Hausdorff measure of sets of dirichlet non-improvable numbers

Abstract

Let Ψ: ℝ+ → ℝ+ be a non-increasing function. A real number x is said to be Ψ -Dirichlet improvable if it admits an improvement to Dirichlet's theorem in the following sense: The system qx - p < Ψ (t) and q < t has a non-trivial integer solution for all large enough t. Denote the collection of such points by D(Ψ) In this paper we prove that the Hausdorff measure of the complement D(Ψ)c (the set of Ψ-Dirichlet non-improvable numbers) obeys a zeroinfinity law for a large class of dimension functions. Together with the Lebesgue measure-theoretic results established by Kleinbock and Wadleigh [A zero-one law for improvements to Dirichlet's theorem. Proc. Amer. Math. Soc. 146 (2018), 1833- 1844], our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.