Metric results of the intersection of sets in Diophantine approximation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909556792098816 |
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| author | Tian, Chen Peng, Liuqing |
| author_facet | Tian, Chen Peng, Liuqing |
| contents | 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 $1\le i \le n.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_14513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric results of the intersection of sets in Diophantine approximation Tian, Chen Peng, Liuqing Number Theory Dynamical Systems 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 $1\le i \le n.$ |
| title | Metric results of the intersection of sets in Diophantine approximation |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2502.14513 |