Recent advances in Diophantine approximation

Chapter

Abstract

A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex number, as well as the simultaneous approximation of powers of a real number by rational numbers with the same denominator. Finally we study generalisations of these questions to higher dimensions. Several recent advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent, T. Rivoal, D. Roy, and W.M. Schmidt, among others. We review some of these works.

Key words

Diophantine approximation rational approximation simultaneous approximation approximation by algebraic numbers approximation by linear forms irrationality measures transcendence criterion criteria for algebraic independence Dirichlet Hurwitz Thue-Siegel-Roth-Schmidt Khintchine Davenport Sprindzuck Laurent Roy 

Notes

Acknowledgements

Many thanks to Boris Adamczewski, Victor Beresnevich, Yann Bugeaud, Maurice Dodson, Michel Laurent, Claude Levesque, Damien Roy for their enlightening remarks and their comments on preliminary versions of this paper. Sections 2.7 and 3.6, as well as part of Section 1.2, have been written by Victor Beresnevich and Maurice Dodson. I wish also to thank Dinakar Ramakrishnan who completed the editorial work in a very efficient way.

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Université Pierre et Marie Curie–Paris 6, UMR 7586 IMJ Institut de, Mathématiques de JussieuParis Cedex 05France

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