Metric results of the intersection of sets in Diophantine approximation

Abstract

Let $\psi :{\mathbb{R}}_{>0}\to {\mathbb{R}}_{>0}$ be a non-increasing function. Denote by $W(\psi )$ the set of $\psi$-well-approximable points and by $E(\psi )$ the set of points $x\in [0,1]$ such that for any $0<\epsilon <1$ there exist infinitely many $(p,q)\in \mathbb{Z}\times \mathbb{N}$ with $$\left(1-\epsilon \right)\psi (q)<\left|x-\frac{p}{q}\right|<\psi (q).$$ In this paper, we investigate the metric properties of the set $E(\psi ).$ Specifically, we compute the $s$-dimensional Hausdorff measure ${\mathcal{H}}^s(E(\psi ))$ of $E(\psi )$ for a large class of $s\in (0,1].$ Additionally, we establish that $${\dim }_{\mathcal{H}}E({\psi }_1)\times \cdots \times E({\psi }_n)=\min \{{\dim }_{\mathcal{H}}E({\psi }_i)+n-1:1\le i\le n\},$$ where ${\psi }_i:{\mathbb{R}}_{>0}\to {\mathbb{R}}_{>0}$ is a non-increasing function satisfying ${\psi }_i(x)=o(x^{-2})$ for $1\le i\le n.$