . . . . . . . . . . . . . .
3.4 On Ryde's
ontinued fra
tion with non de
reasing digits .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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39
4
4.1 Introdu
tion . . . . . . . . . . . . . . . . . . .
4.2 Tong's pre-spe
trum for the NICF . . . . . .
4.3 Tong's spe
trum for Nakada's -expansions .
4.3.1 The
ase g < 1 . . . . . . . . . . .
4.3.2 The
ase 12 g . . . . . . . . . .
4.4 Semi-regular
ontinued fra
tions . . . . . . .
.
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61
Engel Continued Fra
tions
Tong's Spe
trum for SRCF Expansions
xi
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1
17
21
21
23
27
32
34
34
35
37
39
42
47
55
59
61
64
69
70
73
77
CONTENTS
xii
Bibliography
5
A Note on Hurwitzian Numbers
79
81
5.1
Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.2
Hurwitzian numbers for the NICF
82
5.3
Hurwitzian numbers for the ba
kward
ontinued fra
tion . . . . .
84
5.4
Hurwitzian numbers for
. . . . . . . . . . . . . . .
86
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Bibliography
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
-expansions
Mikowski's DCF Expansions of Hurwitzian Numbers
93
6.1
Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.2
Minkowski's DCF . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.3
DCF-Hurwitzian expansion
Bibliography
. . . . . . . . . . . . . . . . . . . . .
95
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
Samenvatting
Bibliography
101
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Curri
ulum Vitae
105
107
Chapter 1
Introdu
tion
Mathemati
s is the queen of the s
ien
e,
and number theory is the queen of mathemati
s.
Carl Friedri
h Gauss (1777 { 1855)
This thesis
onsists of ve papers
on
erning
ontinued fra
tion expansions in
onne
tion with ergodi
theory. Most of them deal with metri
al properties of
ontinued fra
tion algorithms. Other aspe
ts like approximation
oeÆ
ients are
also studied here. Relationships among di erent
ontinued fra
tion expansions
are developed via the singularization and insertion pro
esses.
Throughout this thesis by a
ontinued fra
tion expansion of any real number
we mean an expression of the form
a0
e1
+
a1
;
e2
+
a2
.
+ .. +
(1.1)
en
.
+ ..
where a0 2 Z, an are positive integers and en 2 R for n = 1; 2; : : :. In a mu
h
more
onvenient way we write (1.1) more
ompa
tly as
an
[ a0 ; e1 =a1 ; e2 =a2 ; ; en =an ; ℄ :
The terms a1 ; a2 ; : : : are
alled the partial quotients of the
ontinued fra
tion.
The number of terms may be nite or in nite.
In
ase en = 1 for all n = 1; 2; : : :, we
all (1.1) a regular
ontinued fra
tion
(RCF) expansion and write it as
[ a0 ; a1 ; a2 ; ; an ; ℄ :
x,
(1.2)
In general, let x be a real number, a0 = bx
, the largest integer not ex
eeding
and write x = a0 + . Now = x a0 2 [0; 1), and we write
x
= [ a0 ; a1 ; a2 ; ℄
1
INTRODUCTION
2
if the RCF-expansion of
is given by
= [ 0; a1 ; a2 ; ℄:
In the next two se
tions some histori
al ba
kground and basi
properties of
regular
ontinued fra
tions will be presented, followed by a se
tion on a few
basi
fa
ts in ergodi
theory and another on approximation
oeÆ
ients. A brief
des
ription of the
ontent of the papers in this thesis will
on
lude this
hapter.
1.1
Some histori
al ba
kground and basi
properties
Continued fra
tions have a long history. It starts with the pro
edure known
as Eu
lid's algorithm for nding the greatest
ommon divisor (g.
.d.) of two
integers whi
h o
urs in the seventh book of Eu
lid's Elements (
. 300 b.
.).
This pro
edure is perhaps the earliest step towards the development of the
theory of
ontinued fra
tions.
To see the relation between Eu
lid's algorithm and (regular)
ontinued fra
tions,
onsider Eu
lid's algorithm for nding the g.
.d. of two integers a and b
with a > b > 0. We rst let a0 = ba=b
. Putting
r1
:= a
a0 b;
r0
:= b ;
we have to nd positive integers ai su
h that
ri
1 = ai ri + ri+1 ;
(1.3)
where 0 ri+1 < ri , for i = 1; 2; : : : until the pro
edure stops; that is, when we
have rea
hed an index n su
h that rn 6= 0 and rn+1 = 0. In this
ase, we say
that rn is the g.
.d. of a and b.
Dividing (1.3) through by ri , we get
ri
ri
Writing
ri+1
ri
1 = a + ri+1 ;
i
i
ri
=
1
ai+1
r
+ i+2
;
= 1; 2; : : : n:
i
= 1; 2; : : : ; n;
ri+1
and substituting it into the previous equation for ea
h i yield
a
b
= [ a0 ; a1 ; a2 ;
whi
h is the RCF expansion of
a=b.
; a
n
℄;
Some histori
al ba
kground and basi
properties
3
To generalize Eu
lid's algorithm to irrational numbers x in the unit interval,
onsider the
ontinued fra
tion map T : [0; 1) ! [0; 1) de ned by
1
( ) :=
T x
b1
x
; x
x
6= 0;
(0) := 0:
T
(1.4)
De ne further a1 = a1 (x) = b1=x
and an = a1 (T n 1 (x)); n = 0; 1; : : :, where
0
n
n 1 (x)). We then have by (1.4)
T (x) = x and T (x) = T (T
x
=
1
a
1 + T (x)
= = [ 0; a1 ; a2 ;
n
n + T (x) ℄:
; a
For rational numbers repeated appli
ation of T is in fa
t equivalent to Eu
lid's
algorithm. Hen
e, there exists an n0 2 N su
h that T n0 (x) = 0 and it follows
that a rational number has a nite RCF expansion. This is not the
ase for
irrational numbers. If x is an irrational number, then T n (x) is irrational for all
n 0.
A nite trun
ation in (1.2) gives the so-
alled regular
onvergents
P
n
= [ a0 ;
n
1 2
a ; a ;
Q
n ℄;
; a
n
= 1; 2; : : : ;
(1.5)
where we assume that Qn > 0 and that g
d(Pn ; Qn ) = 1.
The sequen
es (Pn )n 1 and (Qn )n 1 satisfy the following re
ursive formulae
P
1 = 1;
P
0 = a0 ;
P
Q
1 = 0;
Q
n = an P n
0 = 1;
Q
n = an Qn
1 + Pn 2
for n 1;
1 + Qn 2
for n 1;
and the relationship
P
1
n Qn
n
P
1 Qn = ( 1)n 1 :
(1.6)
Moreover, the regular
onvergents satisfy the following inequalities:
0
0
P
Q
<
2
2
P
Q
<
<
P
2n
2n
Q
<
<
P
2n+1
2n+1
Q
<
<
3
3
P
Q
<
P
1:
1
Q
For any irrational number x we say that (1.2) is the RCF expansion of x in
ase
n
= x:
n!1 Qn
lim
P
See, for instan
e, [O℄, [IK℄, and [HW℄ for more properties of RCF and proofs.
In general, (1.1) is
alled a semi-regular
ontinued fra
tion (SRCF) in
ase a0 2
Z, an are positive integers, and en = 1 for all n 1, subje
t to the
ondition
e
n+1 + bn
1
;
for n 1;
INTRODUCTION
4
and with the restri
tion that in the in nite
ase
en+1 + bn 2; in nitely often.
Nakada's -expansions, for 2 [1=2; 1℄, are examples of SRCF expansions.
Clearly, the RCF expansion ( = 1), the nearest integer
ontinued fra
tion
(NICF) expansion ( p= 1=2), and Hurwitz' singular
ontinued fra
tion (g expansion, with g = ( 5 1)=2,) are all SRCF expansions. Other examples
of SRCF expansions
onsidered in this thesis are Minkowski's diagonal
ontinued fra
tion (DCF) and odd
ontinued fra
tion (OddCF).
In the area of appli
ations, the great Dut
h mathemati
ian, me
hani
ian, astronomer, and physi
ist, Christiaan Huygens (1629-1695) used the regular
onvergent to obtain the
orre
t ratio for the rotations of planets when he designed
the toothed wheels of a planetarium. He des
ribed this in his Des
riptio Autamati Planetarii, published posthumously in 1698. This is in fa
t a
onsequen
e
of the fa
t that
ontinued fra
tions give the \best" rational approximations to
irrational numbers.
The modern theory of
ontinued fra
tions began with the writings of Rafael
Bombelli, born in about 1530 in Bologna. He showed, for example, in our
modern notation,
p
13 = [ 3; 4=6; 4=6; ℄:
Pietro Antonio Cataldi (1548-1626) also deserves some
redits in
ontinued fra
tions. He expressed
p
18 = [ 4; 2=8; 2=8; ℄:
Using Eu
lid's algorithm for nding the g.
.d. of 177 and 233, Daniel S
hwenter
(1585-1636) found the
onvergents 79=104; 19=25; 3=4; 1=1; and 0=1. It is probably in Aritmeti
a In nitorum (1655), a book by John Wallis, that the term
ontinued fra
tion was used for the rst time. Great mathemati
ians su
h as Euler
(1707-1783), Lambert (1728- 1777), Lagrange (1736-1813), Gauss (1777-1855),
and many others also made important
ontributions to the earlier development
of the theory of
ontinued fra
tions. It is in parti
ular Euler's great memoir, De
Fra
tionibus Continius (1737), that laid the foundation for the modern theory.
See, for example, [O℄, [K1℄, [S℄, and [Di℄ for more history of
ontinued fra
tions.
The metri
al theory of
ontinued fra
tions started with Gauss' problem. In
his diary on O
tober 25, 1800, Gauss wrote (in modern notation) that
lim Fn (z ) =
!1
n
log (z + 1)
; z 2 [0; 1);
log 2
(1.7)
where Fn (z ) = (T n (x) < z ); z 2 [0; 1). Here T is the
ontinued fra
tion map
de ned in (1.4) and denotes the Lebesgue measure. In a letter dated January
30, 1812, he asked Lapla
e to prove (1.7) and to estimate the error-term
en (z ) := Fn (z )
log (z + 1)
:
log 2
Some histori
al ba
kground and basi
properties
5
More than a
entury later this problem was solved by Kuzmin [Ku℄. He showed
in 1928 that
p
en (z ) = O(q n ) as n ! 1
for some
onstant q 2 (0; 1). His proof is reprodu
ed in Khin
hine [Kh℄. One
year later Levy independently proved that
jen (z )j < qn ; n = 1; 2; : : :
p
with q = 3:5 2 2 = 0:67157 . See Subse
tion 1.3.5 in [IK℄ for an improved
version of Levy's solution to Gauss' problem. In 1961 P. Szusz used Kuzmin's
approa
h to nd that q = 0:485. Gauss' problem was settled by Wirsing [Wi℄
who in 1974 found that q = 0:303 663 002 . Results like these are now known
as Gauss-Kuzmin-Levy Theorems. The following result is a
onsequen
e of these
results.
Theorem 1.1 (Levy, 1929) For almost all x 2 [0; 1) with RCF expansion
(1.2) one has
1 log Qn = 2 ;
lim
n!1 n
12 log2
2
lim
log(
(n )) =
n!1
6 log2 ;
1 x Pn = 2 :
lim
n!1 n
Qn
6 log2
Here denotes the Lebesgue measure and n = n(i1 ; : : : ; in) the so-
alled
fundamental interval de ned by
n = x 2 [0; 1) : ij +1 1 T j 1(x) < i1j ; j = 1; 2; : : : ; n :
Moreover, among other things, Khint
hine [Kh℄ showed the following.
Theorem 1.2 (Khint
hine, 1935) For almost all x 2 [0; 1) with RCF expansion (1.2) one has
log k
1
Y
1
1=n
lim (a1 a2 an ) =
1 + k(k + 1) log 2 = 2:6854 :
n!1
k=1
For proofs of the last two results see [DK℄. One of them is proved in Se
tion 1.3
using some results in ergodi
theory; see page 9.
The limiting distribution of T n(x) in (1.7) leads us to a measure with density
1 1
(1.8)
log 2 1 + x ;
INTRODUCTION
6
today known as Gauss' measure. This measure is invariant under the
ontinued
fra
tion map T de ned in (1.4) (i.e., T is Gauss measure preserving). To see
this, let (a; b) [0; 1). Sin
e
T
1
(a; b) =
1
1
;
;
n+b n+a
we have, with
denoting Gauss measure,
(T
1
(a; b)) =
1 Z n+1 a dx
1 X
log 2 n=1
1
n+b
=
1+x
1
b+1
log
=
((a; b)):
log 2
a+1
See also Theorem 1.2.1 in [IK℄.
