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Proefshrift

terverkrijgingvandegraadvandotor

aandeTehnisheUniversiteit Delft,

opgezagvandeRetorMagnius prof.dr.ir.J.T.Fokkema,

voorzittervanhetCollegevoorPromoties,

in hetopenbaarte verdedigenopmaandag3februari2003om13.30uur

door

Yusuf HARTONO

MasterofSienein AppliedMathematis,

UniversityofMissouri-Rolla,USA

Toegevoegd promotor: Dr. C.Kraaikamp

Samenstellingpromotieommisie:

RetorMagnius voorzitter

Prof.dr.F.M.Dekking TehnisheUniversiteitDelft,promotor

Dr.C.Kraaikamp TehnisheUniversiteitDelft,toegevoegdpromotor

Prof.dr.J.M.Aarts TehnisheUniversiteitDelft

Prof.dr.M.Iosifesu RomanianAademyofSienes,Roemenie

Prof.dr.F.Shweiger UniversitatSalzburg,Oostenrijk

Prof.dr.R.K.Sembiring Institut TeknologiBandung,Indonesie

Dr.W.Bosma KatholiekeUniversiteitNijmegen

Theresearhinthisthesishasbeen

arried out under the auspies of

the Thomas Stieltjes Institute for

Mathematis, at the University of

TehnologyinDelft.

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ISBN90-407-2381-8

Keywords: metri,arithmeti,ontinued frations

Copyright 2002byY.Hartono

Allrightsreserved. Nopartof thematerialprotetedbythis opyrightnotie

maybereproduedorutilizedin anyform orbyanymeans, eletronior

me-hanial,inludingphotoopying,reordingorbyanyinformationstorageand

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theyarenotertain;

andasfarastheyareertain,

theydonotrefertoreality.

Albert Einstein(1879 - 1955)

Thebeginningofknowledgeis

thedisoveryofsomethingwedonotunderstand.

FrankHerbert (1920 - 1986),AmerianWriter

ThefearoftheLordisthebeginningofknowledge,:::.

Forfromhimandthroughhim

andto himareallthings.

Tohimbethegloryforever! Amen.

Iamdeeplyindebtedtomanyinstitutionsandpersonswithoutwhomthisthesiswould

neverexist. AttherstplaeIwouldliketoexpressmygratitudeandappreiationto

mysupervisor CorKraaikamp,who introduedmetothis exitingresearharea,for

hismanyinteresting ideasandonstanthelp thatkeptmeintheright diretion,for

hisenouragementthatkeptmegoing,andforhispatienethatomfortedmeduring

the diÆult timeinthe researh. It hasbeen a greatpleasure to meet Karmaand

Rafael. Also,IwouldliketoexpressmygratefulnesstomypromotorMihelDekking

for inviting meto the Netherlandsand providing me withthe opportunity to arry

outmydotoralresearhatDelftUniversityofTehnology(TUDelft). Ishouldalso

thankmyommitteefortheirvaluableommentsandsuggestions.

Theresearhleadingtothisthesiswasapartofthesientiooperationbetween

DuthandIndonesian governments. Iamverygrateful tothe programoordinators

Prof. Dr. R. K. Sembiring and Dr. A. H. P. van der Burgh who four and a half

years agoorganized a researh workshop inBandung that gave me a hane to be

seletedforthis dotoral programatTUDelft. IthankDr.O.SimbolonandDr.B.

Karyadi,formerprojetmanagersofProyek PengembanganGuru SekolahMenengah

(PGSM) {(Seondary ShoolTeahers Development Projet), inJakarta as wellas

Drs.P.Althuis,diretorofCenterforInternationalCooperationinAppliedTehnology

(CICAT)atTUDelft,fornanialsupportandassistaneduringmyresearhandstay

inHolland.

IpersonallywishtothankallmembersofAfdeling CROSS,inpartiularvakgroup

SSOR,for de gezelligheid andaveryonduiveatmosphere,espeiallyCindy,Ellen,

andDianafor theirbothadministrativeandnon-administrativeassistane,andCarl

forhis omputerassistane. Ialso wishtothankDurk,Christel, Theda, Veronique,

andReneforaverywonderfulfriendshipandforprovidingmewithalmosteverything

Ineededduringmy stayinHolland. Mythanksshouldalsogoto all PGSMfellows

(Abadi,Budi,Caswita,Darmawijoyo,Gede,Happy,Hartono, Kusnandi,Sahid,Siti,

andSuyono)withwhomIstartedthe wholeprojettogetherforeverythingwehave

hadtogetherfromfuntoseriousdisussionsaboutmathematis,partiularlytoSuyono

with whom I shared anoÆe roomand from whomI learned a lot about measure

theory;aswellastoAgus,KomoandJuliuswithwhomIspentsometimesharingan

apartmentinDelft.

A lot of help and enouragement also ame from my olleagues at the Faulty

ofTeaher Training andEduation, partiularly at the Department ofMathematis

Eduation, Sriwijaya University {PakIsmail, PakPurwoko, BuTri, and Somakim,

justtonamesome,aswellasfromKapten O.Tengke,Happyand othermembersof

BalaKeselamatanKorpsKundur,andKakIbrahim. Iwouldliketoaknowledgethem

Ihaveappreiatedallthesuggestions,orretions,andommentsduringthe

prepa-rationofthis thesis. Iamtheonlyoneresponsible foranymistakeremaininginthis

thesis.

IamalsopleasedtoaknowledgeallmembersofHetLegerdesHeilsKorpsDelft,

espeiallyMajoorenmevrouwLoefandKapiteinenmevrouwJansen,andof

Interna-tionalStudentChaplainy,espeiallyFr.BenandRev.Stroh,fortheirgenerosityand

warm hospitality making mereally feel at home and letting mepartiipate in their

many ativities; to mentiona few of them: Daniela, Bernardette, Carla, Riardo,

Yenory, Maro, Bibiana, Sandra, Sarah, Yadira, Susanne, Irek, Fabiana, Carmen,

Ralph,Paul, Poni, Duleep,Niolo,and Fabio. Many thanksto Miekeand Reinifor

onsideringmeaskind aanhuis intheir homeand family. Ikvind het heelleuk om

metSara,David, enNathankennis temaken. Ik wilTilak, Mart, en Jose,met alle

medewerkers ookbedankenvoordesamenwerking eneenhele mooievriendshap.

Moreover, Iertainlyenjoyedwonderfultimes togetherand very warm

ompan-ionshipwithIndonesianstudents,espeiallyfromTUDelftandIHEDelft,duringmy

study Bernadeta, Yusuf, Dwi, Teresia, Sri, Helena, Arief, Elisa, Raymon, Silvia,

Dian,Dedy,Reiza,Evy,Joye, Diah,Hilda,Firdian, Susi,Theresia, Zenlin, and

Su-sana, justto mentiona few. Their presene inmy life shows that strongbonds of

loveinserviewitheahotherareessentialfordevelopingapeaefulandharmonious

ommunity.Theydeserveanaknowledgementtoo.

Iwouldliketodediatethisthesistothememoryofmymotherwhopassedaway

onJanuary 3,2001 attheageof 65. Shetaughtmehowto thankandhavefaith in

God,how topray,andhowtoloveandserveothers. Myfather, who hastaughtme

themeaningofhard-workinglifeand,moreimportantly,howtodoarithmetiduring

myearlyyearsatshool,alsodeservesmyspeialthanksforwithouthimIouldnever

bewhatIamnow;andso doesmyauntwhotookare ofmysonwhilemywifewas

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 memilikiiman kepada Allah, bagaimana berdoa, dan

bagaimanamengasihiserta melayaniorang lain. Ayahsaya yangmengajarsayaarti

kehidupandankerjakerasdan,yanglebihpentinglagi,mengajarsaya berhitung pada

tahun-tahun awalsaya disekolah, juga pantas mendapatuapanterima kasih khusus

karena tanpa dia saya tidak mungkin menjadi seperti saya sekarang, demikian juga

bibisayayangmengurusanaksayasaat istrisaya bekerja.

Finally,asinereexpressionofthanksshould gotomydearestwife Elfaridaand

ourbelovedsonGabriel Ekoputrafor theirlove, patiene,and understandingduring

myabseneinthefamilyduetomystudy.

Terimakasihyangtulussaya sampaikankepadaistrisaya dananakkamiterinta

untukkasih,kesabaran, danpengertianmerekaselamasaya tidak beradadalam

kelu-arga.

Aboveall,IthankGodforeverythingthathashappenedinmylife.

