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 $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in\mathbb{N}$, where $\|\cdot \|$ 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 $$ a_T <q\cdot \|qx_1\| \cdot \|qx_2\|\leq b_T \quad \textrm{and} \quad \|qx_1\|, \|qx_2\|\leq c_T $$ for given sequences $a_T,b_T, c_T$ satisfying certain growth conditions.