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 $1\le i \le n.$
Categories: math.NT math.DS