A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation

Abstract

Let μ be a Gibbs measure of the doubling map T of the circle. For a μ-generic point x and a given sequence {rn} ⊂ ℝ+, consider the intervals (Tnx − rn (mod 1), Tnx + rn (mod 1)). In analogy to the classical Dvoretzky covering of the circle, we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hitting times for moving targets. A mass transference principle is obtained for Gibbs measures that are multifractal. Such a principle was proved by Beresnevich and Velani [Ann. Math. 164 (2006) 971–992] for monofractal measures. In the symbolic language, we completely describe the combinatorial structure of a typical relatively short sequence; in particular, we can describe the occurrence of ‘atypical’ relatively long words. Our results have a direct and deep number-theoretical interpretation via inhomogeneous dyadic Diophantine approximation by numbers belonging to a given (dyadic) Diophantine class.