Report on some recent advances in Diophantine approximation

Details Report on some recent advances in Diophantine approximation Report on some recent advances in Diophantine approximation

2009-08-27

arxiv.org/abs/0908.3973v1

Mathematics

Number Theory

to be published by Springer Verlag, Special volume in honor of Serge Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakri

Scientific paper

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.

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