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Metric Diophantine Approximation on Manifolds : Cambridge Tracts in Mathematics - V. I. Bernik

Metric Diophantine Approximation on Manifolds

Cambridge Tracts in Mathematics


Published: 3rd April 2012
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This book is concerned with Diophantine approximation on smooth manifolds embedded in Euclidean space, and its aim is to develop a coherent body of theory comparable with that which already exists for classical Diophantine approximation. In particular, this book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. After setting out the necessary background material, the authors give a full discussion of Hausdorff dimension and its uses in Diophantine approximation. A wide range of techniques from the number theory arsenal are used to obtain the upper and lower bounds required, and this is an indication of the difficulty of some of the questions considered. The authors go on to consider briefly the p-adic case, and they conclude with a chapter on some applications of metric Diophantine approximation. All researchers with an interest in Diophantine approximation will welcome this book.

'This book is an important addition to the literature from authors who are leading experts in this field.' Glyn Harman, Bulletin of the London Mathematical Society '... carefully written, with an extensive bibliography, and will be of lasting value ...'. Thomas Ward, Zentralblatt fur Mathematik 'This book can be recommended not only to those interested in number-theoretic aspects, but also to those interests lie in topics related to dynamical systems ... self-contained and very readable.' EMS

Prefacep. ix
Notationp. xi
Diophantine approximation and manifoldsp. 1
Introductionp. 1
Diophantine approximation in one dimensionp. 1
Approximation in higher dimensionsp. 5
Euclidean submanifoldsp. 11
Metric Diophantine approximation on manifoldsp. 19
Notesp. 25
Khintchine's and Groshev's theorems for manifoldsp. 27
Introductionp. 27
Extremal manifoldsp. 27
Khintchine and Groshev type manifoldsp. 29
Baker's conjecturep. 30
Higher dimensional manifoldsp. 48
Notesp. 56
Hausdorff measure and dimensionp. 58
Introductionp. 58
Hausdorff measurep. 58
Hausdorff dimensionp. 62
Properties of Hausdorff dimensionp. 65
Determining the Hausdorff dimensionp. 67
Hausdorff dimension on manifoldsp. 73
Notesp. 74
Upper bounds for Hausdorff dimensionp. 77
Introductionp. 77
Diophantine approximation on manifoldsp. 82
Smooth manifolds of dimension at least 2p. 85
Simultaneous Diophantine approximationp. 92
Notesp. 98
Lower bounds for Hausdorff dimensionp. 99
Introductionp. 99
Regular systemsp. 99
Ubiquitous systemsp. 106
Simultaneous Diophantine approximation on manifoldsp. 117
Notesp. 122
Diophantine approximation over the p-adic fieldp. 124
Introduction to p-adic numbersp. 124
Diophantine approximation in Q[subscript p]p. 126
Integral polynomials with small p-adic valuesp. 127
Notesp. 136
Applicationsp. 137
Introductionp. 137
Diophantine type and very well approximable numbersp. 138
A wave equationp. 139
The rotation numberp. 140
Dynamical systemsp. 143
Linearising diffeomorphismsp. 148
Diophantine approximation in hyperbolic spacep. 151
Notesp. 159
Referencesp. 161
Indexp. 171
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521432757
ISBN-10: 0521432758
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 186
Published: 3rd April 2012
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.4
Weight (kg): 0.45