On the metric theory of multiplicative Diophantine approximation

TL;DR

This paper advances the metric theory of multiplicative Diophantine approximation by proving new inhomogeneous and fibred results, extending classical theorems and addressing cases involving Liouville numbers.

Contribution

It introduces an Erdős-Vaaler type theorem for fibred multiplicative Diophantine approximation and extends Chow-Technau's inhomogeneous Gallagher's theorem to Liouville fibers.

Findings

Proved an Erdős-Vaaler type result for fibred multiplicative Diophantine approximation.

Established a weaker version of Chow-Technau's theorem with a Liouville condition.

Extended Chow-Technau's theorem to include Liouville fibers.

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.