1.2
More re
ent developments
Another important development in the theory of
ontinued fra
tions is the introdu
tion of the so-
alled natural extensions by a group of Japanese mathemati
ians; see, e.g., the papers by H. Nakada, S. Ito and S. Tanaka [NIT℄, and
Nakada [N℄. In this last paper the natural extension T of T is de ned by
T(x; y ) = T (x) ;
1
;
a1 (x) + y
(x; y ) 2 [0; 1) [0; 1℄:
(1.9)
It follows immediately that
T n (x; y ) = (Tn ; Vn );
where Tn := T n (x) and Vn := [0; an ; an 1 ; ; a1 ℄ = Qn 1 =Qn . Note that we
might
onsider Tn as the \future" of x at the \
urrent" time n and Vn as the
\past" of x up to time n. The points (Tn ; Vn ) are distributed in the unit square
a
ording to the density fun
tion (log 2) 1 (1 + xy ) 2 . In fa
t, this is a
onsequen
e of the ergodi
system (1.11) in Theorem 1.6 on page 10.
Essential in this thesis are the so-
alled singularization and insertion pro
esses by whi
h we
an obtain other SRCF expansions of x from its RCF expansion, su
h as the nearest integer
ontinued fra
tion and odd
ontinued fra
tion
expansions. We dis
uss some metri
al properties of odd
ontinued fra
tion obtained from the regular
ontinued fra
tion via singularizations and insertions in
Chapter 2. The singularization pro
ess is based on the identity
A+
e
e
;
= A+e +
1
B+1+
1+
B+
while the insertion pro
ess rests on the identity
A+
1
B+
= A+1 +
1
1+
1
B 1+
;
More re
ent developments
7
where 2 [0; 1).
This means, for example, that singularizing an+1 = 1 in an RCF expansion
[ 0; a1 ; a2 ; ; an ; 1; an+2 ; ℄
(A)
with the sequen
e of
onvergents, say, (An =Bn )n1 results in an SRCF expansion
(B)
[ 0; 1=a1 ; 1=a2 ; ; 1=(an + 1); 1=(an+2 + 1); ℄:
On the other hand, inserting 1=1 in the RCF (1.2) at (n + 1)-st position as
an+2 6= 0 results in an SRCF expansion
(C)
[ 0; 1=a1 ; 1=a2 ; ; 1=(an + 1); 1=1; 1=(an+2 1); ℄:
The e e
ts of these two pro
esses on the sequen
e of
onvergents were studied
in [K2℄. It is shown that the sequen
e of
onvergents of the SRCF (B)
an be
obtained from that of the RCF (A) by removing An =Bn . On the other hand, the
sequen
e of
onvergents of the SRCF (C)
an be obtained from that of the RCF
(1.2) by inserting (Pn + Pn 1 )=(Qn + Qn 1 ) between Pn 1 =Qn 1 and Pn =Qn .
In [K2℄ Kraaikamp introdu
ed a new
lass of
ontinued fra
tions
alled S expansions whi
h are obtained from the RCF only by using the singularization
pro
ess. The -expansions (see [N℄) are examples of S -expansions; see [IK℄ for
more examples. Essential to these expansions is the so-
alled singularization
area; that is, a subset S of [0; 1) [0; 1℄ satisfying the following
onditions.
(i) S 2 B and S is a
-
ontinuity set,
(ii) S [1=2; 1) [0; 1℄,
(iii) S \ T(S ) 6= ;.
To obtain the NICF of x, for instan
e, we have to singularize in ea
h blo
k
of m 2 N [ f1g
onse
utive partial quotients equal to 1, the rst, third,: : : et
.
partial quotient. This leads to a singularization area
p
SNICF = [1=2; 1) [0; g℄;
1)=2. Other two examples of S -expansions are Minkowski's
where g := ( 5
diagonal
ontinued fra
tion (DCF), with singularization area
t
1
SDCF = (t; v) 2 [0; 1) [0; 1℄ :
>
1 + tv 2
;
and Bosma's optimal
ontinued fra
tion (OCF), with singularization area
SOCF = (t; v) 2 [0; 1) [0; 1℄ : v < t and v <
2t 1
1 t
:
It was Wolfgang Doeblin [Do℄ who rst dis
overed the ergodi
system underlying the RCF. Unfortunately, his results remained unnoti
ed for a long time.
All
lassi
al results of
ontinued fra
tions were obtained with probabilisti
methods until C. Ryll-Nardzewski showed in 1951 [R-N℄ how metri
al results
an be
obtained in a more elegant way using ergodi
theory. Some results in ergodi
theory are presented in the next se
tion.
8
INTRODUCTION
1.3
Some results in ergodi
theory
Ergodi
theory arose from an attempt in statisti
al me
hani
s to des
ribe a system of a
ertain number of parti
les moving in a three-dimensional spa
e at any
given time. In general, let (
; B; P ) be a probability spa
e. A transformation
T :
!
is
alled measurable if T 1 A 2 B for all A 2 B . We
all T measure
1 A) = P (A) for all A 2 B . A transforpreserving if it is measurable and P (T
mation T is said to be ergodi
if every T -invariant subset of B has measure 0 or
1, that is, T 1A = A ) P (A) 2 f0; 1g: Equivalently, we say that (
; B; P; T )
forms an ergodi
system.
The following result is fundamental in ergodi
theory; see, e.g., [P℄ and [Wa℄
for more details and proofs.
Let (
; B ; P )
:
!
a measure preserving transformation. Fur1
ther, let f :=
! R be su
h that f 2 L (
; B ; P ). Then for almost all x
Theorem 1.3 (Birkho 's
be a probability spa
e and
T
Individual Ergodi
Theorem, 1931)
f (x) :=
Rexists.
Moreover, we have
f dP .
f (x)
1 nX1 f (T k x)
lim
n!1 n
k=0
2 L1(
; B; P )
,
f (x) = f (T x),
and
R
f dP
=
The next theorem is an important
onsequen
e of Birkho 's ergodi
theorem.
Theorem 1.4
f
2 L1(
; B; P )
Let
(
; B; P; T ) be an ergodi
system and f :
! R be su
h that
. Then for almost all
x
we have
Z
n
X1 k
1
lim
f (T (x)) =
n!1 n
k=0
f dP:
The following fundamental result is very important in the development of
the theory of
ontinued fra
tion in
onne
tion with ergodi
theory.
Theorem 1.5
Let
= [0; 1), B be the
olle
tion of all Borel sets of
, and
(1.8). Further, let T be the
ontinued fra
tion map
the Gauss measure given in
(1.4).
(
; B;
; T );
Then
forms an ergodi
system.
The following example illustrates an appli
ation of Theorem 1.5.
Example 1.1
This equivalen
e
an be easily
he
ked for x 2 [0; 1):
1 1
:
;
an (x) = a , T n 1 (x) 2
a+1 a
(1.10)
Some results in ergodi
theory
9
Then the proportion of partial quotients equal to
quotients (an )n0 is for almost all x given by
1
log 2
Z
1
a
dx
=
1 1+x
a+1
a
in the sequen
e of partial
1
(a + 1)2
log
:
log 2
a(a + 2)
This gives, for instan
e, that 2:272 per
ent of the partial quotients equal
to 7.
We now see that the results of Levy and Khint
hine (see Theorems 1.1 and
1.2) are
orollaries of Theorem 1.5, together with Theorem 1.4. As an example,
we give here a proof of Khint
hine's result; see also [DK℄.
. De ne f (x) = log a1 (x) where
(0; 1). Then, due to ergodi
ity of T , we have
Proof of Theorem 1.2
0
11=n
n
Y1
exp(f (T j (x)))A
j =0
0
1
Z
n
1
X
1
j
exp
f (T (x))A = exp
(a1 a2 an )1=n =
=
It remains to show that
n
f
Z
0
and
Z
1
k
1
k+1
f d
=
1
log 2
Z
a1 (x)
0
j =0
= b1=x
,
x
2
1
f d
:
is integrable. Now
1
f d
=
1Z
X
k=1
1
k
1
k+1
f d;
1
log a1 (x)
log k
1
dx =
log 1 +
1
1
+
x
log
2
k (k + 2)
k+1
k
log k
+ 2)
k (k
as k ! 1. Here we have used lim!0 (1 + )=2 = 1: The result follows from the
fa
t that
1 log k
X
k (k + 2)
k=1
is
onvergent and writing
1 log k
X
1
log 1 +
log
2
k (k + 2)
k=1
= log
1
Y
k=1
log k
1
log 2
1+
:
k (k + 2)
2
The natural extension of the system (1.10), whi
h is used several times in
this thesis, is given in the next theorem. More details on this result
an be
found in [NIT℄ and [N℄.
10
INTRODUCTION
=
[0; 1℄, B be the
lass of all Borel sets of
,
be
Let
the extended (two-dimensional) Gauss measure de ned by
Theorem 1.6
Z
1
dx dy
(A) =
log 2 A (1 + xy)2 ;
and T is the natural extension (1.9) of T . Then
(
; B;
; T);
;
A2
(1.11)
forms an ergodi
system.
1.4 Approximation
oeÆ
ients
One of the most important reasons to use (regular)
ontinued fra
tions is that
ontinued fra
tions yield \the best" rational
onvergents to irrational numbers.
In order to express the quality of approximation of an irrational number x by a
rational number p=q, we introdu
e the approximation
oeÆ
ient (x; p=q) by
(x; p=q ) = q jqx
pj:
A
lassi
al theorem by Borel now states that for every
p irrational x there are
in nitely many rationals p=q su
h that (x; p=q) < 1= 5.
For any irrational number x we de ne the approximation
oeÆ
ients n by
n
:= n (x) = Qn jQn x
Pn j;
n = 1; 2; : : : :
(1.12)
They measure how well the rational number Pn =Qn approximates an irrational
number x. Sin
e it
an be shown that
Pn
1;
x
<
Qn
Q2n
we immediately see that 0 < n < 1 for all n 1. Using
Pn + Pn 1 Tn
;
x=
Qn + Qn 1 Tn
in (1.12), we
an show that
n
= 1 +TTn V
n n
;
and
n
Hen
e, de ning :
! R2 by
(x; y ) =
y
;
1
= 1 +VTn V
n n
x
1 + xy 1 + xy
:
(1.13)
Approximation
oeÆ
ients
11
leads to the fa
t that
(n 1 ; n) = (Tn; Vn ):
(1.14)
In fa
t, (
) = , where is a triangle with verti
es (0,0), (1,0), and (0,1). It
then follows immediately that
n 1 + n < 1; n = 1; 2; : : : ;
and hen
e
min(n 1; n ) < 21 ; n = 1; 2; : : : ;
whi
h is a well-known result due to Vahlen [V℄.
Using the fa
t that
(n; n+1) = (T( 1 (n 1 ; n)));
Jager and Kraaikamp [JK℄ were able to show that
p
n+1 = n 1 + an+1 1 4n 1 n a2n+1 n :
From this it easily follows that
min(n 1; n ; n+1) < q 2 1
an+1 + 4
and
max(n
q 21
1 ; n ; n+1 ) >
:
+4
The former
learly generalizes Borel's
lassi
al result, the latter was found by
J.C. Tong [T1℄. As a
orollary we nd the following result.
Theorem 1.7
x
n0
min(n 1; n; n+1) < p15 ;
1=p5
The following result is another
onsequen
e of (1.14) together with Theorem 1.6.
Theorem 1.8 (Jager, 1986)
(n 1; n)
x
1 p 1 :
f (a; b) =
log 2 1 4ab
For all irrational numbers
the
onstant
an+1
and all
annot be repla
ed by a smaller one.
The sequen
e
triangle
one has
a
ording, for almost all
are distributed over the
, to the density fun
tion
12
INTRODUCTION
See [J℄ for details.
Continued fra
tions play an important role in the theory of prime-testing (see,
e.g., Bressoud's book [B℄). In 1981, H. W. Lenstra
onje
tured that for almost
all x
(1.15)
lim 1 #fj : 1 j n; (x) z g; where 0 z 1;
!1 n
n
j
exists and equals F (z ), where
8 z ;
>
< log 2
F (z ) =
>
: 1 (1
log 2
0 z 21 ;
z + log 2z );
1
2
z 1:
In fa
t (1.15) had been
onje
tured in 1940 by Wolfgang Doeblin [Do℄. In 1984
Knuth [Kn℄ showed that
1 #f1 i N : + d g = 1 Z +d `(t) dt;
lim
i
N !1 N
log 2
where
1
`(t) = min 1;
1 :
t
In 1983 Bosma, Jager, and Wiedijk [BJW℄ proved the Lenstra-Doeblin
onje
ture using Nakada's natural extension (
; B;
; T).
1.5
A brief des
ription of the thesis
This thesis
onsists of ve papers dealing with
ontinued fra
tions.
