Multiplicative Diophantine approximation with restricted denominators

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 $\mathrm{\parallel }{a}_{n}x\mathrm{\parallel }\mathrm{\parallel }{b}_{n}y\mathrm{\parallel }<\psi (n)$ infinitely often, and the set of all $x\in [0,1]$ satisfying $\mathrm{\parallel }{a}_{n}x\mathrm{\parallel }\mathrm{\parallel }{b}_{n}x\mathrm{\parallel }<\psi (n)$ 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 $\max \{\mathrm{\parallel }{a}_{n}x\mathrm{\parallel },\mathrm{\parallel }{b}_{n}x\mathrm{\parallel }\}<\psi (n)$ infinitely often.