CAMBRIDGE TRACTS IN MATHEMATICS General Editors
B. BOLLOBAS, F. KIRWAN, P. SARNAK, C.T.C. WALL
137
Metric Diophantin...

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CAMBRIDGE TRACTS IN MATHEMATICS General Editors

B. BOLLOBAS, F. KIRWAN, P. SARNAK, C.T.C. WALL

137

Metric Diophantine Approximation on Manifolds

V. I. Bernik Byelorussian Academy of Sciences

M. M. Dodson University of York

Metric Diophantine Approximation on Manifolds

i CAMBRIDGE 1

UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc6n 13, 28014 Madrid, Spain © Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 . Printed in the United Kingdom at the University Press, Cambridge Typeface Computer Modern 12/14pt System AmsTeX [UPH] A catalogue record for this book is available from the British Library

ISBN 0 521 43275 8 hardback

To Haleh Afshar and Tatiana Bernik

Contents Preface

ix

Notation

xi

Chapter 1. Diophantine approximation and manifolds

1

1.1. Introduction

1

1.2. Diophantine approximation in one dimension

1

1.3. Approximation in higher dimensions

5

1.4. Euclidean submanifolds

11

1.5. Metric Diophantine approximation on manifolds

19

1.6. Notes

25

Chapter 2. Khintchine's and Groshev's theorems for manifolds

27

2.1. Introduction

27

2.2. Extremal manifolds

27

2.3. Khintchine and Groshev type manifolds

29

2.4. Baker's conjecture

30

2.5. Higher dimensional manifolds

48

2.6. Notes

56

Chapter 3. Hausdorff measure and dimension

58

3.1. Introduction

58

3.2. Hausdorff measure

58

3.3. Hausdorff dimension

62

3.4. Properties of Hausdorff dimension

65

3.5. Determining the Hausdorff dimension

67

3.6. Hausdorff dimension on manifolds

73

3.7. Notes

74

Chapter 4. Upper bounds for Hausdorff dimension

77

4.1. Introduction

77

viii

CONTENTS

4.2. Diophantine approximation on manifolds

82

4.3. Smooth manifolds of dimension at least 2

85

4.4. Simultaneous Diophantine approximation

92

4.5. Notes

98

Chapter 5. Lower bounds for Hausdorff dimension

99

5.1. Introduction

99

5.2. Regular systems

99

5.3. Ubiquitous systems

106

5.4. Simultaneous Diophantine approximation on manifolds

117

5.5. Notes

122

Chapter 6. Diophantine approximation over the p-adic field

124

6.1. Introduction to p-adic numbers

124

6.2. Diophantine approximation in Q p

126

6.3. Integral polynomials with small p-adic values

127

6.4. Notes

136

Chapter 7. Applications

137

7.1. Introduction

137

7.2. Diophantine type and very well approximable numbers

138

7.3. A wave equation

139

7.4. The rotation number

140

7.5. Dynamical systems

143

7.6. Linearising diffeomorphisms

148

7.7. Diophantine approximation in hyperbolic space

151

7.8. Notes

159

References Index

161 171

Preface This book is about metric Diophantine approximation on smooth manifolds embedded in Euclidean space. The aim is to develop a coherent body of theory on the lines of that which already exists for the classical theory, corresponding to the manifold being Euclidean space. Although the functional dependence of the coordinates presents serious technical difficulties, there is a surprising degree of interplay between the very different areas of number theory, differential geometry and measure theory. A systematic theory began to emerge in the mid-1960's when V. G. Sprindzuk and W. M. Schmidt established that certain types of curve were extremal (an extremal set enjoys the property that, in a sense that can be made precise, Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved for almost all points in the set; thus the real line is extremal). Sprindzuk conjectured that analytic manifolds satisfying a necessary nondegeneracy condition are extremal. Over the last 30 years, there has been considerable progress in verifying this conjecture for manifolds satisfying various arithmetic and geometric constraints, culminating in its recent proof by D. Y. Kleinbock and G. A. Margulis using ideas of flows on homogeneous spaces of lattices [139]. The greater part of this book is concerned with establishing the counterparts of Khintchine's theorem for manifolds and with the Hausdorff dimension of the associated exceptional sets. It relies very much on