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 \begin{equation*} |qx-p|< \psi (t) \ \ {\text {and}} \ \ |q|<t \end{equation*} 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 sublinear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang [Mathematika 64 (2018), pp. 502–518], for some non-essentially sublinear dimension functions, and for all dimension functions but with a growth condition on the approximating function.