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.