Sprindzuk's important monographs Mahler's problem in metric number theory [208] and Metric theory of Diophantine approximations [210]; indeed to some extent it can be regarded as a sequel. Our approach, like Sprindzuk's, is largely analytic and geometric andflowson lattices axe not used nor, apart from the last chapter, is ergodic theory. These approaches, however, hold great promise even for the more delicate questions of Khintchine type results and Hausdorff dimension on manifolds. Chapter 1 sets out the background required for metric Diophantine approximation on manifolds. Khintchine's theorem on simultaneous Diophantine approximation and its dual form (Groshev's theorem) are considered for manifolds in Chapter 2, which is devoted mainly to the long and demanding proof of a closely related conjecture of A. Baker concerning the rational normal curve {(t,..., tn): t € K}. Chapter 3 begins with a relatively self-contained account of Hausdorff dimension and an introduction to its uses in Diophantine approximation. A fuller discussion is given in Chapters 4 and 5 which deal respectively with the technically different problems of obtaining upper and lower bounds. The range of techniques from the number theory axsenal which axe called upon axe an indication of the level of

x

PREFACE

difficulty of some of the questions. The p-adic case is discussed fairly briefly in Chapter 6; the final chapter is devoted to various applications of metric Diophantine approximation. Theorems, lemmas and so on are numbered consecutively in each chapter. Sections are denoted §l.m and subsections by §l.m.n. The scope and complexity of the material has made notation something of a problem and to help the reader a list precedes Chapter 1. While not complete, the references are nevertheless intended to be reasonably comprehensive and include less well known papers from the former Soviet Union. It is with sadness that we record that our friend and colleague Yuri Melnichuk would have been an author but for his tragic death during a visit to York in 1993. We are very grateful to many people and particularly to Haleh Afshar and Tatiana Bernik for their support and encouragement during this setback and throughout the book's lengthy gestation. Alan Baker has given us constant encouragement, Victor Beresnevich, Detta Dickinson, Sanju Velani, James Vickers and Chris Wood read parts of earlier drafts and made many suggestions and corrections; Peter Jackson read the proofs and removed numerous inconsistencies and typographical errors. They are not, however, responsible for any mistakes remaining. The book was prepared on a Silicon Graphics Personal Iris workstation using WF$(. and GNU Emacs installed by Michael Beaty who with Simon Eveson sorted out our TgX problems with skill and good humour. Roger Astley of the Cambridge University Press has been patient and understanding beyond the call of duty. The collaboration essential to this book would not have been possible without the support that the Royal Society and the Soros Foundation provided for exchanges between the Belorussian Academy of Sciences at Minsk, the Lvov Polytechnic Institute and the University of York. The help which we have had has been invaluable and has ensured that this book will be published this side of the millennium.

V. I. Bernik Institute of Mathematics Academy of Sciences Minsk, Belarus

M. M. Dodson Department of Mathematics University of York York, UK

Notation {£}, [f], 2

i W , 22

( 0 , 2, 6 If I, 2, 6 llf II, 2 If IP, 135 «, », 2 x, 2 |X|, 4 B«(fl),84 B(R,6), 115 | * U , 16 oo. Let X C R. We will write J^(X; ^) for the set of £ € X such that the more general inequality

< M)

(1-2)

holds for infinitely many positive integers q, i.e., JtT{X;ip) = { ( 6 l : ||g£|| < ip(q) for infinitely many q G N} will be called ipfirst studied by A. I. Khintchine [134]. Points in X{X\$) approximable. When the set X is clear from the context, we will usually omit reference to it and write simply J^(4>). The function ip will be called an approximation function and will often be taken to be monotonically decreasing (we will usually omit the term monotonically) as well. Note that ip(q) ^ 1/2 when q is