1 Introdution 1

1.1 Somehistorialbakgroundandbasiproperties . . . 2

1.2 Morereentdevelopments . . . 6

1.3 Someresultsinergoditheory. . . 8

1.4 ApproximationoeÆients. . . 10

1.5 Abriefdesriptionofthethesis . . . 12

Bibliography . . . 15

2 Odd ContinuedFrations 17 2.1 Introdution. . . 17

2.2 Insertions,singularizationsandtheOddCF . . . 21

2.2.1 A singularization/insertionalgorithm. . . 21

2.2.2 MetrialpropertiesoftheOddCF . . . 23

2.2.3 ApproximationoeÆients. . . 27

2.3 Grotesqueontinuedfrations . . . 32

2.4 Otheroddontinuedfrations . . . 34

2.4.1 Maximal OddCF's . . . 34

2.4.2 Non-maximalexpansionswithodddigits . . . 35

Bibliography . . . 37

3 Engel ContinuedFrations 39 3.1 Introdution. . . 39

3.2 Basiproperties. . . 42

3.3 Ergodiproperties . . . 47

3.4 OnRyde'sontinuedfration withnondereasingdigits . . . 55

Bibliography . . . 59

4 Tong'sSpetrum for SRCF Expansions 61 4.1 Introdution. . . 61

4.2 Tong'spre-spetrumfortheNICF . . . 64

4.3 Tong'sspetrumforNakada's -expansions . . . 69

4.3.1 Theaseg<1. . . 70

4.3.2 Thease 1 2 g . . . 73

Bibliography . . . 79

5 A Noteon Hurwitzian Numbers 81 5.1 Introdution. . . 81

5.2 HurwitziannumbersfortheNICF . . . 82

5.3 Hurwitziannumbersforthebakwardontinuedfration. . . 84

5.4 Hurwitziannumbersfor-expansions . . . 86

Bibliography . . . 91

6 Mikowski'sDCF Expansions of Hurwitzian Numbers 93 6.1 Introdution. . . 93

6.2 Minkowski'sDCF. . . 93

6.3 DCF-Hurwitzianexpansion . . . 95

Bibliography . . . 99

Samenvatting 101 Bibliography . . . 105

Introdution

Mathematisisthequeenofthesiene,

andnumbertheoryisthequeenofmathematis.

CarlFriedrihGauss(1777{1855)

This thesisonsists ofvepapersonerningontinued fration expansionsin

onnetionwithergodi theory. Mostof themdealwithmetrial propertiesof

ontinuedfrationalgorithms. OtheraspetslikeapproximationoeÆientsare

alsostudiedhere. Relationshipsamongdierentontinued frationexpansions

aredevelopedviathesingularizationandinsertionproesses.

Throughoutthisthesisbyaontinuedfrationexpansionofanyrealnumber

wemeananexpressionoftheform

a

0 +

e

1

a

1 +

e

2

a

2 +

.

.

.

+ e

n

a

n +

.

.

.

; (1.1)

wherea

0 2Z,a

n

arepositiveintegersand e

n

2R forn=1;2;:::. In amuh

moreonvenientwaywewrite (1.1)moreompatlyas

[a

0 ;e

1 =a

1 ;e

2 =a

2 ; ;e

n =a

n ;℄:

Thetermsa

1 ;a

2

;:::are alledthepartial quotientsoftheontinued fration.

Thenumberoftermsmaybeniteorinnite.

Inasee

n

=1foralln=1;2;:::, weall(1.1)aregularontinuedfration

(RCF) expansionandwriteitas

[a

0 ;a

1 ;a

2 ;;a

n

;℄: (1.2)

Ingeneral,letxbearealnumber,a

0

=bx,thelargestintegernotexeeding

x,andwritex=a

0

+. Now =x a

0

2[0;1), andwewrite

x=[a

0 ;a

1 ;a

iftheRCF-expansionof isgivenby

=[0;a

1 ;a

2 ;℄:

Inthenexttwosetionssomehistorialbakgroundandbasipropertiesof

regular ontinued frations will be presented, followed by a setion on a few

basifatsinergoditheoryandanotheronapproximationoeÆients. Abrief

desriptionoftheontentofthepapersinthisthesiswillonludethishapter.

1.1 Some historial bakground and basi

prop-erties

Continued frations have along history. It starts with the proedure known

as Eulid's algorithm for nding the greatest ommon divisor (g..d.) of two

integerswhih ours in theseventh book of Eulid'sElements (. 300b..).

This proedure is perhaps the earliest step towards the development of the

theoryofontinuedfrations.

ToseetherelationbetweenEulid'salgorithmand(regular)ontinued

fra-tions,onsiderEulid'salgorithm forndingtheg..d.of twointegersaand b

witha>b>0. Werstleta

0

=ba=b. Putting

r

1

:=a a

0 b; r

0 :=b;

wehavetondpositiveintegersa

i

suhthat

r

i 1 =a

i r

i +r

i+1

; (1.3)

where0r

i+1 <r

i

,fori=1;2;:::untiltheproedure stops;thatis,whenwe

havereahed anindex n suh that r

n

6=0and r

n+1

=0. Inthis ase, wesay

thatr

n

istheg..d.ofaandb.

Dividing(1.3)throughbyr

i

, weget

r

i 1

r

i =a

i +

r

i+1

r

i

; i=1;2;:::n:

Writing

r

i+1

r

i =

1

a

i+1 +

r

i+2

r

i+1

; i=1;2;:::;n;

andsubstitutingitintothepreviousequationforeahiyield

a

b = [a

0 ;a

1 ;a

2 ; ;a

n ℄;

To generalize Eulid's algorithm to irrational numbers x in the unit interval,

Dene further a

1

For rationalnumbersrepeatedappliationofT isinfat equivalentto Eulid's

algorithm. Hene, there existsan n

0

2N suhthat T n

0

(x)=0and itfollows

that arational number has a nite RCF expansion. This is notthe ase for

irrationalnumbers. Ifx isanirrationalnumber,thenT n

(x)isirrationalforall

n0.

Anite trunationin(1.2)givestheso-alledregularonvergents

P

whereweassumethatQ

n

The sequenes (P

n

satisfythe followingreursive

for-mulae

andtherelationship

P

Moreover,theregularonvergentssatisfythefollowinginequalities:

P

Foranyirrationalnumberxwesaythat(1.2)istheRCFexpansionofxinase

lim

See,forinstane, [O℄, [IK℄,and[HW ℄formorepropertiesofRCFandproofs.

Ingeneral,(1.1)isalledasemi-regularontinuedfration(SRCF)inasea

0 2

Z,a

n

arepositiveintegers,ande

n

=1foralln1,subjettotheondition

andwith therestritionthatin theinnitease

e

n+1 +b

n

2; innitelyoften.

Nakada's -expansions, for 2 [1=2;1℄, are examples of SRCF expansions.

Clearly, the RCF expansion ( = 1), the nearest integer ontinued fration

(NICF) expansion ( = 1=2), and Hurwitz' singular ontinued fration

(g-expansion, with g = ( p

5 1)=2,) are all SRCF expansions. Other examples

ofSRCFexpansionsonsidered inthis thesisareMinkowski'sdiagonal

ontin-uedfration(DCF)and oddontinuedfration(OddCF).

Intheareaofappliations,thegreatDuthmathematiian,mehaniian,

as-tronomer,andphysiist,ChristiaanHuygens(1629-1695)usedtheregular

on-vergenttoobtaintheorretratiofortherotationsofplanetswhenhedesigned

thetoothedwheelsofaplanetarium. Hedesribedthisin hisDesriptio

Auta-matiPlanetarii,publishedposthumouslyin 1698. Thisisinfataonsequene

ofthefat that ontinued frationsgivethe \best"rationalapproximationsto

irrationalnumbers.

ThemoderntheoryofontinuedfrationsbeganwiththewritingsofRafael

Bombelli, born in about 1530 in Bologna. He showed, for example, in our

modernnotation,

p

13=[3;4=6;4=6;℄:

PietroAntonioCataldi(1548-1626)alsodeservessomereditsinontinued

fra-tions. Heexpressed

p

18=[4;2=8;2=8;℄:

UsingEulid'salgorithmforndingtheg..d.of177and233,DanielShwenter

(1585-1636)foundtheonvergents79=104;19=25;3=4;1=1;and0=1. Itis

prob-ablyinAritmetiaInnitorum(1655),abookbyJohnWallis,thattheterm

on-tinuedfrationwasusedforthersttime. GreatmathematiianssuhasEuler

(1707-1783),Lambert (1728-1777),Lagrange (1736-1813),Gauss (1777-1855),

andmanyothersalsomadeimportantontributionstotheearlierdevelopment

ofthetheoryofontinuedfrations. ItisinpartiularEuler'sgreatmemoir,De

FrationibusContinius (1737),thatlaidthefoundationforthemoderntheory.

See,forexample,[O℄,[K1℄,[S℄,and[Di ℄formorehistoryofontinuedfrations.

ThemetrialtheoryofontinuedfrationsstartedwithGauss'problem. In

hisdiaryonOtober25,1800,Gausswrote(inmodernnotation)that

lim

n!1 F

n (z)=

log(z+1)

log2

; z2[0;1); (1.7)

whereF

n

(z)=(T n

(x)<z); z2[0;1). Here T istheontinuedfration map

denedin (1.4)anddenotestheLebesguemeasure. InaletterdatedJanuary

30,1812,heaskedLaplaeto prove(1.7)andtoestimatetheerror-term

e

n

(z):=F

n (z)

log(z+1)

MorethanaenturylaterthisproblemwassolvedbyKuzmin[Ku℄. Heshowed

in1928that

e

n

(z)=O(q p

n

) as n!1

forsome onstant q2 (0;1). His proofis reprodued in Khinhine [Kh℄. One

yearlaterLevyindependentlyprovedthat

je

n

(z)j<q n

; n=1;2;:::

withq=3:5 2 p

2=0:67157. SeeSubsetion1.3.5in[IK℄ foranimproved

versionofLevy'ssolutiontoGauss'problem. In1961 P.Szuszused Kuzmin's

approah to ndthat q =0:485. Gauss'problem wassettled by Wirsing [Wi ℄

whoin1974foundthatq=0:303663002. Resultslikethesearenowknown

asGauss-Kuzmin-LevyTheorems. Thefollowingresultisaonsequeneofthese

results.

