Hausdorff measure of sets of Dirichlet non-improvable numbers

Abstract

Let $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if it admits an improvement to Dirichlet's theorem in the following sense: the system $$|qx-p|<\psi (t)\text{and}|q|<t$$ has a non-trivial integer solution for all large enough $t$. Denote the collection of such points by $D(\psi )$. In this paper, we prove that the Hausdorff measure of the complement $D(\psi {)}^c$ (the set of $\psi$-Dirichlet non-improvable numbers) obeys a zero-infinity law for a large class of dimension functions. Together with the Lebesgue measure-theoretic results established by Kleinbock \& Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.

15 pages

Cited by in corpus (4)

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• Limit theorems for sums of products of consecutive partial quotients of continued fractions