Metric results of the intersection of sets in Diophantine approximation

Let $\psi : \mathbb{R}_{>0}\rightarrow \mathbb{R}_{>0}$ 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 $\psi_i:\mathbb{R}_{> 0}\rightarrow \mathbb{R}_{> 0}$ is a non-increasing function satisfying $\psi_i(x)=o(x^{-2})$ for $1\le i \le n.$