Theorem1.1 (Levy, 1929) For almost all x 2 [0;1) with RCF expansion

(1.2)onehas

lim

n!1 1

n logQ

n =

2

12log2 ;

lim

n!1

log((

n )) =

2

6log2 ;

lim

n!1 1

n

x

P

n

Q

n

=

2

6log2 :

Here denotes the Lebesgue measure and

n

=

n (i

1 ;:::;i

n

) the so-alled

fundamentalinterval denedby

n =

x2[0;1): 1

i

j +1

T j 1

(x)< 1

i

j

; j=1;2;:::;n

:

Moreover,amongotherthings,Khinthine[Kh℄showedthefollowing.

Theorem1.2 (Khinthine,1935) Foralmostallx2[0;1)withRCF

expan-sion(1.2)onehas

lim

n!1 (a

1 a

2 a

n )

1=n

= 1

Y

k =1

1+ 1

k(k+1)

logk

log2

=2:6854:

Forproofsofthelasttworesultssee[DK℄. OneofthemisprovedinSetion1.3

usingsomeresultsinergoditheory;seepage9.

ThelimitingdistributionofT n

(x)in(1.7)leadsustoameasurewithdensity

1 1

todayknownasGauss'measure. Thismeasureisinvariantundertheontinued

fration map T dened in (1.4) (i.e., T is Gauss measure preserving). To see

this,let(a;b)[0;1). Sine

T 1

(a;b)=

1

n+b ;

1

n+a

;

wehave,with denotingGaussmeasure,

(T 1

(a;b))= 1

log2 1

X

n=1 Z

1

n+a

1

n+b dx

1+x =

1

log2 log

b+1

a+1

=((a;b)):

SeealsoTheorem1.2.1in[IK℄.

1.2 More reent developments

Anotherimportant developmentin thetheoryof ontinuedfrations isthe

in-trodution of the so-alled natural extensions by a group of Japanese

mathe-matiians;see, e.g.,thepapersbyH. Nakada,S. ItoandS. Tanaka[NIT℄,and

Nakada [N℄. Inthis lastpaperthenaturalextension

T ofT isdenedby

T(x;y)=

T(x); 1

a

1 (x)+y

; (x;y)2[0;1)[0;1℄: (1.9)

Itfollowsimmediatelythat

T n

(x;y)=(T

n ;V

n );

whereT

n :=T

n

(x) and V

n

:=[0;a

n ;a

n 1 ;;a

1 ℄=Q

n 1 =Q

n

. Note that we

mightonsider T

n

asthe\future" ofx at the\urrent" time nand V

n asthe

\past"ofxuptotimen. Thepoints(T

n ;V

n

)aredistributedintheunitsquare

aordingto thedensityfuntion (log2) 1

(1+xy) 2

. In fat,this isa

onse-queneoftheergodisystem(1.11)in Theorem1.6onpage10.

Essentialin thisthesisare theso-alledsingularization andinsertion

pro-essesbywhihweanobtainotherSRCFexpansionsofxfromitsRCF

expan-sion,suhasthenearestintegerontinuedfration andoddontinuedfration

expansions. Wedisusssomemetrialpropertiesofoddontinuedfration

ob-tainedfromtheregularontinuedfrationviasingularizationsandinsertionsin

Chapter2. Thesingularizationproessisbasedontheidentity

A + e

1+ 1

B+

= A+e + e

B+1+ ;

whiletheinsertionproessrestsontheidentity

A + 1

B+

= A+1+

1

1+ 1

where2[0;1).

Thismeans,forexample,thatsingularizinga

n+1

=1inanRCFexpansion

(A) [0;a

1 ;a

2 ;;a

n ; 1;a

n+2 ;℄

withthesequeneofonvergents,say,(A

n =B

n )

n1

resultsinanSRCFexpansion

(B) [0;1=a

1 ;1=a

2

;;1=(a

n

+1); 1=(a

n+2

+1);℄:

On theother hand, inserting 1=1in the RCF (1.2)at (n+1)-st position as

a

n+2

6=0resultsinanSRCFexpansion

(C) [0;1=a

1 ;1=a

2

;;1=(a

n

+1); 1=1;1=(a

n+2

1); ℄:

Theeetsof thesetwoproessesonthesequeneofonvergentswerestudied

in [K2℄. It is shownthat the sequeneof onvergentsoftheSRCF(B) an be

obtainedfromthatoftheRCF(A)byremovingA

n =B

n

. Ontheotherhand,the

sequeneofonvergentsoftheSRCF(C)anbeobtainedfromthatoftheRCF

(1.2)byinserting(P

n +P

n 1 )=(Q

n +Q

n 1

)betweenP

n 1 =Q

n 1 andP

n =Q

n .

In [K2℄ Kraaikamp introdued anew lass ofontinued frations alled

S-expansionswhihareobtainedfrom theRCFonlybyusingthesingularization

proess. The-expansions(see[N℄)are examplesof S-expansions;see [IK℄for

more examples. Essential to these expansions is the so-alled singularization

area;thatis,asubsetS of[0;1)[0;1℄satisfyingthefollowingonditions.

(i) S2B andS isa-ontinuityset,

(ii) S[1=2;1)[0;1℄,

(iii) S\

T(S)6=;.

ToobtaintheNICFofx,forinstane, wehavetosingularize ineahblok

ofm2N[f1gonseutivepartialquotientsequalto1,therst,third,:::et.

partialquotient. Thisleadsto asingularizationarea

S

NICF

=[1=2;1)[0;g℄;

where g :=( p

5 1)=2. Other two examplesofS-expansionsare Minkowski's

diagonalontinuedfration(DCF),withsingularizationarea

S

DCF =

(t;v)2[0;1)[0;1℄: t

1+tv >

1

2

;

andBosma'soptimalontinuedfration(OCF),withsingularizationarea

S

OCF =

(t;v)2[0;1)[0;1℄:v<tandv< 2t 1

1 t

:

ItwasWolfgangDoeblin[Do℄whorstdisoveredtheergodisystem

under-lyingthe RCF. Unfortunately, his resultsremained unnotiedfor alongtime.

Alllassialresultsofontinuedfrationswereobtainedwithprobabilisti

meth-odsuntilC.Ryll-Nardzewskishowedin1951[R-N ℄howmetrialresultsanbe

1.3 Some results in ergodi theory

Ergoditheoryarosefromanattemptinstatistialmehanistodesribea

sys-temofaertainnumberofpartilesmovinginathree-dimensionalspaeatany

giventime. Ingeneral, let(;B;P)bea probability spae. A transformation

T :!isalled measurableifT 1

A2B forallA2B. WeallT measure

preservingifitismeasurable andP(T 1

A)=P(A)for allA2B. A

transfor-mationT issaidtobeergodiifeveryT-invariantsubsetofBhasmeasure0or

1,that is, T 1

A =A)P(A)2 f0;1g: Equivalently, wesaythat (;B;P;T)

formsanergodisystem.

Thefollowingresultisfundamentalinergoditheory;see,e.g.,[P℄and[Wa℄

formoredetailsandproofs.

Theorem1.3 (Birkho'sIndividual Ergodi Theorem,1931) Let(;B;P)

beaprobabilityspaeandT:!ameasurepreservingtransformation. F

ur-ther,letf :=!R be suhthatf 2L 1

(;B;P). Then for almost allx

f

(x):= lim

n!1 1

n n 1

X

k =0 f(T

k

x)

exists. Moreover,wehavef

(x)2L 1

(;B;P),f

(x)=f

(Tx),and R

fdP =

R

f

dP.

ThenexttheoremisanimportantonsequeneofBirkho'sergoditheorem.

Theorem1.4 Let(;B;P;T)beanergodisystemandf :!Rbesuhthat

f 2L 1

(;B;P). Then for almostall x wehave

lim

n!1 1

n n 1

X

k =0 f(T

k

(x))= Z

fdP:

The following fundamental result is veryimportant in the development of

thetheoryofontinuedfrationin onnetionwithergoditheory.

Theorem1.5 Let=[0;1), B be the olletion of all Borelsets of , and

theGauss measuregiven in(1.8). Further,letT bethe ontinuedfrationmap

(1.4). Then

(;B;;T); (1.10)

formsanergodi system.

ThefollowingexampleillustratesanappliationofTheorem1.5.

Example1.1 Thisequivaleneanbeeasilyhekedforx2[0;1):

a

n

(x)=a,T n 1

(x)2

1

; 1

Then theproportionof partial quotients equalto a in the sequeneof partial

quotients(a

n )

n0

isforalmostallxgivenby

1

log2 Z

1

a

1

a+1 dx

1+x =

1

log2 log

(a+1) 2

a(a+2) :

This gives, for instane, that 2:272 perent of the partial quotients equal

to7.

Wenowseethat theresultsofLevyandKhinthine(seeTheorems1.1and

1.2)areorollariesofTheorem1.5,togetherwithTheorem1.4. Asanexample,

wegivehereaproofofKhinthine'sresult;seealso[DK℄.

