A uniform metrical theorem in multiplicative Diophantine approximation

Abstract

For Lebesgue generic $(x_1,x_2)\in {\mathbb{R}}^2$, we investigate the distribution of small values of products $q\cdot \Vert q{x}_1\Vert \cdot \Vert q{x}_2\Vert$ with $q\in \mathbb{N}$, where $\Vert \cdot \Vert$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q\le T$ such that $$a_T<q\cdot \Vert q{x}_1\Vert \cdot \Vert q{x}_2\Vert \le b_T\text{and}\Vert q{x}_1\Vert ,\Vert q{x}_2\Vert \le c_T$$ for given sequences $a_T,b_T,c_T$ satisfying certain growth conditions.

The Borel-Cantelli argument in the previous version is wrong, and has been corrected in the present version