SIMULTANEOUS APPROXIMATION BY CONJUGATE ALGEBRAIC NUMBERS IN FIELDS OF TRANSCENDENCE DEGREE ONE

Abstract

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ1, …, ξt by conjugate algebraic numbers of bounded degree over ℚ, provided that the given transcendental numbers ξ1, …, ξt generate over ℚ a field of transcendence degree one. We provide sharper estimates for example when ξ1, …, ξt form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence criterion expressed in terms of polynomials of bounded degree taking small values at ξ1, …, ξt.