Chapter 21 is
on
erned with the
ontinued fra
tion with odd partial quotients
(OddCF). The relation between OddCF and RCF is developed via singularization and insertion pro
esses. Using S
hweiger's natural extension for the OddCF
we show that the sequen
e of
onvergents of the nearest integer
ontinued fra
tion (NICF) is a subsequen
e of that of OddCF. Using the method in [JK℄,
we obtain a result for OddCF approximation
oeÆ
ients whi
h
oin
ides with
Tong's result for NICF [T2℄. Through the relation between RCF and grotesque
ontinued fra
tion (GCF) developed again via singularizations and insertions
we see that the sequen
e of GCF
onvergents forms a subsequen
e of that of
Hurwitz' singular
ontinued fra
tion. Maximal and non-maximal OddCF are
also dis
ussed.
1 a joint work with Cor Kraaikamp in
Rev. Romaine Math. Pures Appl.
47 (2002),
no. 1.
A brief des
ription of the thesis
13
In Chapter 32 we
onsider the map TE : [0; 1) ! [0; 1) given by
1 1
1
TE (x) := 1
b x
x b x
; x 6= 0; TE (0) := 0:
This map yields a (unique)
ontinued fra
tion expansion of x 2 [0; 1) with nonde
reasing partial quotients of the form
1
; bn 2 N ; with bn bn+1 :
b1
b1 +
b2
b2 +
bn 1
.
b3 + . . +
.
bn + . .
We
all this expansion
(ECF) expansion of x sin
e the
map TE is a modi ed version of the Engel series expansion map.
Some basi
properties of RCF also hold for the ECF but they di er in many
ways. For instan
e, ECF
onvergents behave di erently from regular ones. It
turns out that TE is ergodi
with respe
t to Lebesgue measure but has
nite invariant measure, equivalent to Lebesgue. Moreover, it is shown that the
map TE has in nitely many - nite, in nite invariant measures, two of whi
h
are given here. Additionally, we relate the ECF to Ryde's
(MNK) generated by the map TR : ( 12 ; 1) ! ( 21 ; 1),
given by
k k + 1
k
; k 2 N;
TR (x) = SR (x) =
k; for x 2 R(k ) :=
;
x
k+1 k+2
through an isomorphism. From this it follows, for example, that the map TR
is ergodi
with respe
t to Lebesgue measure but no nite TR -invariant measure
equivalent to Lebesgue exists and that not every quadrati
irrational has an
ultimately periodi
ECF expansion.
Engel
ontinued fra
tion
no
monotonen, ni
ht-
abnehmenden Kettenbru
h
A Hurwitz-type spe
trum was studied for the nearest integer
ontinued fra
tion
by Jager and Kraaikamp in [JK℄. With (n )n1 denoting the sequen
e of NICF
approximation
oeÆ
ients they showed that
p
min(n 1 ; n ; n+1 ) < 25 (5 5 11) = 0:4508 :
In [T2℄ Tong extended this result and proved that
p !2k+3
5
3
1
1
:
min(n 1 ; n ; : : : ; n+k ) < p + p
2
5 5
2 a joint work with Cor Kraaikamp and F. S
hweiger in
Bordeaux 14
(2002).
J. de Theorie des Nombres de
14
INTRODUCTION
Chapter 43 gives a proof of Tong's result using the method from [JK℄ whi
h
yields some metri
al observations with respe
t to Tong's spe
trum. Generalizations to a larger
lass of semi-regular
ontinued fra
tion expansions are also
derived.
A number x 2 R is
alled Hurwitzian if its RCF expansion (1.2)
an be written
as
x = [a0 ; a1 ; ; an ; an+1 (k); ; an+p (k)℄1
k=0 ;
where an+1 (k ); : : : ; an+p (k ) ( the so-
alled quasi period of x) are polynomials
with rational
oeÆ
ients whi
h take positive integral values for k = 0; 1; 2; : : :,
and at least one of them is not
onstant. This
learly generalizes periodi
ontinued fra
tions. In Chapter 54 we de ne the Hurwitzian numbers for the NICF,
the `ba
kward'
ontinued fra
tion expansion, and -expansions. We show that
the set of Hurwitzian numbers for su
h
ontinued fra
tions
oin
ides with the
lassi
al set of Hurwitzian numbers.
Chapter 65 is a
ontinuation of the previous
hapter. In this
hapter we de ne
Hurwitzian numbers for Minskowski's diagonal
ontinued fra
tion (DCF). We
also show that the set of DCF-Hurwitzian numbers
oin
ides the
lassi
al set
of Hurwitzian numbers. The situation is more
ompli
ated here than in the
previous paper due to the di eren
e in shape of the singularization area of the
NICF (and other -expansions) on one hand, and that of the DCF on the other.
3 a joint work with Cor Kraaikamp
4 a joint work with Cor Kraaikamp in Tokyo J. Math. 25 (2002), no. 2
5 J. Matematika atau Pembelajarannya VIII (2002), 837{841.
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Theory, Cambridge University Press, Cambridge,
(1997).
[R-N℄ Ryll-Nardzewski, C. - On the ergodi
theorems. II. Ergodi
theory of
ontinued fra
tions, Studia Math. 12 (1951), 74{79.
[S℄ S
hweiger, F. { Ergodi
theory of bred systems and metri
number theory,
Oxford S
ien
e Publi
ations. The Clarendon Press, Oxford University Press,
New York, (1995). MR 97h:11083
[T1℄ Tong, Jing Cheng { The
onjugate property of the Borel theorem on Diophantine approximation, Math. Z. 184 (1983), no. 2, 151{153. MR 85m:11039
[T2℄ Tong, Jing Cheng | Approximation by nearest integer
ontinued fra
tions
(II), Math. S
and. 74 (1994), no. 1, 17{18. MR 95
:11085
[V℄ Vahlen, K. Th. | Uber
Naherungswerte und Kettenbru
he, Journal f. d.
reine und angew. Math. 115 (1895), 221{233.
[Wa℄ Walters, P. { An Introdu
tion to Ergodi
Theory, Springer-Verlag New
York, In
., New York, (2000).
[Wi℄ Wirsing, E. { On the theorem of Gauss-Kuzmin-Lery and a Frobenius type
theorem for fun
tion spa
es, A
ta Arith. 24 (1974), 507{528. MR 49 2637
Chapter 2
Odd Continued Fra
tions
2.1
Introdu
tion
It is well-known that every x 2 [0; 1)
an be written as a nite (in
ase x is
rational) or in nite (when x is irrational)
ontinued fra
tion with odd partial
quotients:
x
e
=
1
=: [ 0; e1 =a1 ; e2 =a2 ;
2
e
1+
a
a
.
2 + .. +
; en =an ;
℄
;
(2.1)
en
an
where e1 = 1; ei = 1 and
.
+ ..
is a positive odd integer, for i 1, and
ai
ai
+ ei+1
>
1;
i
1
:
We
all (2.1) the odd
ontinued fra
tion (OddCF) expansion of x. Apart from
the OddCF-expansion of x one also has the so-
alled grotesque
ontinued fra
tion
(or GCF)
expansion of any x 2 [G 2; G), where G is the golden mean, i.e.,
1 (p5 + 1). The GCF-expansion is also given by (2.1), again with odd
G =
2
partial quotients ai and ei = 1, but now these ai and ei must satisfy
ai
+ ei
>
1;
i
1
;
and e1 = sgn(x).
There is an extended literature on both the OddCF and the GCF. In two
(unpublished) papers F. S
hweiger obtained the ergodi
theorem underlying the
OddCF and its natural extension where, as a by-result he showed that the
GCF is the dual algorithm of the OddCF ([S1℄, [S2℄), and studied the approximation properties of the OddCF ([S2℄). Around the same time G.J. Rieger also
obtained a Gauss-Kuzmin theorem for the OddCF and found the ergodi
systems underlying both the OddCF and the GCF ([R2℄). Also a Heilbron-theorem
17
ODD CONTINUED FRACTIONS
18
was given by Rieger for both expansions in [R1℄. In two re
ent papers G.I. Sebe
returned to the
onvergen
e rate in the Gauss-Kuzmin problem for the OddCF
([Se1℄) and GCF ([Se2℄) using the theory of random systems with
omplete
onne
tions. Sebe also obtained the natural extension for the GCF. More results
on the OddCF and the GCF
an be found in papers by S. Kalpazidou ([Ka1℄,
[Ka2℄) and D. Barbolosi ([B1℄, [B2℄).
At rst sight one might be tempted to say that nothing
an be said anymore
about these expansions! In [B2℄, Barbolosi showed that for any x 2 [0; 1) the
sequen
e of nearest integer
ontinued fra
tion (NICF)
onvergents of x forms
a subsequen
e of the sequen
e of OddCF-
onvergents of x. In order to understand this result we were led to a new
lass of
ontinued fra
tion expansions
with odd partial quotients, of whi
h the OddCF and the GCF are two examples.
In general, a
fra
tion
0+
b
semi-regular
ontinued fra
tion
e
1+
b
1
e
(SRCF) is a nite or in nite
= [ b0 ; e1 =b1 ; e2 =b2 ;
2
.
2 + .. +
n
n
n
; e =b ;
℄;
(2.2)
e
b
n+
b
..
.
with en = 1; b0 2 Z; bn 2 N , for n 1, subje
t to the
ondition
e
n+1 + bn
1; for
n
1;
(2.3)
and with the restri
tion that in the in nite
ase
e
n+1 + bn
2; in nitely often.
(2.4)
A nite trun
ation in (2.2) yields the SRCF-
onvergents
n
n := [ b0 ;
A =B
1 1 2 2
n
e =b ; e =b ; ; e =b
n℄ ;
where it is always assumed that g
d(An ; Bn ) = 1. We say that (2.2) is an
SRCF-expansion of an irrational number x in
ase
x
= lim
!1
n
A
n
n
B
:
Clearly the OddCF is an example of an SRCF-expansion, but the GCF is not.
Other examples of SRCF-expansions are the nearest integer
ontinued fra
tion
(NICF) expansion, satisfying
e
and Hurwitz'
n+1 + bn
2 for
singular
ontinued fra
tion
e
n + bn
n
1;
(HSCF) expansion, whi
h satis es
2 for
n
1:
19
Introdu
tion
Perhaps the best-known example of an SRCF-expansion is the so-
alled regular
ontinued fra
tion expansion (RCF); every real irrational number x has a unique
RCF-expansion
d0
+
1
d1
+
where d0 2 Z is su
h that x
=: [ d0 ; d1 ; d2 ; ℄;
1
.
d2 + . .
d0
(2.5)
2 [0; 1), and dn 2 N for n 2 N .
Obviously the GCF is not an SRCF, but a so-
alled unitary expansion, see
also [G℄. Unitary expansions are de ned in a way similar to SRCF-expansions,
the di eren
e being that (2.3) and (2.4) are repla
ed by
en
+ bn 1; for n 1;
and with the restri
tion that in the in nite
ase
en
+ bn 2; in nitely often.
Essential in our investigations are the notions of insertion and singularization
of a partial quotient equal to 1, whi
h were studied in detail in [K℄.
A singularization is based upon the identity
A
+
e
1 =
1+
B+
A+e
+
e
B
+1+
;
where 2 [0; 1).
To see the e e
t of a singularization, let (2.2) be an SRCF-expansion of x. A
nite trun
ation yields the sequen
e of
onvergents (rk =sk )k 1 : Suppose that
for some n 0 one has
bn+1 = 1; en+2 = 1 ;
i.e.,
[ b0 ; e1 =b1 ; ℄ = [ b0 ; e1 =b1 ; ; en =bn ; en+1 =1; 1=bn+2 ; ℄:
The transformation
ontinued fra
tion
n
(2.6)
whi
h
hanges this
ontinued fra
tion (2.6) into the
[ b0 ; e1 =b1 ; ; en =(bn + en+1 );
en+1 =(bn+2
+ 1); ℄;
(2.7)
whi
h is again a
ontinued fra
tion expansion of x, with
onvergents, say
(pk =qk )k 1 ; is
alled a singularization.It
was shown in [K℄ that the sequen
e
of ve
tors
pk
qk k
from the latter.
1
is obtained from
rk
sk k
1
by removing the term
rn
sn
ODD CONTINUED FRACTIONS
20
An operation whi
h is in some sense the `opposite' of a singularization is a
so-
alled
insertion.
An insertion is either based upon the identity
A
+
1
B
+
A+1
=
1
+
1 +
;
1
B
1+
or on the identity
A+
where
2
1
B
+
1
A
=
1
1+
has
bn+1 >
sr
f-insertion
[
x,
1
1;
is the transformation
b0 ; e1 =b1 ;
; en = bn
(
+1+
and suppose that for some
en+1
n
x,
1=1; 1=(bn
rk
sk k
obtained from
+1
1);
;
℄
with
onvergents, say, (pk =qk )k
1
by
inserting
r
s
1
1
;
r0
s0
;:::;
the term
1
1
rn
sn
;
Now let (2.2) be a unitary-expansion of
1
Let
pk
qk k
rn
sn
rn
sn
+
+
1
1
rn
+ sn
+
rn
sn
1
1
before the
;
rn
sn
;
rn+1
sn+1
1
is
n-th
;::: :
1=1.
