The Hausdorff measure version of Gallagher's theorem -- closing the gap and beyond

TL;DR

This paper establishes a precise Hausdorff measure criterion for multiplicative approximation sets in higher dimensions, resolving a longstanding question about measure zero or full measure based on series convergence, and extends results to a doubly metric context.

Contribution

It provides the first matching upper bound to known lower bounds for Hausdorff measure of multiplicatively approximable sets, closing the gap in Gallagher's theorem and addressing the log factor discrepancy.

Findings

Established a zero-full law for Hausdorff measure of multiplicative approximation sets.

Resolved the log factor discrepancy in convergence/divergence criteria.

Extended results to the multiplicative doubly metric setting.

Abstract

In this paper we prove an upper bound on the "size" of the set of multiplicatively $\psi$-approximable points in $\mathbb R^d$ for $d>1$ in terms of $f$-dimensional Hausdorff measure. This upper bound exactly complements the known lower bound, providing a "zero-full" law which relates the Hausdorff measure to the convergence/divergence of a certain series in both the homogeneous and inhomogeneous settings. This zero-full law resolves a question posed by Beresnevich and Velani (2015) regarding the "log factor" discrepancy in the convergent/divergent sum conditions of their theorem. We further prove the analogous result for the multiplicative doubly metric setup.