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A basic question of Diophantine approximation, which is the first issue wediscuss, is to investigate the rational approximations to a single real number.Next, we consider the algebraic or polynomial approximations to a singlecomplex number, as well as the simultaneous approximation of powers of a realnumber by rational numbers with the same denominator. Finally we studygeneralisations of these questions to higher dimensions. Several recentadvances 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 theseworks.

Report on some recent advances in Diophantine approximation