An sr
f-insertion is denoted by
(rk =sk )k
1.
term of the latter sequen
e, i.e.,
1
0 one
be the sequen
e of
onvergents
onne
ted with (2.2). Using some
pk
qk k
= 1:
matrix-identities it was shown in [K℄ that the sequen
e of ve
tors
n
whi
h
hanges (2.2) into
+ 1);
whi
h is again an SRCF-expansion of
(rk =sk )k
B
[0; 1).
Let (2.2) be an SRCF-expansion of
An
;
1+
. Suppose that for some
bn >
n
1;
x
with the sequen
e of
onvergents
0 one has
en+1
= 1:
Applying the se
ond insertion-identity
hanges (2.2) into
[
b0 ; e1 =b1 ;
; en = bn
(
1); 1=1;
whi
h is again a unitary-expansion of
This kind of insertion is
alled a
x,
+1 + 1); ℄;
1=(bn
with
onvergents, say, (pk =qk )k
unitary-insertion.
1.
In this
ase the sequen
e
21
Insertions, singularizations and the OddCF
of ve
tors
pk
qk
of the new expansion is obtained from
k
1
rk
sk
k
1
by
in-
the term nn nn 11 before the -th term of the latter sequen
e. A
unitary-insertion is denoted by 1 1 .
By
ombining the operations of singularization and sr
f/unitary-insertion one
an obtain any semi-regular/unitary
ontinued fra
tion expansion of a number
from its RCF expansion. In [K℄ a whole
lass of semi-regular
ontinued fra
tions was introdu
ed via singularizations only (some of these SRCF's were new,
some
lassi
al like the
ontinued fra
tion to the nearest integer, or Hurwitz'
(HSCF)), and their ergodi
theory studied (the main
idea in [K℄ is that the operation of singularization is equivalent to having an
indu
ed map on the natural extension of the RCF).
In the next se
tion we will show that the OddCF-expansion
an be obtained
from the RCF via suitable sr
f-insertions and singularizations. We also will
derive some metri
al results for the approximation
oeÆ
ients of the OddCF.
In Se
tion 2.3 we will see that the GCF
an be obtained from the RCF via
singularizations and unitary-insertions. This will lead us in Se
tion 2.4 to a new
lass of semi-regular/unitary
ontinued fra
tion expansions with odd partial
quotients.
serting
r
r
s
s
n
=
x
singular
ontinued fra
tion
2.2
Insertions, singularizations and the OddCF
2.2.1
A singularization/insertion algorithm
The following theorem des
ribes an algorithm whi
h turns the RCF-expansion
of any 2 [0 1) into the OddCF-expansion of . The proof of this theorem
follows easily by inspe
tion, and is therefore omitted.
2 [0 1)
(2.5)
0 =0
(2.5)
x
;
x
Theorem 2.1 Let x
with RCF-expansion
;
starting from the RCF-expansion
, i.e., d
.
Then
of x, the following algorithm yields the
OddCF-expansion of x.
()
I
Let m
()
i
:= inf f 2 N;
1
m+1
If d
n
>
[ 0; 1=d1 ;
( )
ii
If dm
+1
dn is even
11
, insert
=1
;
1=d
=
m 1
, let k
;
g
.
after dm to obtain
m + 1)
1=(d
:= inf f
1=1; 1=(d
;
;
n > m
dn >
m+1
1); 1=d
1g ( = 1
k
singularize in the blo
k of partial quotients
dm
+1 = 1; dm+2 = 1; : : : ; dk 1 = 1
m+2 ℄
;
)
:
is allowed . Now
ODD CONTINUED FRACTIONS
22
the rst, third, fth, et
. partial quotients equal to 1, to arrive at
[ 0; 1=d1 ;
;
1 ; 1=(dm + 1); | 1=3; {z
1=dm
;
1=3;
}
k m 2 times
2
in
ase
k
[ 0; 1=d1 ;
In
ase
k
[ 0; 1=d1 ;
1
m
is odd or
1
m
m
1=d
1; in the latter
ase we nd
( m + 1)
1 3
1 3 ℄
1 ; 1=
d
;
= ;
;
1 ; 1=(dm + 1); | 1=3; {z
In this
ase insert
(I I )
;
;
=
1=3;
k m 3 times
2
[ 0; 1=d1 ;
℄
;
= ;
:
is even we obtain
1=dm
;
;
k
1=(dk + 1); 1=dk+1 ;
1=1
}
1=2; 1=dk ; 1=dk+1 ;
℄
:
to arrive at
1=(dm + 1);
|
1=3;
{z
;
1=1; 1=(dk
1=3;
k m 1 times
2
}
1); 1=dk+1 ;
℄
;
Let m 1 be the rst index in the new SRCF-expansion [
0 ; e1 =
1 ; ℄ of
obtained in (I) for whi
h
m is even. Repeat the pro
edure from (I) to
this new SRCF-expansion of x with this value of m.
x
As soon as
m
1 in (II) we have obtained the OddCF-expansion of
=
.
x
The following example illustrate how to use Theorem 2.1
Example 2.1 Let x
2 [0 1) have RCF-expansion
;
[0; 1=3; 1=1; 1=4; 1=7; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=5;
(i) Apply the algorithm with
m
= 3. Sin
e
d4 >
1, we insert
℄
:
1=1 after 1=4
to obtain
[0; 1=3; 1=1; 1=5;
1=1; 1=6; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=5;
(ii) Apply the algorithm with
d11
= 1, we singularize
[0; 1=3; 1=1; 1=5;
Now insert
1=1; 1=7;
1=1; 1=7;
(iii) Apply the algorithm with
m
=
= 5 in the new expansion. Sin
e
and
d11
1=3;
d6
=
to arrive at
1=3;
1=3;
1=2; 1=5;
℄
:
=
:
1=1 to arrive at
[0; 1=3; 1=1; 1=5;
until
m
d6 , d8 , d10
℄
1.
m
1=3;
1=3;
1=3;
1=3;
1=1; 1=4;
℄
:
= 10 in this new expansion and
o
Properties of Continued
Fra
tion Algorithms
Ergodi
Properties of Continued Fra
tion Algorithms
Proefs
hrift
ter verkrijging van de graad van do
tor
aan de Te
hnis
he Universiteit Delft,
op gezag van de Re
tor Magni
us prof. dr. ir. J. T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 3 februari 2003 om 13.30 uur
door
Yusuf HARTONO
Master of S
ien
e in Applied Mathemati
s,
University of Missouri-Rolla, USA
geboren te Kundur, Indonesie.
Dit proefs
hrift is goedgekeurd door de promotor:
Prof. dr. F. M. Dekking
Toegevoegd promotor: Dr. C. Kraaikamp
Samenstelling promotie
ommisie:
Re
tor Magni
us
Prof. dr. F. M. Dekking
Dr. C. Kraaikamp
Prof. dr. J. M. Aarts
Prof. dr. M. Iosifes
u
Prof. dr. F. S
hweiger
Prof. dr. R. K. Sembiring
Dr. W. Bosma
voorzitter
Te
hnis
he Universiteit Delft, promotor
Te
hnis
he Universiteit Delft, toegevoegd promotor
Te
hnis
he Universiteit Delft
Romanian A
ademy of S
ien
es, Roemenie
Universitat Salzburg, Oostenrijk
Institut Teknologi Bandung, Indonesie
Katholieke Universiteit Nijmegen
The resear
h in this thesis has been
arried out under the auspi
es of
the Thomas Stieltjes Institute for
Mathemati
s, at the University of
Te
hnology in Delft.
Published and distributed by: DUP S
ien
e
DUP S
ien
e is an imprint of
Delft University Press
P.O. Box 98
2600 MG Delft
The Netherlands
Telephone: +31 15 27 85 678
Telefax: +31 15 27 85 706
E-mail: InfoLibrary.TUDelft.NL
ISBN 90-407-2381-8
Keywords: metri
, arithmeti
,
ontinued fra
tions
Copyright
2002 by Y. Hartono
All rights reserved. No part of the material prote
ted by this
opyright noti
e
may be reprodu
ed or utilized in any form or by any means, ele
troni
or me
hani
al, in
luding photo
opying, re
ording or by any information storage and
retrieval system, without written permission from the publisher: Delft University Press.
Cover designed by Silvia Yulianti
Printed in The Netherlands
To the memory of my mother
As far as the laws of mathemati
s refer to reality,
they are not
ertain;
and as far as they are
ertain,
they do not refer to reality.
Albert Einstein (1879 - 1955)
The beginning of knowledge is
the dis
overy of something we do not understand.
Frank Herbert (1920 - 1986),
The fear of the
Ameri
an Writer
Lord is the beginning of knowledge, : : :.
For from him and through him
and to him are all things.
To him be the glory forever! Amen.
Proverbs 1:7; Romans 11:36
A
knowledgments
I am deeply indebted to many institutions and persons without whom this thesis would
never exist. At the rst pla
e I would like to express my gratitude and appre
iation to
my supervisor Cor Kraaikamp, who introdu
ed me to this ex
iting resear
h area, for
his many interesting ideas and
onstant help that kept me in the right dire
tion, for
his en
ouragement that kept me going, and for his patien
e that
omforted me during
the diÆ
ult time in the resear
h. It has been a great pleasure to meet Karma and
Rafael. Also, I would like to express my gratefulness to my promotor Mi
hel Dekking
for inviting me to the Netherlands and providing me with the opportunity to
arry
out my do
toral resear
h at Delft University of Te
hnology (TU Delft). I should also
thank my
ommittee for their valuable
omments and suggestions.
The resear
h leading to this thesis was a part of the s
ienti
ooperation between
Dut
h and Indonesian governments. I am very grateful to the program
oordinators
Prof. Dr. R. K. Sembiring and Dr. A. H. P. van der Burgh who four and a half
years ago organized a resear
h workshop in Bandung that gave me a
han
e to be
sele
ted for this do
toral program at TU Delft. I thank Dr. O. Simbolon and Dr. B.
Karyadi, former proje
t managers of Proyek Pengembangan Guru Sekolah Menengah
(PGSM) { (Se
ondary S
hool Tea
hers Development Proje
t), in Jakarta as well as
Drs. P. Althuis, dire
tor of Center for International Cooperation in Applied Te
hnology
(CICAT) at TU Delft, for nan
ial support and assistan
e during my resear
h and stay
in Holland.
I personally wish to thank all members of Afdeling CROSS, in parti
ular vakgroup
SSOR, for de gezelligheid and a very
ondu
ive atmosphere, espe
ially Cindy, Ellen,
and Diana for their both administrative and non-administrative assistan
e, and Carl
for his
omputer assistan
e. I also wish to thank Durk, Christel, Theda, Veronique,
and Rene for a very wonderful friendship and for providing me with almost everything
I needed during my stay in Holland. My thanks should also go to all PGSM fellows
(Abadi, Budi, Caswita, Darmawijoyo, Gede, Happy, Hartono, Kusnandi, Sahid, Siti,
and Suyono) with whom I started the whole proje
t together for everything we have
had together from fun to serious dis
ussions about mathemati
s, parti
ularly to Suyono
with whom I shared an oÆ
e room and from whom I learned a lot about measure
theory; as well as to Agus, Komo and Julius with whom I spent some time sharing an
apartment in Delft.
A lot of help and en
ouragement also
ame from my
olleagues at the Fa
ulty
of Tea
her Training and Edu
ation, parti
ularly at the Department of Mathemati
s
Edu
ation, Sriwijaya University { Pak Ismail, Pak Purwoko, Bu Tri, and Somakim,
just to name some, as well as from Kapten O. Tengke, Happy and other members of
Bala Keselamatan Korps Kundur, and Kak Ibrahim. I would like to a
knowledge them
here as well.
ix
x
I have appre
iated all the suggestions,
orre
tions, and
omments during the preparation of this thesis. I am the only one responsible for any mistake remaining in this
thesis.
I am also pleased to a
knowledge all members of Het Leger des Heils Korps Delft,
espe
ially Majoor en mevrouw Loef and Kapitein en mevrouw Jansen, and of International Student Chaplain
y, espe
ially Fr. Ben and Rev. Stroh, for their generosity and
warm hospitality making me really feel at home and letting me parti
ipate in their
many a
tivities; to mention a few of them: Daniela, Bernardette, Carla, Ri
ardo,
Yenory, Mar
o, Bibiana, Sandra, Sarah, Yadira, Susanne, Irek, Fabiana, Carmen,
Ralph, Paul, Poni, Duleep, Ni
ol
o, and Fabio. Many thanks to Mieke and Reini for
onsidering me as kind aan huis in their home and family. Ik vind het heel leuk om
met Sara, David, en Nathan kennis te maken. Ik wil Tilak, Mart, en Jose, met alle
medewerkers ook bedanken voor de samenwerking en een hele mooie vriends
hap.