Proof of Theorem 1.2. Dene f(x)= loga

1

(x) where a

1

(x) = b1=x, x 2

(0;1). Then,duetoergodiityofT,wehave

(a

1 a

2 a

n )

1=n

= 0

n 1

Y

j=0

exp(f(T j

(x))) 1

A 1=n

= exp 0

1

n n 1

X

j=0 f(T

j

(x)) 1

A

=exp Z

1

0 fd

:

Itremainstoshowthat f isintegrable. Now

Z

1

0

fd= 1

X

k =1 Z 1

k

1

k +1 fd;

and

Z 1

k

1

k +1 fd=

1

log2 Z 1

k

1

k +1 loga

1 (x)

1+x dx=

logk

log2 log

1+ 1

k(k+2)

logk

k(k+2)

ask!1. Herewehaveusedlim

!0

(1+)=2

=1:Theresultfollowsfromthe

fatthat

1

X

k =1 logk

k(k+2)

isonvergentandwriting

1

X

k =1 logk

log2 log

1+ 1

k(k+2)

=log 1

Y

k =1

1+ 1

k(k+2)

logk

log2

:

2

The naturalextension of the system(1.10), whih is used several times in

Theorem1.6 Let

= [0;1℄,

B be the lass of all Borel sets of

, be

theextended(two-dimensional)Gauss measuredenedby

(A)=

1

log2 Z

A

dxdy

(1+xy) 2

; A2

;

and

T isthe natural extension(1.9)of T. Then

(

;

B;;

T); (1.11)

formsanergodi system.

1.4 Approximation oeÆients

Oneof themostimportantreasonsto use(regular)ontinuedfrationsis that

ontinuedfrationsyield\thebest"rationalonvergentstoirrationalnumbers.

Inordertoexpressthequalityofapproximationofanirrationalnumberxbya

rationalnumberp=q,weintroduetheapproximation oeÆient (x;p=q)by

(x;p=q)=qjqx pj:

A lassial theorem by Borel now states that for every irrational x there are

innitelymanyrationalsp=q suhthat (x;p=q)<1= p

5.

Foranyirrationalnumberx wedenetheapproximationoeÆients

n by

n :=

n

(x)=Q

n jQ

n

x P

n

j; n=1;2;:::: (1.12)

TheymeasurehowwelltherationalnumberP

n =Q

n

approximatesanirrational

numberx. Sineitanbeshownthat

x

P

n

Q

n

<

1

Q 2

n ;

weimmediately seethat0<

n

<1foralln1. Using

x= P

n +P

n 1 T

n

Q

n +Q

n 1 T

n ;

in(1.12),wean showthat

n =

T

n

1+T

n V

n

; and

n 1 =

V

n

1+T

n V

n

: (1.13)

Hene,dening:

!R 2

by

(x;y)=

y

; x

leadstothefat that

(

n 1 ;

n )=(T

n ;V

n

): (1.14)

Infat,(

)=,where isatrianglewithverties(0,0),(1,0),and(0,1). It

thenfollowsimmediatelythat

n 1 +

n

<1; n=1;2;:::;

andhene

min(

n 1 ;

n )<

1

2

; n=1;2;:::;

whih isawell-knownresultdue toVahlen[V ℄.

Usingthefatthat

(

n ;

n+1 )=(

T( 1

(

n 1 ;

n )));

JagerandKraaikamp[JK℄wereabletoshowthat

n+1 =

n 1 +a

n+1 p

1 4

n 1

n a

2

n+1

n :

Fromthisiteasilyfollowsthat

min(

n 1 ;

n ;

n+1 )<

1

q

a 2

n+1 +4

and

max(

n 1 ;

n ;

n+1 )>

1

q

a 2

n+1 +4

:

Theformer learlygeneralizes Borel's lassialresult, thelatterwasfound by

J.C.Tong[T1℄. Asaorollarywendthefollowingresult.

Theorem1.7 Forallirrational numbersx andall n0one has

min (

n 1 ;

n ;

n+1 )<

1

p

5 ;

the onstant1= p

5annotbereplaedbya smallerone.

The followingresult is another onsequeneof (1.14) together with

Theo-rem1.6.

Theorem 1.8 (Jager,1986) Thesequene(

n 1 ;

n

)aredistributedoverthe

triangle aording, for almostall x,tothe densityfuntion

f(a;b)= 1

log2 1

See[J℄fordetails.

Continuedfrationsplayanimportantrole inthetheoryofprime-testing(see,

e.g.,Bressoud'sbook[B℄). In1981,H. W. Lenstraonjeturedthat foralmost

allx

lim

n!1 1

n

#fj:1jn;

j

(x)zg; where 0z1; (1.15)

existsandequalsF(z),where

F(z)= 8

>

<

>

: z

log2

; 0z

1

2 ;

1

log2

(1 z+log2z); 1

2

z1:

Infat(1.15)hadbeenonjeturedin 1940byWolfgangDoeblin[Do℄. In1984

Knuth[Kn℄ showedthat

lim

N!1 1

N

#f1iN:

i

+dg= 1

log2 Z

+d

`(t)dt;

where

`(t)=min

1; 1

t 1

:

In1983 Bosma,Jager,andWiedijk[BJW℄provedtheLenstra-Doeblin

onje-tureusingNakada'snaturalextension(

;

B;;

T).

1.5 A brief desription of the thesis

Thisthesisonsistsofvepapersdealingwithontinuedfrations.

Chapter2 1

isonernedwiththeontinuedfrationwithoddpartialquotients

(OddCF).TherelationbetweenOddCFandRCFisdevelopedvia

singulariza-tionandinsertionproesses. UsingShweiger'snaturalextensionfortheOddCF

weshowthatthesequeneofonvergentsofthenearestintegerontinued

fra-tion (NICF) is a subsequene of that of OddCF. Using the method in [JK℄,

weobtainaresultfor OddCFapproximationoeÆientswhih oinides with

Tong'sresultforNICF[T2℄. ThroughtherelationbetweenRCFandgrotesque

ontinued fration (GCF) developed again via singularizations and insertions

we see that the sequene of GCF onvergents forms a subsequeneof that of

Hurwitz' singularontinued fration. Maximal and non-maximal OddCF are

alsodisussed.

InChapter 3 2

weonsiderthemapT

E

:[0;1)![0;1) givenby

T

E (x) :=

1

b 1

x

1

x b

1

x

;x6=0; T

E

(0) := 0:

Thismapyieldsa(unique)ontinuedfration expansionofx2[0;1)with

non-dereasingpartialquotientsoftheform

1

b

1 +

b

1

b

2 +

b

2

b

3 +

.

.

.

+ b

n 1

b

n +

.

.

. ; b

n

2N; withb

n b

n+1 :

WeallthisexpansionEngelontinuedfration(ECF)expansionofxsinethe

mapT

E

isamodiedversionoftheEngelseriesexpansionmap.

SomebasipropertiesofRCFalsoholdfortheECFbuttheydierinmany

ways. For instane, ECFonvergentsbehavedierentlyfrom regular ones. It

turns out that T

E

is ergodi with respet to Lebesgue measure but has no

niteinvariantmeasure,equivalenttoLebesgue. Moreover,itisshownthatthe

map T

E

hasinnitely many-nite, innite invariantmeasures,twoof whih

are given here. Additionally, we relatethe ECF to Ryde'smonotonen,

niht-abnehmendenKettenbruh (MNK) generated bythe mapT

R :(

1

2

;1) !( 1

2 ;1),

givenby

T

R

(x) = S

R (x) =

k

x

k; forx2R (k) :=

k

k+1 ;

k+1

k+2

; k2N;

throughan isomorphism. From this it follows, for example, that the map T

R

isergodiwithrespettoLebesguemeasurebutnoniteT

R

-invariantmeasure

equivalent to Lebesgue exists and that not every quadrati irrational has an

ultimatelyperiodiECFexpansion.

AHurwitz-typespetrumwasstudiedforthenearestintegerontinuedfration

byJagerandKraaikampin[JK℄. With(

n )

n1

denotingthesequeneofNICF

approximationoeÆientstheyshowedthat

min(

n 1 ;

n ;

n+1 )<

5

2 (5

p

5 11)=0:4508:

In[T2℄Tongextendedthisresultandprovedthat

min(

n 1 ;

n ;:::;

n+k )<

1

p

5 +

1

p

5 3

p

5

2 !

2k +3

:

2

Chapter 4 3

gives a proof of Tong's result using the method from [JK℄ whih

yields some metrial observations with respet to Tong's spetrum.

General-izationstoalargerlassof semi-regularontinuedfrationexpansionsarealso

derived.

Anumberx2RisalledHurwitzianifitsRCFexpansion(1.2)anbewritten

as

x = [a

0 ;a

1 ; ;a

n ;a

n+1

(k);;a

n+p (k)℄

1

k =0 ;

where a

n+1

(k);:::;a

n+p

(k) ( the so-alled quasi period of x) are polynomials

withrationaloeÆientswhih takepositiveintegralvaluesfork =0;1;2;:::,

andatleastoneof themisnotonstant. Thislearlygeneralizesperiodi

on-tinuedfrations. InChapter5 4

wedenetheHurwitziannumbersfortheNICF,

the`bakward'ontinuedfration expansion,and-expansions. Weshowthat

theset of Hurwitziannumbersfor suh ontinued frations oinides withthe

lassialsetofHurwitziannumbers.

Chapter6 5

isaontinuationoftheprevioushapter. Inthishapter wedene

Hurwitziannumbersfor Minskowski's diagonalontinued fration (DCF).We

alsoshowthat theset of DCF-Hurwitziannumbersoinides thelassial set

of Hurwitzian numbers. The situation is more ompliated here than in the

previouspaperdue tothedierenein shapeofthesingularizationareaofthe

NICF(andother-expansions)ononehand,andthatoftheDCFontheother.

