Diophantine Approximation by Numbers of Small Height

Abstract

As is well known, Roth’s Theorem says that given algebraic ξ ∈ ℝ, and given δ > 0, there are only finitely many rationals p/q with |ξ − p/q| < q^−2−δ. More generally, it has been shown that given a number field K ⊂ ℂ, and given algebraic ξ ∈ ℂ, there are only finitely many elements α ∈ K with

$$|\xi - \alpha | < H{(\alpha )^{ - 2 - \delta }}$$

((1))

where H(α) denotes the absolute multiplicative Height of α. Even more generally, there is a version due to Lang involving a finite number of absolute values of K. Let us call α an exceptional approximation if it does satisfy (1). The number of exceptional approximations was first estimated by Davenport and Roth [3], then by Bombieri and Van der Poorten [1], also by Luckhardt [8] and Schmidt [10]. See also Section 6.5 in Bombieri and Gubler [2]. In all these works good estimates were given only for approximations α of large Height.