Autour du th�or�me de Roth

Abstract

On Roth's theorem. The celebrated theorem of Roth, together with its generalizations given by Mahler and Ridout, gives a lower bound for the degree of approximation of one or more algebraic numbers with respect to a fixed set of valuations by elements of a fixed number field. An analogous result holds for function fields in characteristic zero. In this paper we do the following: (1) generalize Roth's theorem to the case of fields with a product formula in characteristic zero, removing any technical hypothesis from a previous result of Lang: (2) give a unified proof of Roth's theorem in the number field and function field cases; (3) provide a quantitative version of the general Roth's theorem, extending, even in the number field case, previous results of Bombieri and Van Der Poorten.