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 $\psi:\mathbb{N}\to (0,1/2)$ with $\sum_{q=1}^{\infty} \psi(q)\log q=\infty$ and $\gamma=\gamma'=0$ the following set \[ \{(x,y)\in [0,1]^2: \|qx-\gamma\|\|qy-\gamma'\|<\psi(q) \text{ infinitely often}\} \] has full Lebesgue measure. Recently, Chow and Technau proved a fully inhomogeneous version (without restrictions on $\gamma,\gamma'$) of the above result. In this paper, we prove an Erd\H{o}s-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 $\gamma,\gamma'$ is not Liouville. We also extend Chow-Technau's result for fibred inhomogeneous Gallagher's theorem for Liouville fibres.