Report on some recent advances in Diophantine approximation

Michel Waldschmidt (1)

1 IMJ - Institut de Mathématiques de Jussieu

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.

Domains: Number Theory [math.NT]

Michel Waldschmidt. Report on some recent advances in Diophantine approximation. 2009. ⟨hal-00407199⟩