Approximation By Algebraic Numbers

292 Pages·2004·1.43 MB·english

by  YANN BUGEAUD

This page intentionally left blank
CAMBRIDGETRACTSINMATHEMATICS
GeneralEditors
B.BOLLOBA´S.W.FULTON,A.KATOK,F.KIRWAN,P.SARNAK,B.SIMON
160 Approximation by Algebraic Numbers

Approximation by Algebraic Numbers
YANN BUGEAUD
Universite´ LouisPasteur,Strasbourg
CAMBRIDGEUNIVERSITYPRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo
Cambridge University Press
TheEdinburghBuilding,CambridgeCB22RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationo nthistitle:www.cambridge.org/9780521823296
© Cambridge University Press 2004
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof
relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace
withoutthewrittenpermissionofCambridgeUniversityPress.
Firstpublishedinprintformat 2004
ISBN-13 978-0-511-26591-4 eBook (NetLibrary)
ISBN-10 0-511-26591-3 eBook (NetLibrary)
ISBN-13 978-0-521-82329-6 hardback
ISBN-10 0-521-82329-3 hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls
forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
Contents
Preface pageix
Frequentlyusednotation xiv
1 Approximationbyrationalnumbers 1
1.1 DirichletandLiouville 1
1.2 Continuedfractions 5
1.3 ThetheoremofKhintchine 12
1.4 TheDuffin–SchaefferConjecture 18
1.5 Complementaryresultsoncontinuedfractions 19
1.6 Exercises 21
1.7 Notes 23
2 Approximationtoalgebraicnumbers 27
2.1 Rationalapproximation 27
2.2 Effectiverationalapproximation 29
2.3 Approximationbyalgebraicnumbers 31
2.4 Effectiveapproximationbyalgebraicnumbers 33
2.5 Remarksonirrationalityandtranscendencestatements 39
2.6 Notes 40
3 TheclassificationsofMahlerandKoksma 41
3.1 Mahler’sclassification 42
3.2 SomepropertiesofMahler’sclassification 45
3.3 Koksma’sclassification 47
3.4 Comparisonbetweenbothclassifications 52
3.5 Someexamples 61
3.6 ExponentsofDiophantineapproximation 62
v
vi Contents
3.7 Exercises 68
3.8 Notes 70
4 Mahler’sConjectureonS-numbers 74
4.1 Statementsofthetheorems 75
4.2 Anauxiliaryresult 78
4.3 ProofofTheorem4.3 80
4.4 Exercise 87
4.5 Notes 88
5 Hausdorffdimensionofexceptionalsets 90
5.1 HausdorffmeasureandHausdorffdimension 90
5.2 UpperboundfortheHausdorffdimension 93
5.3 Themassdistributionprinciple 95
5.4 Regularsystems 98
5.5 ThetheoremofJarn´ık–Besicovitch 103
∗
5.6 HausdorffdimensionofsetsofS -numbers 105
5.7 HausdorffdimensionofsetsofS-numbers 110
5.8 RestrictedDiophantineapproximation 113
5.9 Exercises 114
5.10 Notes 117
6 Deeperresultsonthemeasureofexceptionalsets 122
6.1 Optimalregularsystems 123
6.2 AKhintchine-typeresult 125
6.3 Hausdorffdimensionofexceptionalsets 129
6.4 Hausdorffmeasureofexceptionalsets 130
6.5 Setswithlargeintersectionproperties 130
6.6 Applicationtotheapproximationbyalgebraicnumbers 131
6.7 Exercises 136
6.8 Notes 137
7 OnT-numbersandU-numbers 139
7.1 T-numbersdoexist 140
7.2 Theinductiveconstruction 141
7.3 CompletionoftheproofofTheorem7.1 149
7.4 Onthegapbetweenw∗andw 151
n n
7.5 HausdorffdimensionandHausdorffmeasure 152
7.6 OnU-numbers 153
7.7 AmethodofGu¨ting 159
Contents vii
7.8 BriefsummaryoftheresultstowardstheMainProblem 161
7.9 Exercises 162
7.10 Notes 163
8 Otherclassificationsofrealandcomplexnumbers 166
8.1 Sprindzˇuk’sclassification 166
8.2 AnotherclassificationproposedbyMahler 171
8.3 Transcendencemeasuresandmeasuresofalgebraic
approximation 180
8.4 Exercises 184
8.5 Notes 188
9 Approximationinotherfields 191
9.1 Approximationinthefieldofcomplexnumbers 191
9.2 ApproximationinthefieldofGaussianintegers 193
9.3 Approximationinthe p-adicfields 194
9.4 Approximationinfieldsofformalpowerseries 199
9.5 Notes 201
10 Conjecturesandopenquestions 204
10.1 TheLittlewoodConjecture 204
10.2 Openquestions 206
10.3 Notes 217
AppendixA Lemmasonpolynomials 219
A.1 Definitionsandusefullemmas 219
A.2 Liouville’sinequality 222
A.3 Zerosofpolynomials 227
A.4 Exercises 233
A.5 Notes 233
AppendixB Geometryofnumbers 235
References 240
Index 273