Moreover, I
ertainly enjoyed wonderful times together and very warm
ompanionship with Indonesian students, espe
ially from TU Delft and IHE Delft, during my
study
Bernadeta, Yusuf, Dwi, Teresia, Sri, Helena, Arief, Elisa, Raymon, Silvia,
Dian, Dedy, Reiza, Evy, Joy
e, Diah, Hilda, Firdian, Susi, Theresia, Zenlin, and Susana, just to mention a few. Their presen
e in my life shows that strong bonds of
love in servi
e with ea
h other are essential for developing a pea
eful and harmonious
ommunity. They deserve an a
knowledgement too.
I would like to dedi
ate this thesis to the memory of my mother who passed away
on January 3, 2001 at the age of 65. She taught me how to thank and have faith in
God, how to pray, and how to love and serve others. My father, who has taught me
the meaning of hard-working life and, more importantly, how to do arithmeti
during
my early years at s
hool, also deserves my spe
ial thanks for without him I
ould never
be what I am now; and so does my aunt who took
are of my son while my wife was
working.
Saya ingin mempersembahkan tesis ini sebagai peringatan pada ibu saya yang
meninggal dunia pada tanggal 3 Januari 2001 pada usia 65. Dia telah mengajar
saya bagaimana bersyukur dan memiliki iman kepada Allah, bagaimana berdoa, dan
bagaimana mengasihi serta melayani orang lain. Ayah saya yang mengajar saya arti
kehidupan dan kerja keras dan, yang lebih penting lagi, mengajar saya berhitung pada
tahun-tahun awal saya di sekolah, juga pantas mendapat u
apan terima kasih khusus
karena tanpa dia saya tidak mungkin menjadi seperti saya sekarang, demikian juga
bibi saya yang mengurus anak saya saat istri saya bekerja.
Finally, a sin
ere expression of thanks should go to my dearest wife Elfarida and
our beloved son Gabriel Ekoputra for their love, patien
e, and understanding during
my absen
e in the family due to my study.
Terima kasih yang tulus saya sampaikan kepada istri saya dan anak kami ter
inta
untuk kasih, kesabaran, dan pengertian mereka selama saya tidak berada dalam keluarga.
Above all, I thank God for everything that has happened in my life.
Delft, O
tober 17, 2002
YH
:
:
Contents
1
Introdu
tion
1.1 Some histori
al ba
kground and basi
properties .
1.2 More re
ent developments . . . . . . . . . . . . . .
1.3 Some results in ergodi
theory . . . . . . . . . . . .
1.4 Approximation
oeÆ
ients . . . . . . . . . . . . . .
1.5 A brief des
ription of the thesis . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
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.
2
2.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . .
2.2 Insertions, singularizations and the OddCF . . . .
2.2.1 A singularization/insertion algorithm . . . .
2.2.2 Metri
al properties of the OddCF . . . . .
2.2.3 Approximation
oeÆ
ients . . . . . . . . . .
2.3 Grotesque
ontinued fra
tions . . . . . . . . . . . .
2.4 Other odd
ontinued fra
tions . . . . . . . . . . . .
2.4.1 Maximal OddCF's . . . . . . . . . . . . . .
2.4.2 Non-maximal expansions with odd digits .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . .
2
6
8
10
12
15
Odd Continued Fra
tions
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17
3
3.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Basi
properties . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Ergodi
properties . . . . . . . . . . . . . . . . . . . . . .
3.4 On Ryde's
ontinued fra
tion with non de
reasing digits .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
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.
.
.
39
4
4.1 Introdu
tion . . . . . . . . . . . . . . . . . . .
4.2 Tong's pre-spe
trum for the NICF . . . . . .
4.3 Tong's spe
trum for Nakada's -expansions .
4.3.1 The
ase g < 1 . . . . . . . . . . .
4.3.2 The
ase 12 g . . . . . . . . . .
4.4 Semi-regular
ontinued fra
tions . . . . . . .
.
.
.
.
.
.
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.
.
.
.
.
.
61
Engel Continued Fra
tions
Tong's Spe
trum for SRCF Expansions
xi
.
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1
17
21
21
23
27
32
34
34
35
37
39
42
47
55
59
61
64
69
70
73
77
CONTENTS
xii
Bibliography
5
A Note on Hurwitzian Numbers
79
81
5.1
Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.2
Hurwitzian numbers for the NICF
82
5.3
Hurwitzian numbers for the ba
kward
ontinued fra
tion . . . . .
84
5.4
Hurwitzian numbers for
. . . . . . . . . . . . . . .
86
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Bibliography
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
-expansions
Mikowski's DCF Expansions of Hurwitzian Numbers
93
6.1
Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.2
Minkowski's DCF . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.3
DCF-Hurwitzian expansion
Bibliography
. . . . . . . . . . . . . . . . . . . . .
95
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
Samenvatting
Bibliography
101
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Curri
ulum Vitae
105
107
Chapter 1
Introdu
tion
Mathemati
s is the queen of the s
ien
e,
and number theory is the queen of mathemati
s.
Carl Friedri
h Gauss (1777 { 1855)
This thesis
onsists of ve papers
on
erning
ontinued fra
tion expansions in
onne
tion with ergodi
theory. Most of them deal with metri
al properties of
ontinued fra
tion algorithms. Other aspe
ts like approximation
oeÆ
ients are
also studied here. Relationships among di erent
ontinued fra
tion expansions
are developed via the singularization and insertion pro
esses.
Throughout this thesis by a
ontinued fra
tion expansion of any real number
we mean an expression of the form
a0
e1
+
a1
;
e2
+
a2
.
+ .. +
(1.1)
en
.
+ ..
where a0 2 Z, an are positive integers and en 2 R for n = 1; 2; : : :. In a mu
h
more
onvenient way we write (1.1) more
ompa
tly as
an
[ a0 ; e1 =a1 ; e2 =a2 ; ; en =an ; ℄ :
The terms a1 ; a2 ; : : : are
alled the partial quotients of the
ontinued fra
tion.
The number of terms may be nite or in nite.
In
ase en = 1 for all n = 1; 2; : : :, we
all (1.1) a regular
ontinued fra
tion
(RCF) expansion and write it as
[ a0 ; a1 ; a2 ; ; an ; ℄ :
x,
(1.2)
In general, let x be a real number, a0 = bx
, the largest integer not ex
eeding
and write x = a0 + . Now = x a0 2 [0; 1), and we write
x
= [ a0 ; a1 ; a2 ; ℄
1
INTRODUCTION
2
if the RCF-expansion of
is given by
= [ 0; a1 ; a2 ; ℄:
In the next two se
tions some histori
al ba
kground and basi
properties of
regular
ontinued fra
tions will be presented, followed by a se
tion on a few
basi
fa
ts in ergodi
theory and another on approximation
oeÆ
ients. A brief
des
ription of the
ontent of the papers in this thesis will
on
lude this
hapter.
1.1
Some histori
al ba
kground and basi
properties
Continued fra
tions have a long history. It starts with the pro
edure known
as Eu
lid's algorithm for nding the greatest
ommon divisor (g.
.d.) of two
integers whi
h o
urs in the seventh book of Eu
lid's Elements (
. 300 b.
.).
This pro
edure is perhaps the earliest step towards the development of the
theory of
ontinued fra
tions.
To see the relation between Eu
lid's algorithm and (regular)
ontinued fra
tions,
onsider Eu
lid's algorithm for nding the g.
.d. of two integers a and b
with a > b > 0. We rst let a0 = ba=b
. Putting
r1
:= a
a0 b;
r0
:= b ;
we have to nd positive integers ai su
h that
ri
1 = ai ri + ri+1 ;
(1.3)
where 0 ri+1 < ri , for i = 1; 2; : : : until the pro
edure stops; that is, when we
have rea
hed an index n su
h that rn 6= 0 and rn+1 = 0. In this
ase, we say
that rn is the g.
.d. of a and b.
Dividing (1.3) through by ri , we get
ri
ri
Writing
ri+1
ri
1 = a + ri+1 ;
i
i
ri
=
1
ai+1
r
+ i+2
;
= 1; 2; : : : n:
i
= 1; 2; : : : ; n;
ri+1
and substituting it into the previous equation for ea
h i yield
a
b
= [ a0 ; a1 ; a2 ;
whi
h is the RCF expansion of
a=b.
; a
n
℄;
Some histori
al ba
kground and basi
properties
3
To generalize Eu
lid's algorithm to irrational numbers x in the unit interval,
onsider the
ontinued fra
tion map T : [0; 1) ! [0; 1) de ned by
1
( ) :=
T x
b1
x
; x
x
6= 0;
(0) := 0:
T
(1.4)
De ne further a1 = a1 (x) = b1=x
and an = a1 (T n 1 (x)); n = 0; 1; : : :, where
0
n
n 1 (x)). We then have by (1.4)
T (x) = x and T (x) = T (T
x
=
1
a
1 + T (x)
= = [ 0; a1 ; a2 ;
n
n + T (x) ℄:
; a
For rational numbers repeated appli
ation of T is in fa
t equivalent to Eu
lid's
algorithm. Hen
e, there exists an n0 2 N su
h that T n0 (x) = 0 and it follows
that a rational number has a nite RCF expansion. This is not the
ase for
irrational numbers. If x is an irrational number, then T n (x) is irrational for all
n 0.
A nite trun
ation in (1.2) gives the so-
alled regular
onvergents
P
n
= [ a0 ;
n
1 2
a ; a ;
Q
n ℄;
; a
n
= 1; 2; : : : ;
(1.5)
where we assume that Qn > 0 and that g
d(Pn ; Qn ) = 1.
The sequen
es (Pn )n 1 and (Qn )n 1 satisfy the following re
ursive formulae
P
1 = 1;
P
0 = a0 ;
P
Q
1 = 0;
Q
n = an P n
0 = 1;
Q
n = an Qn
1 + Pn 2
for n 1;
1 + Qn 2
for n 1;
and the relationship
P
1
n Qn
n
P
1 Qn = ( 1)n 1 :
(1.6)
Moreover, the regular
onvergents satisfy the following inequalities:
0
0
P
Q
<
2
2
P
Q
<
<
P
2n
2n
Q
<
<
P
2n+1
2n+1
Q
<
<
3
3
P
Q
<
P
1:
1
Q
For any irrational number x we say that (1.2) is the RCF expansion of x in
ase
n
= x:
n!1 Qn
lim
P
See, for instan
e, [O℄, [IK℄, and [HW℄ for more properties of RCF and proofs.
In general, (1.1) is
alled a semi-regular
ontinued fra
tion (SRCF) in
ase a0 2
Z, an are positive integers, and en = 1 for all n 1, subje
t to the
ondition
e
n+1 + bn
1
;
for n 1;
INTRODUCTION
4
and with the restri
tion that in the in nite
ase
en+1 + bn 2; in nitely often.
Nakada's -expansions, for 2 [1=2; 1℄, are examples of SRCF expansions.
Clearly, the RCF expansion ( = 1), the nearest integer
ontinued fra
tion
(NICF) expansion ( p= 1=2), and Hurwitz' singular
ontinued fra
tion (g expansion, with g = ( 5 1)=2,) are all SRCF expansions. Other examples
of SRCF expansions
onsidered in this thesis are Minkowski's diagonal
ontinued fra
tion (DCF) and odd
ontinued fra
tion (OddCF).
In the area of appli
ations, the great Dut
h mathemati
ian, me
hani
ian, astronomer, and physi
ist, Christiaan Huygens (1629-1695) used the regular
onvergent to obtain the
orre
t ratio for the rotations of planets when he designed
the toothed wheels of a planetarium. He des
ribed this in his Des
riptio Autamati Planetarii, published posthumously in 1698. This is in fa
t a
onsequen
e
of the fa
t that
ontinued fra
tions give the \best" rational approximations to
irrational numbers.
The modern theory of
ontinued fra
tions began with the writings of Rafael
Bombelli, born in about 1530 in Bologna. He showed, for example, in our
modern notation,
p
13 = [ 3; 4=6; 4=6; ℄:
Pietro Antonio Cataldi (1548-1626) also deserves some
redits in
ontinued fra
tions. He expressed
p
18 = [ 4; 2=8; 2=8; ℄:
Using Eu
lid's algorithm for nding the g.
.d. of 177 and 233, Daniel S
hwenter
(1585-1636) found the
onvergents 79=104; 19=25; 3=4; 1=1; and 0=1. It is probably in Aritmeti
a In nitorum (1655), a book by John Wallis, that the term
ontinued fra
tion was used for the rst time. Great mathemati
ians su
h as Euler
(1707-1783), Lambert (1728- 1777), Lagrange (1736-1813), Gauss (1777-1855),
and many others also made important
ontributions to the earlier development
of the theory of
ontinued fra
tions. It is in parti
ular Euler's great memoir, De
Fra
tionibus Continius (1737), that laid the foundation for the modern theory.
