The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

TL;DR

This paper establishes a zero-infinity law for the generalized Hausdorff measure of sets of Dirichlet non-improvable numbers, extending previous results to broader classes of dimension functions under natural conditions.

Contribution

It proves a comprehensive zero-infinity law for all dimension functions for sets of Dirichlet non-improvable numbers, generalizing prior results and including new classes of dimension functions.

Findings

Zero-infinity law valid for all dimension functions under natural conditions

Extension of previous results to non-essentially sub-linear dimension functions

Applicability to a broad class of growth conditions on the approximating function

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.