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.