See, for example, [O℄, [K1℄, [S℄, and [Di℄ for more history of
ontinued fra
tions.
The metri
al theory of
ontinued fra
tions started with Gauss' problem. In
his diary on O
tober 25, 1800, Gauss wrote (in modern notation) that
lim Fn (z ) =
!1
n
log (z + 1)
; z 2 [0; 1);
log 2
(1.7)
where Fn (z ) = (T n (x) < z ); z 2 [0; 1). Here T is the
ontinued fra
tion map
de ned in (1.4) and denotes the Lebesgue measure. In a letter dated January
30, 1812, he asked Lapla
e to prove (1.7) and to estimate the error-term
en (z ) := Fn (z )
log (z + 1)
:
log 2
Some histori
al ba
kground and basi
properties
5
More than a
entury later this problem was solved by Kuzmin [Ku℄. He showed
in 1928 that
p
en (z ) = O(q n ) as n ! 1
for some
onstant q 2 (0; 1). His proof is reprodu
ed in Khin
hine [Kh℄. One
year later Levy independently proved that
jen (z )j < qn ; n = 1; 2; : : :
p
with q = 3:5 2 2 = 0:67157 . See Subse
tion 1.3.5 in [IK℄ for an improved
version of Levy's solution to Gauss' problem. In 1961 P. Szusz used Kuzmin's
approa
h to nd that q = 0:485. Gauss' problem was settled by Wirsing [Wi℄
who in 1974 found that q = 0:303 663 002 . Results like these are now known
as Gauss-Kuzmin-Levy Theorems. The following result is a
onsequen
e of these
results.
Theorem 1.1 (Levy, 1929) For almost all x 2 [0; 1) with RCF expansion
(1.2) one has
1 log Qn = 2 ;
lim
n!1 n
12 log2
2
lim
log(
(n )) =
n!1
6 log2 ;
1 x Pn = 2 :
lim
n!1 n
Qn
6 log2
Here denotes the Lebesgue measure and n = n(i1 ; : : : ; in) the so-
alled
fundamental interval de ned by
n = x 2 [0; 1) : ij +1 1 T j 1(x) < i1j ; j = 1; 2; : : : ; n :
Moreover, among other things, Khint
hine [Kh℄ showed the following.
Theorem 1.2 (Khint
hine, 1935) For almost all x 2 [0; 1) with RCF expansion (1.2) one has
log k
1
Y
1
1=n
lim (a1 a2 an ) =
1 + k(k + 1) log 2 = 2:6854 :
n!1
k=1
For proofs of the last two results see [DK℄. One of them is proved in Se
tion 1.3
using some results in ergodi
theory; see page 9.
The limiting distribution of T n(x) in (1.7) leads us to a measure with density
1 1
(1.8)
log 2 1 + x ;
INTRODUCTION
6
today known as Gauss' measure. This measure is invariant under the
ontinued
fra
tion map T de ned in (1.4) (i.e., T is Gauss measure preserving). To see
this, let (a; b) [0; 1). Sin
e
T
1
(a; b) =
1
1
;
;
n+b n+a
we have, with
denoting Gauss measure,
(T
1
(a; b)) =
1 Z n+1 a dx
1 X
log 2 n=1
1
n+b
=
1+x
1
b+1
log
=
((a; b)):
log 2
a+1
See also Theorem 1.2.1 in [IK℄.
1.2
More re
ent developments
Another important development in the theory of
ontinued fra
tions is the introdu
tion of the so-
alled natural extensions by a group of Japanese mathemati
ians; see, e.g., the papers by H. Nakada, S. Ito and S. Tanaka [NIT℄, and
Nakada [N℄. In this last paper the natural extension T of T is de ned by
T(x; y ) = T (x) ;
1
;
a1 (x) + y
(x; y ) 2 [0; 1) [0; 1℄:
(1.9)
It follows immediately that
T n (x; y ) = (Tn ; Vn );
where Tn := T n (x) and Vn := [0; an ; an 1 ; ; a1 ℄ = Qn 1 =Qn . Note that we
might
onsider Tn as the \future" of x at the \
urrent" time n and Vn as the
\past" of x up to time n. The points (Tn ; Vn ) are distributed in the unit square
a
ording to the density fun
tion (log 2) 1 (1 + xy ) 2 . In fa
t, this is a
onsequen
e of the ergodi
system (1.11) in Theorem 1.6 on page 10.
Essential in this thesis are the so-
alled singularization and insertion pro
esses by whi
h we
an obtain other SRCF expansions of x from its RCF expansion, su
h as the nearest integer
ontinued fra
tion and odd
ontinued fra
tion
expansions. We dis
uss some metri
al properties of odd
ontinued fra
tion obtained from the regular
ontinued fra
tion via singularizations and insertions in
Chapter 2. The singularization pro
ess is based on the identity
A+
e
e
;
= A+e +
1
B+1+
1+
B+
while the insertion pro
ess rests on the identity
A+
1
B+
= A+1 +
1
1+
1
B 1+
;
More re
ent developments
7
where 2 [0; 1).
This means, for example, that singularizing an+1 = 1 in an RCF expansion
[ 0; a1 ; a2 ; ; an ; 1; an+2 ; ℄
(A)
with the sequen
e of
onvergents, say, (An =Bn )n1 results in an SRCF expansion
(B)
[ 0; 1=a1 ; 1=a2 ; ; 1=(an + 1); 1=(an+2 + 1); ℄:
On the other hand, inserting 1=1 in the RCF (1.2) at (n + 1)-st position as
an+2 6= 0 results in an SRCF expansion
(C)
[ 0; 1=a1 ; 1=a2 ; ; 1=(an + 1); 1=1; 1=(an+2 1); ℄:
The e e
ts of these two pro
esses on the sequen
e of
onvergents were studied
in [K2℄. It is shown that the sequen
e of
onvergents of the SRCF (B)
an be
obtained from that of the RCF (A) by removing An =Bn . On the other hand, the
sequen
e of
onvergents of the SRCF (C)
an be obtained from that of the RCF
(1.2) by inserting (Pn + Pn 1 )=(Qn + Qn 1 ) between Pn 1 =Qn 1 and Pn =Qn .
In [K2℄ Kraaikamp introdu
ed a new
lass of
ontinued fra
tions
alled S expansions whi
h are obtained from the RCF only by using the singularization
pro
ess. The -expansions (see [N℄) are examples of S -expansions; see [IK℄ for
more examples. Essential to these expansions is the so-
alled singularization
area; that is, a subset S of [0; 1) [0; 1℄ satisfying the following
onditions.
(i) S 2 B and S is a
-
ontinuity set,
(ii) S [1=2; 1) [0; 1℄,
(iii) S \ T(S ) 6= ;.
To obtain the NICF of x, for instan
e, we have to singularize in ea
h blo
k
of m 2 N [ f1g
onse
utive partial quotients equal to 1, the rst, third,: : : et
.
partial quotient. This leads to a singularization area
p
SNICF = [1=2; 1) [0; g℄;
1)=2. Other two examples of S -expansions are Minkowski's
where g := ( 5
diagonal
ontinued fra
tion (DCF), with singularization area
t
1
SDCF = (t; v) 2 [0; 1) [0; 1℄ :
>
1 + tv 2
;
and Bosma's optimal
ontinued fra
tion (OCF), with singularization area
SOCF = (t; v) 2 [0; 1) [0; 1℄ : v < t and v <
2t 1
1 t
:
It was Wolfgang Doeblin [Do℄ who rst dis
overed the ergodi
system underlying the RCF. Unfortunately, his results remained unnoti
ed for a long time.
All
lassi
al results of
ontinued fra
tions were obtained with probabilisti
methods until C. Ryll-Nardzewski showed in 1951 [R-N℄ how metri
al results
an be
obtained in a more elegant way using ergodi
theory. Some results in ergodi
theory are presented in the next se
tion.
8
INTRODUCTION
1.3
Some results in ergodi
theory
Ergodi
theory arose from an attempt in statisti
al me
hani
s to des
ribe a system of a
ertain number of parti
les moving in a three-dimensional spa
e at any
given time. In general, let (
; B; P ) be a probability spa
e. A transformation
T :
!
is
alled measurable if T 1 A 2 B for all A 2 B . We
all T measure
1 A) = P (A) for all A 2 B . A transforpreserving if it is measurable and P (T
mation T is said to be ergodi
if every T -invariant subset of B has measure 0 or
1, that is, T 1A = A ) P (A) 2 f0; 1g: Equivalently, we say that (
; B; P; T )
forms an ergodi
system.
The following result is fundamental in ergodi
theory; see, e.g., [P℄ and [Wa℄
for more details and proofs.
Let (
; B ; P )
:
!
a measure preserving transformation. Fur1
ther, let f :=
! R be su
h that f 2 L (
; B ; P ). Then for almost all x
Theorem 1.3 (Birkho 's
be a probability spa
e and
T
Individual Ergodi
Theorem, 1931)
f (x) :=
Rexists.
Moreover, we have
f dP .
f (x)
1 nX1 f (T k x)
lim
n!1 n
k=0
2 L1(
; B; P )
,
f (x) = f (T x),
and
R
f dP
=
The next theorem is an important
onsequen
e of Birkho 's ergodi
theorem.
Theorem 1.4
f
2 L1(
; B; P )
Let
(
; B; P; T ) be an ergodi
system and f :
! R be su
h that
. Then for almost all
x
we have
Z
n
X1 k
1
lim
f (T (x)) =
n!1 n
k=0
f dP:
The following fundamental result is very important in the development of
the theory of
ontinued fra
tion in
onne
tion with ergodi
theory.
Theorem 1.5
Let
= [0; 1), B be the
olle
tion of all Borel sets of
, and
(1.8). Further, let T be the
ontinued fra
tion map
the Gauss measure given in
(1.4).
(
; B;
; T );
Then
forms an ergodi
system.
The following example illustrates an appli
ation of Theorem 1.5.
Example 1.1
This equivalen
e
an be easily
he
ked for x 2 [0; 1):
1 1
:
;
an (x) = a , T n 1 (x) 2
a+1 a
(1.10)
Some results in ergodi
theory
9
Then the proportion of partial quotients equal to
quotients (an )n0 is for almost all x given by
1
log 2
Z
1
a
dx
=
1 1+x
a+1
a
in the sequen
e of partial
1
(a + 1)2
log
:
log 2
a(a + 2)
This gives, for instan
e, that 2:272 per
ent of the partial quotients equal
to 7.
We now see that the results of Levy and Khint
hine (see Theorems 1.1 and
1.2) are
orollaries of Theorem 1.5, together with Theorem 1.4. As an example,
we give here a proof of Khint
hine's result; see also [DK℄.
. De ne f (x) = log a1 (x) where
(0; 1). Then, due to ergodi
ity of T , we have
Proof of Theorem 1.2
0
11=n
n
Y1
exp(f (T j (x)))A
j =0
0
1
Z
n
1
X
1
j
exp
f (T (x))A = exp
(a1 a2 an )1=n =
=
It remains to show that
n
f
Z
0
and
Z
1
k
1
k+1
f d
=
1
log 2
Z
a1 (x)
0
j =0
= b1=x
,
x
2
1
f d
:
is integrable. Now
1
f d
=
1Z
X
k=1
1
k
1
k+1
f d;
1
log a1 (x)
log k
1
dx =
log 1 +
1
1
+
x
log
2
k (k + 2)
k+1
k
log k
+ 2)
k (k
as k ! 1. Here we have used lim!0 (1 + )=2 = 1: The result follows from the
fa
t that
1 log k
X
k (k + 2)
k=1
is
onvergent and writing
1 log k
X
1
log 1 +
log
2
k (k + 2)
k=1
= log
1
Y
k=1
log k
1
log 2
1+
:
k (k + 2)
2
The natural extension of the system (1.10), whi
h is used several times in
this thesis, is given in the next theorem. More details on this result
an be
found in [NIT℄ and [N℄.
10
INTRODUCTION
=
[0; 1℄, B be the
lass of all Borel sets of
,
be
Let
the extended (two-dimensional) Gauss measure de ned by
Theorem 1.6
Z
1
dx dy
(A) =
log 2 A (1 + xy)2 ;
and T is the natural extension (1.9) of T . Then
(
; B;
; T);
;
A2
(1.11)
forms an ergodi
system.
1.4 Approximation
oeÆ
ients
One of the most important reasons to use (regular)
ontinued fra
tions is that
ontinued fra
tions yield \the best" rational
onvergents to irrational numbers.
In order to express the quality of approximation of an irrational number x by a
rational number p=q, we introdu
e the approximation
oeÆ
ient (x; p=q) by
(x; p=q ) = q jqx
pj:
A
lassi
al theorem by Borel now states that for every
p irrational x there are
in nitely many rationals p=q su
h that (x; p=q) < 1= 5.
