On the metric theory of multiplicative Diophantine approximation

Abstract

In 1962, Gallagher proved an higher dimensional version of Khintchine's theorem on Diophantine approximation. Gallagher's theorem states that for any non-increasing approximation function $\ddot{I}ˆ:\mathbb{N}\to (0,1/2)$ with ${\sum }_{q=1}^{\infty }\ddot{I}ˆ(q)\log q=\infty$ and $γ=γ'=0$ the following set \[ \{(x,y)\in [0,1]^2: \|qx-γ\|\|qy-γ'\|<ψ(q) \text{ infinitely often}\} \] has full Lebesgue measure. Recently, Chow and Technau proved a fully inhomogeneous version (without restrictions on $γ,γ'$) of the above result. In this paper, we prove an Erdős-Vaaler type result for fibred multiplicative Diophantine approximation. Along the way, via a different method, we prove a slightly weaker version of Chow-Technau's theorem with the condition that at least one of $γ,γ'$ is not Liouville. We also extend Chow-Technau's result for fibred inhomogeneous Gallagher's theorem for Liouville fibres.

36 pages; version to appear in Journal d'Analyse Mathématique