Metric results of the intersection of sets in Diophantine approximation

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 $$(1-\epsilon )\psi (q)<\mid x-\frac{p}{q}\mid <\psi (q)\mathrm{.}$$ In this paper, we investigate the metric properties of the set $E(\psi )\mathrm{.}$ Specifically, we compute the $s$-dimensional Hausdorff measure ${\mathcal{H}}^s(E(\psi ))$ of $E(\psi )$ for a large class of $s\in (0,1]\mathrm{.}$ 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 $1\le i\le n\mathrm{.}$