Multiplicative Diophantine approximation with restricted denominators

Abstract: Let ${a_n}{n\in\mathbb{N}}$, ${b_n}{n\in \mathbb{N}}$ be two infinite subsets of positive integers and $\psi:\mathbb{N}\to \mathbb{R}_{>0}$ be a positive function. We completely determine the Hausdorff dimensions of the set of all points $(x,y)\in [0,1]^2$ which satisfy $|a_nx||b_ny|<\psi(n)$ infinitely often, and the set of all $x\in [0,1]$ satisfying $|a_nx||b_nx|<\psi(n)$ infinitely often. This is based on establishing general convergence results for Hausdorff measures of these two sets. We also obtain some results on the set of all $x\in [0,1]$ such that $\max{|a_nx|, |b_nx|}<\psi(n)$ infinitely often.