Metric results of the intersection of sets in Diophantine approximation

Abstract

Let $ψ: \mathbb{R}_{>0}\rightarrow \mathbb{R}_{>0}$ be a non-increasing function. Denote by $W(ψ)$ the set of $ψ$-well-approximable points and by $E(ψ)$ the set of points $x\in[0,1]$ such that for any $0 < ε< 1$ there exist infinitely many $(p,q)\in\mathbb{Z}\times\mathbb{N} $ with $$\left(1-ε\right)ψ(q)< \left| x-\frac{p}{q}\right|< ψ(q) .$$ In this paper, we investigate the metric properties of the set $E(ψ).$ Specifically, we compute the $s$-dimensional Hausdorff measure $\mathcal{H}^s(E(ψ))$ of $E(ψ)$ for a large class of $s \in (0,1].$ Additionally, we establish that $$\dim_{\mathcal H} E(ψ_1) \times \cdots \times E(ψ_n) =\min \{ \dim_{\mathcal H} E(ψ_i)+n-1: 1\le i \le n \},$$ where $ψ_i:\mathbb{R}_{> 0}\rightarrow \mathbb{R}_{> 0} $ is a non-increasing function satisfying $ψ_i(x)=o(x^{-2}) $ for $1\le i \le n.$