1.2. DIOPHANTINE APPROXIMATION IN ONE DIMENSION

3

sufficiently large. We will make much use of the observation that the set Jf(X; ip) and its generalisations are 'lim-sup' sets as ) = {£ 6 X: £ € %,)(

B. BOLLOBAS, F. KIRWAN, P. SARNAK, C.T.C. WALL

137

Metric Diophantine Approximation on Manifolds

V. I. Bernik Byelorussian Academy of Sciences

M. M. Dodson University of York

Metric Diophantine Approximation on Manifolds

i CAMBRIDGE 1

UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc6n 13, 28014 Madrid, Spain © Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 . Printed in the United Kingdom at the University Press, Cambridge Typeface Computer Modern 12/14pt System AmsTeX [UPH] A catalogue record for this book is available from the British Library

ISBN 0 521 43275 8 hardback

To Haleh Afshar and Tatiana Bernik

Contents Preface

ix

Notation

xi

Chapter 1. Diophantine approximation and manifolds

1

1.1. Introduction

1

1.2. Diophantine approximation in one dimension

1

1.3. Approximation in higher dimensions

5

1.4. Euclidean submanifolds

11

1.5. Metric Diophantine approximation on manifolds

19

1.6. Notes

25

Chapter 2. Khintchine's and Groshev's theorems for manifolds

27

2.1. Introduction

27

2.2. Extremal manifolds

27

2.3. Khintchine and Groshev type manifolds

29

2.4. Baker's conjecture

30

2.5. Higher dimensional manifolds

48

2.6. Notes

56

Chapter 3. Hausdorff measure and dimension

58

3.1. Introduction

58

3.2. Hausdorff measure

58

3.3. Hausdorff dimension

62

3.4. Properties of Hausdorff dimension

65

3.5. Determining the Hausdorff dimension

67

3.6. Hausdorff dimension on manifolds

73

3.7. Notes

74

Chapter 4. Upper bounds for Hausdorff dimension

77

4.1. Introduction

77

viii

CONTENTS

4.2. Diophantine approximation on manifolds

82

4.3. Smooth manifolds of dimension at least 2

85

4.4. Simultaneous Diophantine approximation

92

4.5. Notes

98

Chapter 5. Lower bounds for Hausdorff dimension

99

5.1. Introduction

99

5.2. Regular systems

99

5.3. Ubiquitous systems

106

5.4. Simultaneous Diophantine approximation on manifolds

117

5.5. Notes

122

Chapter 6. Diophantine approximation over the p-adic field

124

6.1. Introduction to p-adic numbers

124

6.2. Diophantine approximation in Q p

126

6.3. Integral polynomials with small p-adic values

127

6.4. Notes

136

Chapter 7. Applications

137

7.1. Introduction

137

7.2. Diophantine type and very well approximable numbers

138

7.3. A wave equation

139

7.4. The rotation number

140

7.5. Dynamical systems

143

7.6. Linearising diffeomorphisms

148

7.7. Diophantine approximation in hyperbolic space

151

7.8. Notes

159

References Index

161 171

Preface This book is about metric Diophantine approximation on smooth manifolds embedded in Euclidean space. The aim is to develop a coherent body of theory on the lines of that which already exists for the classical theory, corresponding to the manifold being Euclidean space. Although the functional dependence of the coordinates presents serious technical difficulties, there is a surprising degree of interplay between the very different areas of number theory, differential geometry and measure theory. A systematic theory began to emerge in the mid-1960's when V. G. Sprindzuk and W. M. Schmidt established that certain types of curve were extremal (an extremal set enjoys the property that, in a sense that can be made precise, Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved for almost all points in the set; thus the real line is extremal). Sprindzuk conjectured that analytic manifolds satisfying a necessary nondegeneracy condition are extremal. Over the last 30 years, there has been considerable progress in verifying this conjecture for manifolds satisfying various arithmetic and geometric constraints, culminating in its recent proof by D. Y. Kleinbock and G. A. Margulis using ideas of flows on homogeneous spaces of lattices [139]. The greater part of this book is concerned with establishing the counterparts of Khintchine's theorem for manifolds and with the Hausdorff dimension of the associated exceptional sets. It relies very much on Sprindzuk's important monographs Mahler's problem in metric number theory [208] and Metric theory of Diophantine approximations [210]; indeed