Hausdorff measure of sets of Dirichlet non-improvable numbers

By Mumtaz Hussain, Dmitry Kleinbock, Nick Wadleigh, Bao-Wei Wang. 2017

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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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/83e8a0bdd8c1286c58216ebc9f6ca4e314d4a320