For any irrational number x we de ne the approximation
oeÆ
ients n by
n
:= n (x) = Qn jQn x
Pn j;
n = 1; 2; : : : :
(1.12)
They measure how well the rational number Pn =Qn approximates an irrational
number x. Sin
e it
an be shown that
Pn
1;
x
<
Qn
Q2n
we immediately see that 0 < n < 1 for all n 1. Using
Pn + Pn 1 Tn
;
x=
Qn + Qn 1 Tn
in (1.12), we
an show that
n
= 1 +TTn V
n n
;
and
n
Hen
e, de ning :
! R2 by
(x; y ) =
y
;
1
= 1 +VTn V
n n
x
1 + xy 1 + xy
:
(1.13)
Approximation
oeÆ
ients
11
leads to the fa
t that
(n 1 ; n) = (Tn; Vn ):
(1.14)
In fa
t, (
) = , where is a triangle with verti
es (0,0), (1,0), and (0,1). It
then follows immediately that
n 1 + n < 1; n = 1; 2; : : : ;
and hen
e
min(n 1; n ) < 21 ; n = 1; 2; : : : ;
whi
h is a well-known result due to Vahlen [V℄.
Using the fa
t that
(n; n+1) = (T( 1 (n 1 ; n)));
Jager and Kraaikamp [JK℄ were able to show that
p
n+1 = n 1 + an+1 1 4n 1 n a2n+1 n :
From this it easily follows that
min(n 1; n ; n+1) < q 2 1
an+1 + 4
and
max(n
q 21
1 ; n ; n+1 ) >
:
+4
The former
learly generalizes Borel's
lassi
al result, the latter was found by
J.C. Tong [T1℄. As a
orollary we nd the following result.
Theorem 1.7
x
n0
min(n 1; n; n+1) < p15 ;
1=p5
The following result is another
onsequen
e of (1.14) together with Theorem 1.6.
Theorem 1.8 (Jager, 1986)
(n 1; n)
x
1 p 1 :
f (a; b) =
log 2 1 4ab
For all irrational numbers
the
onstant
an+1
and all
annot be repla
ed by a smaller one.
The sequen
e
triangle
one has
a
ording, for almost all
are distributed over the
, to the density fun
tion
12
INTRODUCTION
See [J℄ for details.
Continued fra
tions play an important role in the theory of prime-testing (see,
e.g., Bressoud's book [B℄). In 1981, H. W. Lenstra
onje
tured that for almost
all x
(1.15)
lim 1 #fj : 1 j n; (x) z g; where 0 z 1;
!1 n
n
j
exists and equals F (z ), where
8 z ;
>
< log 2
F (z ) =
>
: 1 (1
log 2
0 z 21 ;
z + log 2z );
1
2
z 1:
In fa
t (1.15) had been
onje
tured in 1940 by Wolfgang Doeblin [Do℄. In 1984
Knuth [Kn℄ showed that
1 #f1 i N : + d g = 1 Z +d `(t) dt;
lim
i
N !1 N
log 2
where
1
`(t) = min 1;
1 :
t
In 1983 Bosma, Jager, and Wiedijk [BJW℄ proved the Lenstra-Doeblin
onje
ture using Nakada's natural extension (
; B;
; T).
1.5
A brief des
ription of the thesis
This thesis
onsists of ve papers dealing with
ontinued fra
tions.
Chapter 21 is
on
erned with the
ontinued fra
tion with odd partial quotients
(OddCF). The relation between OddCF and RCF is developed via singularization and insertion pro
esses. Using S
hweiger's natural extension for the OddCF
we show that the sequen
e of
onvergents of the nearest integer
ontinued fra
tion (NICF) is a subsequen
e of that of OddCF. Using the method in [JK℄,
we obtain a result for OddCF approximation
oeÆ
ients whi
h
oin
ides with
Tong's result for NICF [T2℄. Through the relation between RCF and grotesque
ontinued fra
tion (GCF) developed again via singularizations and insertions
we see that the sequen
e of GCF
onvergents forms a subsequen
e of that of
Hurwitz' singular
ontinued fra
tion. Maximal and non-maximal OddCF are
also dis
ussed.
1 a joint work with Cor Kraaikamp in
Rev. Romaine Math. Pures Appl.
47 (2002),
no. 1.
A brief des
ription of the thesis
13
In Chapter 32 we
onsider the map TE : [0; 1) ! [0; 1) given by
1 1
1
TE (x) := 1
b x
x b x
; x 6= 0; TE (0) := 0:
This map yields a (unique)
ontinued fra
tion expansion of x 2 [0; 1) with nonde
reasing partial quotients of the form
1
; bn 2 N ; with bn bn+1 :
b1
b1 +
b2
b2 +
bn 1
.
b3 + . . +
.
bn + . .
We
all this expansion
(ECF) expansion of x sin
e the
map TE is a modi ed version of the Engel series expansion map.
Some basi
properties of RCF also hold for the ECF but they di er in many
ways. For instan
e, ECF
onvergents behave di erently from regular ones. It
turns out that TE is ergodi
with respe
t to Lebesgue measure but has
nite invariant measure, equivalent to Lebesgue. Moreover, it is shown that the
map TE has in nitely many - nite, in nite invariant measures, two of whi
h
are given here. Additionally, we relate the ECF to Ryde's
(MNK) generated by the map TR : ( 12 ; 1) ! ( 21 ; 1),
given by
k k + 1
k
; k 2 N;
TR (x) = SR (x) =
k; for x 2 R(k ) :=
;
x
k+1 k+2
through an isomorphism. From this it follows, for example, that the map TR
is ergodi
with respe
t to Lebesgue measure but no nite TR -invariant measure
equivalent to Lebesgue exists and that not every quadrati
irrational has an
ultimately periodi
ECF expansion.
Engel
ontinued fra
tion
no
monotonen, ni
ht-
abnehmenden Kettenbru
h
A Hurwitz-type spe
trum was studied for the nearest integer
ontinued fra
tion
by Jager and Kraaikamp in [JK℄. With (n )n1 denoting the sequen
e of NICF
approximation
oeÆ
ients they showed that
p
min(n 1 ; n ; n+1 ) < 25 (5 5 11) = 0:4508 :
In [T2℄ Tong extended this result and proved that
p !2k+3
5
3
1
1
:
min(n 1 ; n ; : : : ; n+k ) < p + p
2
5 5
2 a joint work with Cor Kraaikamp and F. S
hweiger in
Bordeaux 14
(2002).
J. de Theorie des Nombres de
14
INTRODUCTION
Chapter 43 gives a proof of Tong's result using the method from [JK℄ whi
h
yields some metri
al observations with respe
t to Tong's spe
trum. Generalizations to a larger
lass of semi-regular
ontinued fra
tion expansions are also
derived.
A number x 2 R is
alled Hurwitzian if its RCF expansion (1.2)
an be written
as
x = [a0 ; a1 ; ; an ; an+1 (k); ; an+p (k)℄1
k=0 ;
where an+1 (k ); : : : ; an+p (k ) ( the so-
alled quasi period of x) are polynomials
with rational
oeÆ
ients whi
h take positive integral values for k = 0; 1; 2; : : :,
and at least one of them is not
onstant. This
learly generalizes periodi
ontinued fra
tions. In Chapter 54 we de ne the Hurwitzian numbers for the NICF,
the `ba
kward'
ontinued fra
tion expansion, and -expansions. We show that
the set of Hurwitzian numbers for su
h
ontinued fra
tions
oin
ides with the
lassi
al set of Hurwitzian numbers.
Chapter 65 is a
ontinuation of the previous
hapter. In this
hapter we de ne
Hurwitzian numbers for Minskowski's diagonal
ontinued fra
tion (DCF). We
also show that the set of DCF-Hurwitzian numbers
oin
ides the
lassi
al set
of Hurwitzian numbers. The situation is more
ompli
ated here than in the
previous paper due to the di eren
e in shape of the singularization area of the
NICF (and other -expansions) on one hand, and that of the DCF on the other.
3 a joint work with Cor Kraaikamp
4 a joint work with Cor Kraaikamp in Tokyo J. Math. 25 (2002), no. 2
5 J. Matematika atau Pembelajarannya VIII (2002), 837{841.
Bibliography
[B℄ Bressoud, D. M. {
Fa
torization and Primality Testing, Sringer-Verlag, New
York, (1989). MR 91e:11150
[BJW℄ Bosma, W., H. Jager and F. Wiedijk. -
the expansion by
ontinued fra
tions,
[DK℄ Dajani, K. and C. Kraaikamp. {
Some metri
al observations on
Indag. Math.
45 (1983), 353{379.
Ergodi
Theory of Numbers, Carus Math-
emati
al Monographs, No. 29, (2002).
[Di℄ Di
kson, L. E. {
History of the Theory of Numbers, Vols I, II, III, Carnegie
Institution of Washington, Washington, (1991-1923).
[Do℄ Doeblin, W. |
Compositio Math.
Remarque sur la theorie metrique des fra
tions
ontinues,
7 (1940), 353{371. MR 2,107e
[HW℄ Hardy, G. H. and Wright, E. M. { An introdu
tion to the theory of
numbers. Fifth edition.
York,
The Clarendon Press, Oxford University Press, New
(1979). MR 81i:10002
[IK℄ Iosifes
u, M. and C. Kraaikamp. -
tions,
The Metri
al Theory of Continued Fra
-
Kluwer A
ademi
Press, Dordre
ht, The Netherlands, (2002).
[J℄ Jager, H. -
Continued fra
tion and ergodi
theory,
and Related Topi
s, RIMS Kokyuroku,
Trans
endental Numbers
599, Kyoto University, Kyoto, Japan,
(1986), 55{59.
[JK℄ Jager, H. and C. Kraaikamp. |
tions,
Indag. Math.
[Kh℄ Khint
hine, A. Ya. -
1 (1935), 361{382.
[Kn℄ Knuth, D. E. Number Theory
Metris
he kettenbru
hprablemen
, Compositio Math.
The distribution of
ontinued fra
tion approximations,
19 (1984), no 3, 443{448. MR 86d:11058
[K1℄ Kraaikamp, C. -
Expansions,
On the approximation by
ontinued fra
-
51 (1989), no. 2, 289-307. MR 90k:11084
J.
Metri
and Arithmeti
Results for Continued Fra
tion
Ph.D. Thesis (1990), Universiteit van Amsterdam, Amsterdam.
15
16
INTRODUCTION
[K2℄ Kraaikamp, C. - A new
lass of
ontinued fra
tion expansions, A
ta Arith.
57 (1991), no. 1, 1{39. MR 92a:11090
[Ku℄ Kuzmin, R. O. - On a problem of Gauss , Dokl. Akad. Nauk. SSSR Ser.
A, (1928), 375{380.
[N℄ Nakada, H. | Metri
al theory for a
lass of
ontinued fra
tion transformations and their natural extensions, Tokyo J. Math. 4 (1981), no. 2, 399{426.
MR 83k:10095
[NIT℄ Nakada, H., S. Ito and S. Tanaka | On the invariant measure for the
transformations asso
iated with some real
ontinued fra
tions, Keio Engrg
Rep., 30 (1977), no. 13, 159{175. MR 58 16574
[O℄ Olds, C. D. { Continued fra
tions, Random House, New York, (1963). MR
26#3672
[P℄ Petersen, K. { Ergodi
Theory, Cambridge University Press, Cambridge,
(1997).
[R-N℄ Ryll-Nardzewski, C. - On the ergodi
theorems. II. Ergodi
theory of
ontinued fra
tions, Studia Math. 12 (1951), 74{79.
[S℄ S
hweiger, F. { Ergodi
theory of bred systems and metri
number theory,
Oxford S
ien
e Publi
ations. The Clarendon Press, Oxford University Press,
New York, (1995). MR 97h:11083
[T1℄ Tong, Jing Cheng { The
onjugate property of the Borel theorem on Diophantine approximation, Math. Z. 184 (1983), no. 2, 151{153. MR 85m:11039
[T2℄ Tong, Jing Cheng | Approximation by nearest integer
ontinued fra
tions
(II), Math. S
and. 74 (1994), no. 1, 17{18. MR 95
:11085
[V℄ Vahlen, K. Th. | Uber
Naherungswerte und Kettenbru
he, Journal f. d.
reine und angew. Math. 115 (1895), 221{233.
[Wa℄ Walters, P. { An Introdu
tion to Ergodi
Theory, Springer-Verlag New
York, In
., New York, (2000).
[Wi℄ Wirsing, E. { On the theorem of Gauss-Kuzmin-Lery and a Frobenius type
theorem for fun
tion spa
es, A
ta Arith. 24 (1974), 507{528. MR 49 2637
Chapter 2
Odd Continued Fra
tions
2.1
Introdu
tion
It is well-known that every x 2 [0; 1)
an be written as a nite (in
ase x is
rational) or in nite (when x is irrational)
ontinued fra
tion with odd partial
quotients:
x
e
=
1
=: [ 0; e1 =a1 ; e2 =a2 ;
2
e
1+
a
a
.