3

ajointworkwithCorKraaikamp

4

ajointworkwithCorKraaikampinTokyoJ.Math.25(2002),no. 2

[B℄ Bressoud,D.M.{FatorizationandPrimalityTesting,Sringer-Verlag,New

York,(1989).MR91e:11150

[BJW℄ Bosma, W.,H. Jager andF. Wiedijk. -Some metrial observationson

theexpansionby ontinuedfrations,Indag.Math. 45(1983),353{379.

[DK℄ Dajani,K.andC.Kraaikamp.{ErgodiTheoryofNumbers,Carus

Math-ematialMonographs,No.29, (2002).

[Di℄ Dikson,L.E.{Historyof theTheoryofNumbers,VolsI,II,III,Carnegie

InstitutionofWashington,Washington, (1991-1923).

[Do℄ Doeblin,W. |Remarquesurlatheorie metrique desfrationsontinues,

CompositioMath.7(1940),353{371.MR2,107e

[HW℄ Hardy, G. H. and Wright, E. M. { An introdution to the theory of

numbers.Fifth edition.The ClarendonPress,OxfordUniversityPress,New

York,(1979).MR81i:10002

[IK℄ Iosifesu,M.andC.Kraaikamp.-TheMetrialTheoryofContinued

Fra-tions,KluwerAademiPress,Dordreht,TheNetherlands,(2002).

[J℄ Jager,H. -Continuedfration andergoditheory,TransendentalNumbers

andRelatedTopis,RIMSKokyuroku,599,KyotoUniversity,Kyoto,Japan,

(1986),55{59.

[JK℄ Jager,H. andC.Kraaikamp.|Onthe approximation byontinued

fra-tions,Indag.Math. 51(1989),no.2,289-307.MR90k:11084

[Kh℄ Khinthine,A.Ya.-Metrishe kettenbruhprablemen ,CompositioMath.

1(1935),361{382.

[Kn℄ Knuth, D. E. - The distribution of ontinued fration approximations, J.

NumberTheory19(1984),no3,443{448.MR86d:11058

[K1℄ Kraaikamp, C. - Metri and Arithmeti Results for Continued Fration

[K2℄ Kraaikamp,C.-Anewlassofontinuedfrationexpansions,AtaArith.

57(1991),no.1,1{39.MR92a:11090

[Ku℄ Kuzmin, R.O. - Ona problem of Gauss , Dokl. Akad. Nauk. SSSRSer.

A, (1928),375{380.

[N℄ Nakada,H.|Metrial theory foralassofontinuedfration

transforma-tionsand their natural extensions,TokyoJ.Math. 4(1981),no.2,399{426.

MR83k:10095

[NIT℄ Nakada, H., S. Ito and S. Tanaka| On the invariant measure for the

transformations assoiated with some real ontinued frations, Keio Engrg

Rep.,30 (1977),no.13,159{175.MR5816574

[O℄ Olds,C.D. {Continuedfrations, RandomHouse,NewYork,(1963).MR

26#3672

[P℄ Petersen, K. { Ergodi Theory, Cambridge University Press, Cambridge,

(1997).

[R-N℄ Ryll-Nardzewski,C.-Ontheergoditheorems.II.Ergoditheory of

on-tinuedfrations, StudiaMath.12 (1951),74{79.

[S℄ Shweiger,F.{Ergoditheory of bredsystemsandmetrinumber theory,

OxfordSienePubliations.TheClarendon Press,OxfordUniversityPress,

NewYork,(1995).MR97h:11083

[T1℄ Tong,JingCheng{The onjugate property of the Borel theoremon

Dio-phantineapproximation,Math.Z.184(1983),no.2,151{153.MR85m:11039

[T2℄ Tong,JingCheng|Approximation bynearestintegerontinuedfrations

(II),Math. Sand.74(1994),no.1,17{18.MR95:11085

[V℄ Vahlen, K. Th. | 

Uber Naherungswerte und Kettenbr uhe, Journalf. d.

reineundangew.Math. 115(1895),221{233.

[Wa℄ Walters, P. { An Introdution to Ergodi Theory, Springer-Verlag New

York,In.,NewYork,(2000).

[Wi℄ Wirsing,E.{Onthe theoremofGauss-Kuzmin-LeryandaFrobeniustype

Odd Continued Frations

2.1 Introdution

It is well-known that every x 2 [0;1) an be written as a nite (in ase x is

rational)or innite (when x is irrational) ontinued fration with oddpartial

quotients:

x =

e

1

a

1 +

e

2

a

2 +

.

.

.

+ e

n

a

n +

.

.

.

=: [0;e

1 =a

1 ;e

2 =a

2 ; ;e

n =a

n

;℄; (2.1)

wheree

1 =1;e

i

=1anda

i

isapositiveoddinteger,fori1,and

a

i +e

i+1

> 1; i1:

Weall (2.1)theodd ontinued fration(OddCF) expansionofx. Apartfrom

theOddCF-expansionofxonealsohastheso-alledgrotesqueontinuedfration

(or GCF) expansionof any x 2 [G 2;G), where G is the golden mean, i.e.,

G = 1

2 (

p

5+1). The GCF-expansion is also given by (2.1), again with odd

partialquotientsa

i ande

i

=1,butnowthese a

i ande

i

mustsatisfy

a

i +e

i

> 1; i1;

ande

1

=sgn(x).

There is an extendedliterature onboth theOddCF and theGCF. Intwo

(unpublished)papersF.Shweigerobtainedtheergoditheoremunderlyingthe

OddCFanditsnaturalextension where,asaby-result heshowedthatthe

GCFisthedualalgorithmoftheOddCF([S1℄,[S2℄),andstudied the

approxi-mationpropertiesoftheOddCF([S2℄). AroundthesametimeG.J.Riegeralso

obtainedaGauss-Kuzmintheorem for theOddCF andfound theergodi

wasgivenbyRiegerforbothexpansionsin[R1℄. IntworeentpapersG.I.Sebe

returnedtotheonvergenerateintheGauss-KuzminproblemfortheOddCF

([Se1℄)andGCF([Se2℄)usingthetheoryofrandomsystemswithomplete

on-netions. Sebe alsoobtained thenaturalextension for theGCF. More results

ontheOddCFand theGCFanbefound in papersbyS. Kalpazidou([Ka1℄,

[Ka2℄)andD. Barbolosi ([B1℄,[B2℄).

Atrstsightonemightbetemptedtosaythatnothinganbesaidanymore

aboutthese expansions! In[B2℄, Barbolosi showed that forany x 2[0;1) the

sequene of nearestinteger ontinued fration (NICF) onvergentsof x forms

a subsequene of the sequene of OddCF-onvergents of x. In order to

un-derstandthisresultwewereledtoanewlassofontinuedfrationexpansions

withoddpartialquotients,ofwhihtheOddCFandtheGCFaretwoexamples.

In general, a semi-regular ontinued fration (SRCF) is anite orinnite

fration

b

0 +

e

1

b

1 +

e

2

b

2 +

.

.

.

+ e

n

b

n +

.

.

. = [b

0 ;e

1 =b

1 ;e

2 =b

2 ;;e

n =b

n

;℄; (2.2)

withe

n

=1;b

0 2Z;b

n

2N, forn1,subjetto theondition

e

n+1 +b

n

1; forn1; (2.3)

andwith therestritionthatin theinnitease

e

n+1 +b

n

2; innitelyoften. (2.4)

Anitetrunationin (2.2)yieldstheSRCF-onvergents

A

n =B

n := [b

0 ;e

1 =b

1 ;e

2 =b

2 ;; e

n =b

n ℄;

where it is always assumed that gd(A

n ;B

n

) = 1. We say that (2.2) is an

SRCF-expansionofanirrationalnumberx inase

x = lim

n!1 A

n

B

n :

ClearlytheOddCFisanexampleofanSRCF-expansion,but theGCFisnot.

OtherexamplesofSRCF-expansionsarethenearest integerontinuedfration

(NICF)expansion,satisfying

e

n+1 +b

n

2 for n1;

andHurwitz' singularontinuedfration(HSCF) expansion,whihsatises

Perhapsthebest-knownexampleofanSRCF-expansionistheso-alledregular

ontinuedfrationexpansion(RCF);everyrealirrationalnumberxhasaunique

RCF-expansion

d

0 +

1

d

1 +

1

d

2 +

.

.

. =: [d

0 ;d

1 ;d

2

;℄; (2.5)

whered

0

2Zissuhthat x d

0

2[0;1),andd

n

2N forn2N.

ObviouslytheGCFis notanSRCF,but aso-alledunitary expansion,see

also[G℄.Unitaryexpansionsaredened inawaysimilartoSRCF-expansions,

thedierenebeingthat(2.3)and(2.4)arereplaedby

e

n +b

n

1; forn1;

andwiththerestritionthat intheinnitease

e

n +b

n

2; innitelyoften.

Essentialin ourinvestigationsare thenotions ofinsertion and singularization

ofapartialquotientequalto1,whih werestudiedin detailin [K℄.

Asingularizationisbasedupontheidentity

A + e

1+ 1

B+

= A+e+ e

B+1+ ;

where2[0;1).

