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 $\ddot{I}ˆ:\mathbb{N}\to {\mathbb{R}}_{>0}$ be a positive function. We completely determine the Hausdorff dimensions of the set of all points $(x,y)\in [0,1{]}^2$ which satisfy $\Vert a_{n}x\Vert \Vert b_{n}y\Vert <\ddot{I}ˆ(n)$ infinitely often, and the set of all $x\in [0,1]$ satisfying $\Vert a_{n}x\Vert \Vert b_{n}x\Vert <\ddot{I}ˆ(n)$ 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 $\max \{\Vert a_{n}x\Vert ,\Vert b_{n}x\Vert \}<\ddot{I}ˆ(n)$ infinitely often.

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