Dynamics and Analytic Number Theory

doi:10.1017/CBO9781316402696

Written by leading experts, this book explores several directions of current research at the interface between dynamics and analytic number theory. Topics include Diophantine approximation, exponential sums, Ramsey theory, ergodic theory and homogeneous dynamics. The origins of this material lie in the 'Dynamics and Analytic Number Theory' Easter School held at Durham University in 2014. Key concepts, cutting-edge results, and modern techniques that play an essential role in contemporary research are presented in a manner accessible to young researchers, including PhD students. This book will also be useful for established mathematicians. The areas discussed include ubiquitous systems and Cantor-type sets in Diophantine approximation, flows on nilmanifolds and their connections with exponential sums, multiple recurrence and Ramsey theory, counting and equidistribution problems in homogeneous dynamics, and applications of thin groups in number theory. Both dynamical and 'classical' approaches towards number theoretical problems are also provided.

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Dynamics and Analytic Number Theory

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

List of contributors
pp vii-viii

Preface
pp ix-xviii

1 - Metric Diophantine Approximation: Aspects of Recent Work
pp 1-95

2 - Exponents of Diophantine Approximation
pp 96-135

3 - Effective Equidistribution of Nilflows and Bounds onWeyl Sums
pp 136-188

4 - Multiple Recurrence and Finding Patterns in Dense Sets
pp 189-257

5 - Diophantine Problems and Homogeneous Dynamics
pp 258-288

6 - Applications of Thin Orbits
pp 289-317

Index
pp 318-321

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Dynamics and Analytic Number Theory

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