Multiplicative Diophantine approximation with restricted denominators

TL;DR

This paper determines the Hausdorff dimensions of sets of points in the unit square and the unit interval that satisfy certain multiplicative Diophantine approximation inequalities infinitely often, based on convergence results for Hausdorff measures.

Contribution

It provides a complete characterization of Hausdorff dimensions for these sets under restricted denominators, extending previous results in multiplicative Diophantine approximation.

Findings

Determined Hausdorff dimensions for points satisfying multiplicative inequalities infinitely often.

Established general convergence results for Hausdorff measures of these sets.

Provided results on points where the maximum of the two norms is small infinitely often.

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.