Toseetheeetofasingularization,let(2.2)beanSRCF-expansionofx. A

nitetrunation yieldsthesequeneofonvergents(r

k =s

k )

k 1

:Supposethat

forsomen0onehas

b

n+1 =1;e

n+2 =1;

i.e.,

[b

0 ;e

1 =b

1

;℄ = [b

0 ;e

1 =b

1 ;;e

n =b

n ;e

n+1 =1; 1=b

n+2

;℄: (2.6)

The transformation

n

whih hanges this ontinued fration (2.6) into the

ontinuedfration

[b

0 ; e

1 =b

1 ;;e

n =(b

n +e

n+1 ); e

n+1 =(b

n+2

+1);℄; (2.7)

whih is again a ontinued fration expansion of x, with onvergents, say

(p

k =q

k )

k 1

;isalled asingularization. It wasshownin [K℄thatthesequene

ofvetors

p

k

q

k

k 1

isobtainedfrom

r

k

s

k

k 1

by removingtheterm

r

n

s

An operationwhihisin somesense the`opposite'ofasingularizationis a

so-alledinsertion. Aninsertioniseitherbasedupontheidentity

A +

orontheidentity

A+

Let(2.2)beanSRCF-expansionofx,andsupposethatforsomen0one

has

Ansrf-insertionisthetransformation

n

whihhanges(2.2)into

[b

whihisagainanSRCF-expansionofx,withonvergents,say,(p

k

bethesequene ofonvergentsonnetedwith(2.2). Using some

matrix-identities itwasshownin [K℄thatthesequeneof vetors

obtainedfrom

by inserting the term

before the n-th

termofthelattersequene,i.e.,

Ansrf-insertionis denotedby 1=1.

Nowlet(2.2)beaunitary-expansionofx withthesequeneofonvergents

(r

Applyingtheseondinsertion-identityhanges(2.2)into

[b

whih is again a unitary-expansion of x, with onvergents, say, (p

ofvetors

p

k

q

k

k 1

of thenewexpansionis obtainedfrom

r

k

s

k

k 1 by

in-serting theterm

r

n r

n 1

s

n s

n 1

before the n-th term ofthe latter sequene. A

unitary-insertionisdenoted by1=1 .

Byombining the operations of singularizationand srf/unitary-insertion one

anobtainanysemi-regular/unitaryontinuedfration expansionofanumber

xfrom itsRCFexpansion. In[K℄awhole lassofsemi-regularontinued

fra-tionswasintroduedviasingularizationsonly(someoftheseSRCF'swerenew,

somelassial liketheontinuedfration tothenearestinteger,orHurwitz'

singularontinuedfration(HSCF)),andtheirergoditheorystudied(themain

idea in [K℄ is that the operation of singularizationis equivalent to having an

induedmaponthenaturalextension oftheRCF).

In the next setion we will show that the OddCF-expansion anbe obtained

from the RCF via suitable srf-insertions and singularizations. We also will

derive somemetrial results forthe approximationoeÆients of theOddCF.

In Setion 2.3 we will see that the GCF an be obtained from the RCF via

singularizationsandunitary-insertions.ThiswillleadusinSetion2.4toanew

lass of semi-regular/unitary ontinued fration expansions with odd partial

quotients.

2.2 Insertions, singularizations and the OddCF

2.2.1 A singularization/insertion algorithm

Thefollowingtheorem desribesanalgorithmwhihturns theRCF-expansion

of any x 2 [0;1) into the OddCF-expansion of x. The proof of this theorem

followseasilybyinspetion,andisthereforeomitted.

Theorem2.1 Let x 2 [0;1) with RCF-expansion (2.5), i.e., d

0

= 0. Then

starting from the RCF-expansion (2.5)of x, the following algorithm yields the

OddCF-expansionof x.

(I) Letm:=inffn2N; d

n

iseveng.

(i)If d

m+1

>1, insert 1=1after d

m

toobtain

[0;1=d1;;1=dm

1

;1=(dm+1); 1=1;1=(dm+1 1);1=dm+2;℄:

(ii) If d

m+1

=1, letk:=inffn>m;d

n

>1g(k =1 isallowed). Now

singularize inthe blokofpartialquotients

the rst,third, fth,et. partialquotientsequal to1,toarrive at

[0;1=d1;;1=dm 1;1=(dm+1); 1=3;; 1=3

| {z }

k m 2

2

times

; 1=(dk+1);1=dk +1;℄;

in asek m 1isodd ork=1; in thelatterasewe nd

[0;1=d

1

;;1=d

m 1 ;1=(d

m

+1); 1=3;; 1=3;℄:

Inasek m 1iseven weobtain

[0;1=d

1

;;1=d

m 1 ;1=(d

m

+1); 1=3;; 1=3

| {z }

k m 3

2

times

; 1=2;1=d

k ;1=d

k +1 ;℄:

In thisaseinsert 1=1toarrive at

[0;1=d1;;1=(dm+1); 1=3;; 1=3

| {z }

k m 1

2

times

; 1=1;1=(dk 1);1=dk +1;℄;

(II) Letm1betherstindexinthenewSRCF-expansion[

0 ;e

1 =

1 ;℄of

x obtained in (I) for whih

m

is even. Repeat the proedure from(I) to

this newSRCF-expansionofx withthis value ofm.

Assoonasm=1in(II)we haveobtainedthe OddCF-expansionof x.

ThefollowingexampleillustratehowtouseTheorem2.1

Example2.1 Letx2[0;1)haveRCF-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) Applythealgorithmwithm=3. Sined

4

>1,weinsert 1=1after 1=4

toobtain

[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) Applythealgorithmwithm=5inthenewexpansion. Sined

6

==

d

11

=1,wesingularized

6 , d

8 ,d

10 andd

11

toarriveat

[0;1=3;1=1;1=5; 1=1;1=7; 1=3; 1=3; 1=3; 1=2;1=5;℄:

Nowinsert 1=1toarriveat

[0;1=3;1=1;1=5; 1=1;1=7; 1=3; 1=3; 1=3; 1=3; 1=1;1=4;℄:

2.2.2 Metrial properties of the OddCF

In [S2℄ (and impliitlyin [S1℄), Shweiger introdued and studied the natural

extension of the ergodi system underlying the OddCF. See also [Se2℄, where

SebeobtainedthenaturalextensionoftheGCF.

Setting

B(+;k) =

1

2k ;

1

2k 1 i

; k=1;2;:::;

B( ;k) =

1

2k 1 ;

1

2k 2 i

; k=2;3;:::;

themap

T(x) = e

1

x

(2k 1)

; x2B(e;k);e=1;

generatesthe OddCF-expansion(2.1) ofx. Notie that e

1

=1andthat e

n =

e

n (x);a

n =a

n

(x)aregivenby

(e

n+1 ;a

n

) = (e;2k 1) , T n 1

(x)2B(e;k); forn1:

Thedual-algorithmT

ofT isgivenby

T

(x) = (x)

x

(2k 1); (x) = sgn(x);

onanappropriate partition of[G 2;G℄. Thisdual-algorithmis themap

un-derlyingtheGCF. Setting

= [0;1)[G 2;G℄

anddeningT :!by

T(x;y) =

T(x); e

a+y

;

wheree=1anda=2k 1aresuh thatx2B(e;k),Shweigershowedthat

(;B;;T)

forms anergodi system. Here B is theolletionof Borelsets of , and is

aprobabilitymeasure on(;B), withdensity (3logG) 1

(1+xy) 2

on, see

alsoFigure2.1.

In[K℄itwasshownthatthenearestintegerontinuedfration(NICF)expansion

ofanyx2[0;1)anbeobtainedfromtheRCF-expansionofxbyapplyingthe

rststepin(I)(ii)ofTheorem2.1toanyblokofregularpartialquotientsequal

to1,whihispreededandfollowedbyaregularpartialquotientdierentfrom

1(thisrestritiondoesnotapply iftheexpansionofxstartswith1's,orwhen

theblokof1'sisinnite). ButthenBarbolosi'sresultfrom[B2℄,whihstates

thatthesequeneofNICF-onvergentsofxformsasubsequeneofthesequene

g 2

1

2

1 0

g 1 G

Legends:

=NICF-singularizationarea

=insertionarea

Figure2.1: Shweiger'sNaturalExtension(;B;;T)

Theorem2.2 Let x 2 [0;1) be an irrational number with OddCF-expansion

(2.1),with RCF-expansion(2.5),andlet

x = [b

0 ; f

1 =b

1 ;℄

bethe NICF-expansionof x. Say

(p

n =q

n )

n 1 ; (P

n =Q

n )

n 1

; and (A

n =B

n )

n 1

arethe sequenes ofthe OddCF, RCFresp. the NICF-onvergentsof x.

(i) Then thereexistsanarithmetial funtion k=k(x):N !N, suhthat

A

n

B

n =

p

k (n)

q

k (n)

; n1;

andonehas for almostall 1

xthat

lim

n!1 k(n)

n =

3

2 :

(ii) In the OddCF-expansion (2.1) of x, singularize every digit a

i

= 1 for

whih e

i

= 1 (i.e., remove all the inserted mediant onvergents). In

doingsowe ndan SRCF-expansion

x = [0;"

1 =u

1 ;"

2 =u

2 ;℄

of x, with onvergents, say, C

n =D

n

,n 1. Then (C

n =D

n )

n 1 forms

a subsequene of (P

n =Q

n )

n 1

and has (A

n =B

n )

n 1

as asubsequene.

