Mesures de transcendance et aspects quantitatifs de la méthode de Thue-Siegel-Roth-Schmidt

Abstract

A proof of the transcendence of a real number ξ based on the Thue-Siegel-Roth-Schmidt method involves generally a sequence (αn)n≥1 of algebraic numbers of bounded degree or a sequence (xn)n≥1 of integer r-tuples. In the present paper, we show how such a proof can produce a transcendence measure for ξ, if one is able to quantify the growth of the heights of the algebraic numbers αn or of the points xn. Our method rests on the quantitative Schmidt subspace theorem. We further give several applications, including to certain normal numbers and to the extremal numbers introduced by Roy. © 2010 London Mathematical Society.