to some extent it can be regarded as a sequel. Our approach, like Sprindzuk's, is largely analytic and geometric andflowson lattices axe not used nor, apart from the last chapter, is ergodic theory. These approaches, however, hold great promise even for the more delicate questions of Khintchine type results and Hausdorff dimension on manifolds. Chapter 1 sets out the background required for metric Diophantine approximation on manifolds. Khintchine's theorem on simultaneous Diophantine approximation and its dual form (Groshev's theorem) are considered for manifolds in Chapter 2, which is devoted mainly to the long and demanding proof of a closely related conjecture of A. Baker concerning the rational normal curve {(t,..., tn): t € K}. Chapter 3 begins with a relatively self-contained account of Hausdorff dimension and an introduction to its uses in Diophantine approximation. A fuller discussion is given in Chapters 4 and 5 which deal respectively with the technically different problems of obtaining upper and lower bounds. The range of techniques from the number theory axsenal which axe called upon axe an indication of the level of

x

PREFACE

difficulty of some of the questions. The p-adic case is discussed fairly briefly in Chapter 6; the final chapter is devoted to various applications of metric Diophantine approximation. Theorems, lemmas and so on are numbered consecutively in each chapter. Sections are denoted §l.m and subsections by §l.m.n. The scope and complexity of the material has made notation something of a problem and to help the reader a list precedes Chapter 1. While not complete, the references are nevertheless intended to be reasonably comprehensive and include less well known papers from the former Soviet Union. It is with sadness that we record that our friend and colleague Yuri Melnichuk would have been an author but for his tragic death during a visit to York in 1993. We are very grateful to many people and particularly to Haleh Afshar and Tatiana Bernik for their support and encouragement during this setback and throughout the book's lengthy gestation. Alan Baker has given us constant encouragement, Victor Beresnevich, Detta Dickinson, Sanju Velani, James Vickers and Chris Wood read parts of earlier drafts and made many suggestions and corrections; Peter Jackson read the proofs and removed numerous inconsistencies and typographical errors. They are not, however, responsible for any mistakes remaining. The book was prepared on a Silicon Graphics Personal Iris workstation using WF$(. and GNU Emacs installed by Michael Beaty who with Simon Eveson sorted out our TgX problems with skill and good humour. Roger Astley of the Cambridge University Press has been patient and understanding beyond the call of duty. The collaboration essential to this book would not have been possible without the support that the Royal Society and the Soros Foundation provided for exchanges between the Belorussian Academy of Sciences at Minsk, the Lvov Polytechnic Institute and the University of York. The help which we have had has been invaluable and has ensured that this book will be published this side of the millennium.

V. I. Bernik Institute of Mathematics Academy of Sciences Minsk, Belarus

M. M. Dodson Department of Mathematics University of York York, UK

Notation {£}, [f], 2

i W , 22

( 0 , 2, 6 If I, 2, 6 llf II, 2 If IP, 135 «, », 2 x, 2 |X|, 4 B«(fl),84 B(R,6), 115 | * U , 16 oo. Let X C R. We will write J^(X; ^) for the set of £ € X such that the more general inequality

< M)

(1-2)

holds for infinitely many positive integers q, i.e., JtT{X;ip) = { ( 6 l : ||g£|| < ip(q) for infinitely many q G N} will be called ipfirst studied by A. I. Khintchine [134]. Points in X{X\$) approximable. When the set X is clear from the context, we will usually omit reference to it and write simply J^(4>). The function ip will be called an approximation function and will often be taken to be monotonically decreasing (we will usually omit the term monotonically) as well. Note that ip(q) ^ 1/2 when q is

1.2. DIOPHANTINE APPROXIMATION IN ONE DIMENSION

3

sufficiently large. We will make much use of the observation that the set Jf(X; ip) and its generalisations are 'lim-sup' sets as ) = {£ 6 X: £ € %,)(

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