Simultaneous Approximation by Conjugate Algebraic Numbers in Fields of Transcendence Degree One

Abstract

Abstract. We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ 1 , ..., ξ t by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers ξ 1 , ..., ξ t generate over Q a field of transcendence degree one. We provide sharper estimates for example when ξ 1 , ..., ξ t 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 ξ 1 , ..., ξ t .

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“…It is well known that the standard Khintchine and Dirichlet theorems [28] only deal with the approximation of real numbers by rationals, which are algebraic numbers of degree one. Although some results on approximating a real number by an algebraic number are available in literature [29], [30], these results come with various restrictions which unfortunately do not lend themselves to our problem at hand. For instance, [29] only addresses simultaneous approximation of one number by algebraic conjugates or multiple numbers by non-conjugates of a bounded degree, while [30] requires the real numbers to be approximated lie in a field of transcendence degree one.…”