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

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.

arXiv commentIn the revised version we have fixed some typos, added an example on page 2 and stated the slicing lemma on page 6