Thereexistarithmetial funtions`;`

:N!N suhthat

C

n

D

n =

P

`(n)

Q

`(n) ;

A

n

B

n =

C

`

(n)

D

`

(n)

; n1;

andonehas for almost allx that

lim

n!1 `(n)

n =

log4

log2G ; lim

n!1 `

(n)

n =

log2G

2logG :

Remark2.1 Let x 2 [0;1) be an irrational number with RCF-onvergents

(P

n =Q

n )

n 1

andNICF-onvergents(A

n =B

n )

n 1

. Sine(A

n =B

n )

n 1 forms

asubsequeneof(P

n =Q

n )

n 1

,thereexistsanarithmetialfuntionw=w(x):

N !N suhthat

A

n

B

n =

P

w(n)

Q

w(n)

; n1:

In[A℄,W.W.Adamsshowedthat

lim

n!1 w(n)

n =

log2

logG

=1:4404 a:e:;

see also [J℄ and [K℄. In words, this result states that for almost all x about

30.58%of the regularonvergents ofx were removed(in an appropriateway)

fromtheRCF-expansionofx toobtaintheNICF-expansionofx.

Proof.

(i)Let S

nif and S

ins

betheNICF-singularization and insertionareas,

respe-tively;seeFigure2.1. Let

S=S

nif [S

ins :

Thenasimplealulationyieldsthat

(S)= 1

3 :

The result follows from taking S asthe singularization area and using

Theo-rem(4.13)in[K℄.

(ii)Consider=nS

ins

andS: ! denedby

S(x;y)= 8

<

:

T(x;y); T(x;y)2=S

ins ;

T 2

(x;y); T(x;y)2S

ins :

ItfollowsfromergodiityofT that

where

(E)=

1

(1 (S

ins

))3logG Z

E

dxdy

(1+xy) 2

;

formsanergodisystem. TakingS

nif

asthesingularizationareainthissystem,

togetherwith

(S

nif )=

log2g

log2G

=0:18046;

oneobtains,see Theorem(4.13)in[K℄,

lim

n !1 `

(n)

n =

1

1

(S

nif )

= log2G

2logG

=1:2202:

Byonstrution(C

n =D

n )

n 1

formsasubsequeneofthesequeneof

RCF-onvergents. Thusthereexistarithmetialfuntions `;`

:N !N suhthat

C

m

D

m =

P

`(m)

Q

`(m) ;

A

m

B

m =

C

`

(m)

D

`

(m)

; form2N:

FromRemark2.1wehave

w(n)

n =

`(`

(n))

`

(n) `

(n)

n ;

whihgives

lim

n !1 `(`

(n))

`

(n) =

log4

log2G

=1:1804

foralmostallx. Itnowremainstoshowthat`(m)=monvergestothesamelimit

log4=log2Gasm !1,wherem2N isfromthesetofindiesofonvergents

(C

n =D

n

)whihwereremovedtoobtain(A

n =B

n )

n 1

. Tothisend, letm2N

besuh thatforalln2N

m6=`

(n):

Butthenthereexists ann2N suhthat

m 1=`

(n) and m+1=`

(n+1) (2.8)

sineotherwisetwoonseutiveonvergentsin(C

n =D

n )

n 1

(orrespondingto

insertions) would havebeen singularized (whih is impossible, sine we never

havetwoonseutiveinsertions). Due to thefat that twoonseutive

NICF-onvergents annot be toofar apart in the sequene of RCF-onvergents, we

musthavethat

`(m 1)=`(m) 1 and `(m+1)=`(m)+1:

Hene,wehave

`(m 1)

m 1

m 1

m =

`(m) 1

m <

`(m)

m <

`(m)+1

m =

`(m+1)

m+1

m+1

m

Remark2.2 1. LetS bedened asintheproofofTheorem2.2(i). Thendue

toAbramov's formula(see[P℄,pp.257-258)onehas

h(S) = h(T)

(S) ;

It is well-known, see e.g. [N℄, that the entropy h(T

nif

) of the NICF equals

2

=6logG,andsine(nS; 1

(S)

;S)forms anergodisystemwhihis

metri-allyisomorphitotheergodisystemunderlying theNICF, wendthat

h(S) =

2

6logG

= 3:418;

2. InTheorem2.2(ii)wesawthat about15.29%oftheRCF-onvergentswere

removed in the algorithm from Theorem 2.1 to get the OddCF-expansion of

x, and that 18.64% of the OddCF-onvergentswere obtained by appropriate

insertions.

Taking S

ins

as the singularization area, we immediately see that

(C

n =D

n )

n 1

forms a subsequene of (p

n =q

n )

n 1

and hene there exists an

arithmetialfuntionw

:N !N suhthat

C

n

D

n =

p

w

(n)

q

w

(n)

; n1:

Itfollowsfromasimplealulationthat

(S

ins )=

log(G 2

=2)

3logG ;

andhene

lim

n!1 w

(n)

n =

3logG

log2G

=1:2292 a:e:

2.2.3 Approximation oeÆients

Approximation oeÆients for the OddCF-expansion of an irrational number

x2[0;1)aredened by

n :=q

2

n

x

p

n

q

n

; n0: (2.9)

Theapproximation oeÆients(

n (x))

n0

of anirrationalnumber x indiate

the quality of approximationof x by its (rational) onvergents. For instane,

fortheRCFtheapproximationoeÆients(

n (x))

n0

whiharedened by

n :=Q

2

n

x

P

n

Q

n

; n0;

where(P

N =Q

n )

n0

(Vahlen, 1895)min(

n 1 ;

n

)<1=2forallirrationalxandn1.

(E.Borel,1903)min(

n 1 ;

n ;

n+1 )<1=

p

5forallirrationalxandn1.

(Legendre) Suppose that p and q > 0 are integers whih are relatively

prime,suhthatq 2

jx p=qj 1=2:Thenp=qisanRCFonvergentofx.

Forproofsofthese results,seee.g.,[S ℄.

Inthissetionwewillobtaina`Borel-type'resultfortheOddCF.Itanbe

shown(see,e.g.,[S2℄)that

n =

1

T n

x +

e

n+1 q

n 1

q

n

1

=

e

n+2 T

n+1

x+ q

n+1

q

n

1

: (2.10)

Considerthefuntion : !R 2

denedby

(t

n ;v

n ):=

jv

n j

1+t

n v

n ;

t

n

1+t

n v

n

;

wheret

n =T

n

(x)andv

n =e

n+1 q

n 1 =q

n

. Itfollowsfrom (2.10)that

(t

n ;v

n )=(

n 1 ;

n ):

Hene,(

n 1 ;

n )

n1

isdistributedoverthetworegions inasee

n+1

= 1

and +

inasee

n+1

=+1,where isaquadranglewithverties(0,0),(g 2

;0),

(g;G), and (0,1)and +

aquadrangle with verties(0,0), (G;0), (g;g 2

), and

(0,1);see Figure2.2.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(i) : from Shweiger'snatural (ii) +

: fromShweiger'snatural

extensionwithy<0 extensionwithy>0

Figure 2.2:

OddCF

Moreover,both

with2f+; garedividedintoregionsaordingtothe

valueofthepartialquotienta

n

. Wedenotetheseregionsby

in asea

Notealsothat fora3,eah

a

isdividedintotworegionsaordingtoe

n+2

bythelines

= e

whih are thedotted lines in Figure 2.2. We denote theseregions by

aordingly.

Theinverseof isgivenby

1

Nowonsiderthe operatorF :

OddCF

Itfollowsfromthedenitionof F that

F(;)=

)wethereforeseethat

Toinvestigatethepointsin

OddCF

Clearly,for(

n 1

onehasthat

min(

Itfollowsfrom(2.11)witha

n =1,e

n+1

=+1ande

n+2

=+1thatthefuntion

bywhihthenextisalulatedon +

1

isgivenby

h(;)=+ p

1 4 :

ThisfuntionattainsitsmaximumvalueonD +

at(1= p

5;1= p

5),themaximum

Nowonsider points(

liesonthe boundary

of

regionsD

U

thenextisgivenbythefuntion

h(;) = + p

1+4 ;

whih attainsits maximumon theboundary ofD

U

at thepoint (1= p

5;(10+

p

5)=20),themaximumbeing

1

fromwhih itatonefollowsthatfor(

n 1 ;

n )2D

U

onehasthat

min(

ThusweonlyneedtofousourattentiontoD

D

=D \

3; 1

. Thisisatriangle

withvertiesA:(1=

(2.11)weseethatthenextonthisregionisgivenbythefuntion

h(;) = 3 p

1+4+9;

whihattainsitsmaximumvalueontheboundaryofD

D

. Itanbehekedthat

thismaximumvalueisGandisattainedat(2g 3

wehave

min(

Notie that 2g 3

is only slightly larger than the lassialvalue 1= p

5. Taking

everythingtogetherwehavefounda`Borel-typetheorem'.

However,there is moreoneansay! InD

D

we annd anestedsequene

oftrianglesD

k

whereD

k

hasvertiesF k

(C);seeFigure2.3.

Here

whereF

Figure 2.3: D

D

Rening theargumentfromabove, wesee that foreahk1,the next

onD

k

isgivenbythefuntionh(;) = 3 p

1+4+9,whihattainsits

maximumvalueon theboundary ofD

k

; itsmaximal valuebeingF (k 1)

(C),

attainedatF k

),itfollowsthat

min(

Wehavethefollowingresult.

