# Hausdorff measure of sets of Dirichlet non-improvable numbers

## TL;DR

This paper investigates the size of the set of real numbers that cannot be improved upon Dirichlet's approximation theorem, establishing a zero-infinity law for their Hausdorff measure under broad conditions.

## Contribution

It extends the metric theory of Dirichlet non-improvable numbers by determining the Hausdorff measure of their complement for a wide class of dimension functions.

## Key findings

- Hausdorff measure of non-improvable set obeys zero-infinity law
- Complements of Dirichlet non-improvable sets have measure zero or infinity
- Results complement Lebesgue measure findings by Kleinbock & Wadleigh

## 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.