2 + .. +
; en =an ;
℄
;
(2.1)
en
an
where e1 = 1; ei = 1 and
.
+ ..
is a positive odd integer, for i 1, and
ai
ai
+ ei+1
>
1;
i
1
:
We
all (2.1) the odd
ontinued fra
tion (OddCF) expansion of x. Apart from
the OddCF-expansion of x one also has the so-
alled grotesque
ontinued fra
tion
(or GCF)
expansion of any x 2 [G 2; G), where G is the golden mean, i.e.,
1 (p5 + 1). The GCF-expansion is also given by (2.1), again with odd
G =
2
partial quotients ai and ei = 1, but now these ai and ei must satisfy
ai
+ ei
>
1;
i
1
;
and e1 = sgn(x).
There is an extended literature on both the OddCF and the GCF. In two
(unpublished) papers F. S
hweiger obtained the ergodi
theorem underlying the
OddCF and its natural extension where, as a by-result he showed that the
GCF is the dual algorithm of the OddCF ([S1℄, [S2℄), and studied the approximation properties of the OddCF ([S2℄). Around the same time G.J. Rieger also
obtained a Gauss-Kuzmin theorem for the OddCF and found the ergodi
systems underlying both the OddCF and the GCF ([R2℄). Also a Heilbron-theorem
17
ODD CONTINUED FRACTIONS
18
was given by Rieger for both expansions in [R1℄. In two re
ent papers G.I. Sebe
returned to the
onvergen
e rate in the Gauss-Kuzmin problem for the OddCF
([Se1℄) and GCF ([Se2℄) using the theory of random systems with
omplete
onne
tions. Sebe also obtained the natural extension for the GCF. More results
on the OddCF and the GCF
an be found in papers by S. Kalpazidou ([Ka1℄,
[Ka2℄) and D. Barbolosi ([B1℄, [B2℄).
At rst sight one might be tempted to say that nothing
an be said anymore
about these expansions! In [B2℄, Barbolosi showed that for any x 2 [0; 1) the
sequen
e of nearest integer
ontinued fra
tion (NICF)
onvergents of x forms
a subsequen
e of the sequen
e of OddCF-
onvergents of x. In order to understand this result we were led to a new
lass of
ontinued fra
tion expansions
with odd partial quotients, of whi
h the OddCF and the GCF are two examples.
In general, a
fra
tion
0+
b
semi-regular
ontinued fra
tion
e
1+
b
1
e
(SRCF) is a nite or in nite
= [ b0 ; e1 =b1 ; e2 =b2 ;
2
.
2 + .. +
n
n
n
; e =b ;
℄;
(2.2)
e
b
n+
b
..
.
with en = 1; b0 2 Z; bn 2 N , for n 1, subje
t to the
ondition
e
n+1 + bn
1; for
n
1;
(2.3)
and with the restri
tion that in the in nite
ase
e
n+1 + bn
2; in nitely often.
(2.4)
A nite trun
ation in (2.2) yields the SRCF-
onvergents
n
n := [ b0 ;
A =B
1 1 2 2
n
e =b ; e =b ; ; e =b
n℄ ;
where it is always assumed that g
d(An ; Bn ) = 1. We say that (2.2) is an
SRCF-expansion of an irrational number x in
ase
x
= lim
!1
n
A
n
n
B
:
Clearly the OddCF is an example of an SRCF-expansion, but the GCF is not.
Other examples of SRCF-expansions are the nearest integer
ontinued fra
tion
(NICF) expansion, satisfying
e
and Hurwitz'
n+1 + bn
2 for
singular
ontinued fra
tion
e
n + bn
n
1;
(HSCF) expansion, whi
h satis es
2 for
n
1:
19
Introdu
tion
Perhaps the best-known example of an SRCF-expansion is the so-
alled regular
ontinued fra
tion expansion (RCF); every real irrational number x has a unique
RCF-expansion
d0
+
1
d1
+
where d0 2 Z is su
h that x
=: [ d0 ; d1 ; d2 ; ℄;
1
.
d2 + . .
d0
(2.5)
2 [0; 1), and dn 2 N for n 2 N .
Obviously the GCF is not an SRCF, but a so-
alled unitary expansion, see
also [G℄. Unitary expansions are de ned in a way similar to SRCF-expansions,
the di eren
e being that (2.3) and (2.4) are repla
ed by
en
+ bn 1; for n 1;
and with the restri
tion that in the in nite
ase
en
+ bn 2; in nitely often.
Essential in our investigations are the notions of insertion and singularization
of a partial quotient equal to 1, whi
h were studied in detail in [K℄.
A singularization is based upon the identity
A
+
e
1 =
1+
B+
A+e
+
e
B
+1+
;
where 2 [0; 1).
To see the e e
t of a singularization, let (2.2) be an SRCF-expansion of x. A
nite trun
ation yields the sequen
e of
onvergents (rk =sk )k 1 : Suppose that
for some n 0 one has
bn+1 = 1; en+2 = 1 ;
i.e.,
[ b0 ; e1 =b1 ; ℄ = [ b0 ; e1 =b1 ; ; en =bn ; en+1 =1; 1=bn+2 ; ℄:
The transformation
ontinued fra
tion
n
(2.6)
whi
h
hanges this
ontinued fra
tion (2.6) into the
[ b0 ; e1 =b1 ; ; en =(bn + en+1 );
en+1 =(bn+2
+ 1); ℄;
(2.7)
whi
h is again a
ontinued fra
tion expansion of x, with
onvergents, say
(pk =qk )k 1 ; is
alled a singularization.It
was shown in [K℄ that the sequen
e
of ve
tors
pk
qk k
from the latter.
1
is obtained from
rk
sk k
1
by removing the term
rn
sn
ODD CONTINUED FRACTIONS
20
An operation whi
h is in some sense the `opposite' of a singularization is a
so-
alled
insertion.
An insertion is either based upon the identity
A
+
1
B
+
A+1
=
1
+
1 +
;
1
B
1+
or on the identity
A+
where
2
1
B
+
1
A
=
1
1+
has
bn+1 >
sr
f-insertion
[
x,
1
1;
is the transformation
b0 ; e1 =b1 ;
; en = bn
(
+1+
and suppose that for some
en+1
n
x,
1=1; 1=(bn
rk
sk k
obtained from
+1
1);
;
℄
with
onvergents, say, (pk =qk )k
1
by
inserting
r
s
1
1
;
r0
s0
;:::;
the term
1
1
rn
sn
;
Now let (2.2) be a unitary-expansion of
1
Let
pk
qk k
rn
sn
rn
sn
+
+
1
1
rn
+ sn
+
rn
sn
1
1
before the
;
rn
sn
;
rn+1
sn+1
1
is
n-th
;::: :
1=1.
An sr
f-insertion is denoted by
(rk =sk )k
1.
term of the latter sequen
e, i.e.,
1
0 one
be the sequen
e of
onvergents
onne
ted with (2.2). Using some
pk
qk k
= 1:
matrix-identities it was shown in [K℄ that the sequen
e of ve
tors
n
whi
h
hanges (2.2) into
+ 1);
whi
h is again an SRCF-expansion of
(rk =sk )k
B
[0; 1).
Let (2.2) be an SRCF-expansion of
An
;
1+
. Suppose that for some
bn >
n
1;
x
with the sequen
e of
onvergents
0 one has
en+1
= 1:
Applying the se
ond insertion-identity
hanges (2.2) into
[
b0 ; e1 =b1 ;
; en = bn
(
1); 1=1;
whi
h is again a unitary-expansion of
This kind of insertion is
alled a
x,
+1 + 1); ℄;
1=(bn
with
onvergents, say, (pk =qk )k
unitary-insertion.
1.
In this
ase the sequen
e
21
Insertions, singularizations and the OddCF
of ve
tors
pk
qk
of the new expansion is obtained from
k
1
rk
sk
k
1
by
in-
the term nn nn 11 before the -th term of the latter sequen
e. A
unitary-insertion is denoted by 1 1 .
By
ombining the operations of singularization and sr
f/unitary-insertion one
an obtain any semi-regular/unitary
ontinued fra
tion expansion of a number
from its RCF expansion. In [K℄ a whole
lass of semi-regular
ontinued fra
tions was introdu
ed via singularizations only (some of these SRCF's were new,
some
lassi
al like the
ontinued fra
tion to the nearest integer, or Hurwitz'
(HSCF)), and their ergodi
theory studied (the main
idea in [K℄ is that the operation of singularization is equivalent to having an
indu
ed map on the natural extension of the RCF).
In the next se
tion we will show that the OddCF-expansion
an be obtained
from the RCF via suitable sr
f-insertions and singularizations. We also will
derive some metri
al results for the approximation
oeÆ
ients of the OddCF.
In Se
tion 2.3 we will see that the GCF
an be obtained from the RCF via
singularizations and unitary-insertions. This will lead us in Se
tion 2.4 to a new
lass of semi-regular/unitary
ontinued fra
tion expansions with odd partial
quotients.
serting
r
r
s
s
n
=
x
singular
ontinued fra
tion
2.2
Insertions, singularizations and the OddCF
2.2.1
A singularization/insertion algorithm
The following theorem des
ribes an algorithm whi
h turns the RCF-expansion
of any 2 [0 1) into the OddCF-expansion of . The proof of this theorem
follows easily by inspe
tion, and is therefore omitted.
2 [0 1)
(2.5)
0 =0
(2.5)
x
;
x
Theorem 2.1 Let x
with RCF-expansion
;
starting from the RCF-expansion
, i.e., d
.
Then
of x, the following algorithm yields the
OddCF-expansion of x.
()
I
Let m
()
i
:= inf f 2 N;
1
m+1
If d
n
>
[ 0; 1=d1 ;
( )
ii
If dm
+1
dn is even
11
, insert
=1
;
1=d
=
m 1
, let k
;
g
.
after dm to obtain
m + 1)
1=(d
:= inf f
1=1; 1=(d
;
;
n > m
dn >
m+1
1); 1=d
1g ( = 1
k
singularize in the blo
k of partial quotients
dm
+1 = 1; dm+2 = 1; : : : ; dk 1 = 1
m+2 ℄
;
)
:
is allowed . Now
ODD CONTINUED FRACTIONS
22
the rst, third, fth, et
. partial quotients equal to 1, to arrive at
[ 0; 1=d1 ;
;
1 ; 1=(dm + 1); | 1=3; {z
1=dm
;
1=3;
}
k m 2 times
2
in
ase
k
[ 0; 1=d1 ;
In
ase
k
[ 0; 1=d1 ;
1
m
is odd or
1
m
m
1=d
1; in the latter
ase we nd
( m + 1)
1 3
1 3 ℄
1 ; 1=
d
;
= ;
;
1 ; 1=(dm + 1); | 1=3; {z
In this
ase insert
(I I )
;
;
=
1=3;
k m 3 times
2
[ 0; 1=d1 ;
℄
;
= ;
:
is even we obtain
1=dm
;
;
k
1=(dk + 1); 1=dk+1 ;
1=1
}
1=2; 1=dk ; 1=dk+1 ;
℄
:
to arrive at
1=(dm + 1);
|
1=3;
{z
;
1=1; 1=(dk
1=3;
k m 1 times
2
}
1); 1=dk+1 ;
℄
;
Let m 1 be the rst index in the new SRCF-expansion [
0 ; e1 =
1 ; ℄ of
obtained in (I) for whi
h
m is even. Repeat the pro
edure from (I) to
this new SRCF-expansion of x with this value of m.
x
As soon as
m
1 in (II) we have obtained the OddCF-expansion of
=
.
x
The following example illustrate how to use Theorem 2.1
Example 2.1 Let x
2 [0 1) have RCF-expansion
;
[0; 1=3; 1=1; 1=4; 1=7; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=5;
(i) Apply the algorithm with
m
= 3. Sin
e
d4 >
1, we insert
℄
:
1=1 after 1=4
to obtain
[0; 1=3; 1=1; 1=5;
1=1; 1=6; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=1; 1=5;
(ii) Apply the algorithm with
d11
= 1, we singularize
[0; 1=3; 1=1; 1=5;
Now insert
1=1; 1=7;
1=1; 1=7;
(iii) Apply the algorithm with
m
=
= 5 in the new expansion. Sin
e
and
d11
1=3;
d6
=
to arrive at
1=3;
1=3;
1=2; 1=5;
℄
:
=
:
1=1 to arrive at
[0; 1=3; 1=1; 1=5;
until
m
d6 , d8 , d10
℄
1.
m
1=3;
1=3;
1=3;
1=3;
1=1; 1=4;
℄
:
= 10 in this new expansion and
o