On the metric theory of multiplicative Diophantine approximation

Abstract

In 1962, Gallagher proved a higher-dimensional version of Khintchine’s theorem on Diophantine approximation. Gallagher’s theorem states that for any non-increasing approximation function ψ: ℕ → (0, 1/2) with $\sum\nolimits_{q = 1}^\infty {\psi \left( q \right)}$ and γ = γ′ = 0 the following set

$$\left\{ {\left( {x,y} \right) \in {{\left[ {0,\,1} \right]}^2}:\left\| {qx - \gamma } \right\|\left\| {qy - {\gamma ^\prime }} \right\| < \psi \left( q \right)\,\,{\rm{infinitely}}\,{\rm{often}}} \right\}$$

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.