The generalised 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 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 a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sub-linear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang (2018), for some non-essentially sub-linear dimension functions, and for all dimension functions but with a growth condition on the approximating function.

arXiv commentIn Press: Proceedings of the American Mathematical Society. arXiv admin note: text overlap with arXiv:1704.03089