Theorem2.3 Let

n

begiven by (2.9). Then

min(

whereC

k

UsingtheexpliitformulaforFibonainumbersandanalysisofdierenes(see

setion2.5in[SP℄fordetails),weobtain

andtheresultfollowsfrom(2.12). 2

Remark2.3 This result oinideswith Tong's resultfor NICF, see [T℄. This

isdue to thefat that thepart ofD whih liesbelowthe line = 4

25 +

2

5

is exatlythe triangle D in [JK℄and Tong's result is a generalizationon this

triangle. Thefollowingorollaryfollowsdiretlyfrom Theorem2.3.

Corollary2.1 (Shweiger,1984)Foranyirrational x the inequality

n (x) <

1

p

5

isvalidfor innitely many n

Inthenextsetionwewillseethatthegrotesqueontinuedfrationsanbe

obtainedfrom theRCFviasingularizationsand unitary-insertionsand similar

resultswillfollow;seeRemark2.4.

2.3 Grotesque ontinued frations

ItisobviousthattheGCFanneverbeobtainedviaanalgorithmsinwhihan

srf-insertionisapplied; onewill alwayshave 1=1somewhere,whih violates

oneoftherulesoftheGCF.

ThefollowingtheoremgivesanalgorithmwhihturnstheRCF-expansionof

x2[0;1)into theGCF-expansionofx. Theproof ofthistheorem alsofollows

easilybyinspetion, andisthereforeomitted.

Theorem2.4 Let x 2 [0;1) with RCF-expansion (2.5), i.e., d

0

= 0. Then

startingfrom the RCF-expansion(2.5) of x, the following algorithm yields the

GCF-expansion ofx.

(I) Let m:=inffn2N; d

n

iseven g.

(i)If d

m+1

>1,insert 1=1 after d

m

toobtain

[0;1=d1;;1=dm 1;1=(dm 1);1=1; 1=(dm+1+1);1=dm+2;℄:

(ii)If d

m+1

=1, let k:=inffn>m;d

n

>1g (k =1 is allowed). Now

singularizein the blokof partialquotients

d

m+1 =1;d

m+2

=1;:::;d

k 1 =1

the last,seondfromlast,et. partialquotientsequal to1,toarrive at

[0;1=d1;;1=dm 1;1=(dm+1); 1=3;; 1=3

| {z }

k m 2

times

inasek m 1isoddor k=1; inthe latterasewend

[0;1=d

1

; ;1=d

m 1 ;1=(d

m

+1); 1=3;; 1=3;℄:

In asek m 1isevenweobtain

[0;1=d

1

;;1=d

m 1 ;1=d

m

;1=2; 1=3;; 1=3

| {z }

k m 3

2

times

; 1=(d

k

+1);1=d

k +1 ;℄:

Inthis aseinsert1=1 after1=d

m

toarrive at

[0;1=d

1

;;1=(d

m

1);1=1; 1=3;; 1=3

| {z }

k m 1

2

times

; 1=(d

k

+1);1=d

k +1 ;℄:

(II) Letm1be the rstindex in the new unitary-expansion[

0 ;e

1 =

1 ;℄

of x obtained in (I) for whih

m

iseven. Repeat the proedurefrom (I)

tothis newunitary-expansionofx with thisvalue ofm.

Assoon asm=1in(II) wehaveobtainedthe GCF-expansionof x.

Remark2.4 It was shown in [K℄ that Hurwitz' singular ontinued fration

(HSCF) expansionof x is obtainedfrom the RCF-expansion of x byapplying

this`new'step(I)(ii)fromTheorem2.4toanyblokofregularpartialquotients

equalto1,whihispreededandfollowedbyaregularpartialquotientdierent

from 1(againthis restrition doesnotapply ifthe expansionof x startswith

1's,orwhentheblokof1'sisinnite). Thusweseethatthesequeneof

HSCF-onvergentsofxformsasubsequeneofthesequeneofGCF-onvergentsofx.

Duetothis,Theorem2.2hasanobviousanaloguefortheGCF,whihweomit

here.

Example2.2 Applyingthe`new'step(I)(ii)fromTheorem2.4toxin

Exam-ple2.1,weobtain

[0;1=3;1=1;1=3;1=1; 1=7;1=1; 1=3; 1=3; 1=3; 1=3; 1=6;℄:

Notethatwehadinserted1=1 after1=4.

Instead of inserting 1=1, we ould also insert 1=1 `at the appropriate

plae'in anyalgorithmin theprevioussubsetion. Thisleadstoanotherlass

2.4 Other odd ontinued frations

2.4.1 Maximal OddCF's

NotiethatinTheorem2.2(I)(ii),b k m

2

1'saresingularized,whihisadiret

onsequeneofthefatthattheNICFisamaximalS-expansion,seealsoSetion

4in[K℄.Inasek m 1isoddoneisforedtosingularizetherst,third,et.

1tosingularizethemaximalnumberof 1'sintheblok.

However,inase k m 1iseventhereis onsiderablefreedomto hoose

the k m 1

2

1's whih should be singularized (in order to singularize asmany

1'saspossible),andoneouldalsodothefollowing: singularizeintheblokof

partialquotients

d

m+1 =1;d

m+2

=1;:::;d

k 1 =1

k m 1

2

1'swhih arenotonseutive(twoonseutiveregularpartialquotients

equalto 1an neverbesingularizedsimultaneously), andthen insert 1=1`at

theappropriateplae'. Forinstane,singularizingtheseond,fourth,sixth,et.

partialquotientsequalto1yields

[0;1=d

1

;;1=d

m 1 ;1=d

m

;1=2; 1=3;; 1=3

| {z }

k m 3

2

times

; 1=(d

k

+1);1=d

k +1 ;℄:

Inthisaseweneedtoinsert 1=1`inbetween'thetwoevenpartialquotients

d

m andd

m+1

+1=2atthebeginningoftheblokwejustobtainedvia

singu-larizations,toarriveat

[0;1=d

1

; ;1=(d

m

+1); 1=1;1=1; 1=3;; 1=3

| {z }

k m 3

2

times

; 1=(d

k

+1);1=d

k +1 ;℄:

Ifwe applythis new`algorithm'to the RCF-expansion ofx wegetfor almost

every 2

xaontinuedfration expansionofx withoddpartialquotients,whih

is dierent from both the OddCF- and the GCF-expansions of x. In general

suhanoddexpansionisalledamaximalexpansionwithodd partialquotients

ofx,sineamaximalnumberofpossiblesingularizationsisusedtoobtainthis

expansionfromtheRCF-expansionof x.

Example2.3 Applyingthis`new'algorithmtox inExample2.1, weobtain

[0;1=3;1=1;1=5; 1=1;1=7; 1=1;1=1; 1=3; 1=3; 1=3; 1=6;℄:

Notethatweinserted 1=1after1=4in Example2.1(i).

Maximal ontinued frations with odd partial quotients are only `loally

dif-ferent'from one-another. Insertionsin (I)(i) in Theorem2.1 will alwaysbe at

the same plae in the RCF-expansion of x if one wants to obtain an

SRCF-expansionofx with oddpartial quotients,andbloksof1'sof oddlengthan

besingularizedonlyinonewayifone wantsto singularizeasmany1'sas

pos-sible. Dierenesan onlyourwhenanevenblokof RCFpartialquotients

equalto 1 must besingularized in (I)(ii). As we sawin the example, in that

aseonealwaysneedsanextrainsertion.

2.4.2 Non-maximal expansions with odd digits

Intheprevioussub-setionwesawthatthereexisttwolassesofmaximal

on-tinuedfrationexpansionswithodddigits. Onelasswasobtainedbyinserting

1=1andtheotheronebyinserting1=1 ,andinbothlassesonesingularized

themaximalnumberof1'spossible. (Athirdlassanbeobtainedby`mixing'

bothtypesofinsertions.) However,it isnotneessarytosingularize the

maxi-mumamountof1'spossible,asthefollowingexampleshows.

Example2.4 Let x 2 [0;1) be as in Example 2.1. We must insert either

1=1 or 1=1 after d

3

= 4. In the thus obtained expansion of x, the fth

partial quotient is even, followed by eight partial quotients equal to 1. Now

singularizingtherst,fourthandseventh1yields

[0;1=3;1=1;1=5; 1=1;1=7; 1=2;1=2; 1=2;1=2; 1=2;1=5;℄;

inaseweinserted 1=1afterd

3 =4,or

[0;1=3;1=1;1=3;1=1; 1=9; 1=2;1=2; 1=2;1=2; 1=2;1=5;℄;

inaseweinserted1=1 afterd

3

=4. Nowinserting 1=1resp. 1=1 yields

[0;1=3;1=1;1=5; 1=1;1=7; 1=3; 1=1;1=1; 1=3; 1=1;1=1; 1=3; 1=1;1=4;℄;

resp.

[0;1=3;1=1;1=5; 1=1;1=9; 1=1;1=1; 1=3; 1=1;1=1; 1=3; 1=1;1=1; 1=6;℄:

Insteadofsingularizingthemaximumnumberof1's,inExample2.4we

singu-larizedtheminimumnumberof1'sintheblok. Againthisleadstotwolasses

ofminimalexpansionswithoddpartialquotientsofx,dependingonwhih

inser-tionused. Notie,thatifthenumberof1'sintheblok`thatneedstodisappear'

equals3`+i,withi=0;1;2,theminimalnumberofsingularizationsneededis