On the metric theory of multiplicative Diophantine approximation (Q6113034)

This paper deals with metric multiplicative Diophantine approximations. One can begin with author's 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 \(\psi :\mathbb N\to (0, 1/2)\) with \(\sum^{\infty} _{q=1}{\psi (q) \log q } =\infty \) and \(\gamma = \gamma^{'}= 0\) the following set \[ \left\{(x, y) \in [0, 1]^2 : \|qx-\gamma\|\|qy-\gamma^{'}\| < \psi(q) \text{ infinitely often } \right\} \] 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 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 \(\gamma, \gamma^{'}\) is not Liouville. We also extend Chow-Technau's result for fibred inhomogeneous Gallagher's theorem for Liouville fibres.'' The techniques used in proofs are explained. Special attention is given to known results for inhomogeneous and multiplicative metric Diophantine approximations, as well as to such statements as the Beresnevich, Haynes, and Velani conjecture, the Borel-Cantelli lemma, etc. Also, several auxiliary results are proven. A certain conjecture is formulated, some its cases are proven, and open questions are formulated.