Diophantine近似中集合交集的度量结果
数论
2025-04-01 v2 动力系统
摘要
设ψ : R > 0 → R > 0 \psi : \mathbb{R}_{>0}\rightarrow \mathbb{R}_{>0} ψ : R > 0 → R > 0 W ( ψ ) W(\psi) W ( ψ ) ψ \psi ψ E ( ψ ) E(\psi) E ( ψ ) 0 < ϵ < 1 0 < \epsilon < 1 0 < ϵ < 1 ( p , q ) ∈ Z × N (p,q)\in\mathbb{Z}\times\mathbb{N} ( p , q ) ∈ Z × N ( 1 − ϵ ) ψ ( q ) < ∣ x − p q ∣ < ψ ( q ) \left(1-\epsilon\right)\psi(q)< \left| x-\frac{p}{q}\right|< \psi(q) ( 1 − ϵ ) ψ ( q ) < x − q p < ψ ( q ) x ∈ [ 0 , 1 ] x\in[0,1] x ∈ [ 0 , 1 ] E ( ψ ) E(\psi) E ( ψ ) W ( ψ ) W(\psi) W ( ψ ) s ∈ ( 0 , 1 ] s \in (0,1] s ∈ ( 0 , 1 ] s s s H s ( E ( ψ ) ) \mathcal{H}^s(E(\psi)) H s ( E ( ψ )) dim H E ( ψ 1 ) × ⋯ × E ( ψ n ) = min { dim H E ( ψ i ) + n − 1 : 1 ≤ i ≤ n } , \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\}, dim H E ( ψ 1 ) × ⋯ × E ( ψ n ) = min { dim H E ( ψ i ) + n − 1 : 1 ≤ i ≤ n } , ψ i : R > 0 → R > 0 \psi_i:\mathbb{R}_{> 0}\rightarrow \mathbb{R}_{> 0} ψ i : R > 0 → R > 0 ψ i ( x ) = o ( x − 2 ) \psi_i(x)=o(x^{-2}) ψ i ( x ) = o ( x − 2 )
引用
@article{arxiv.2502.14513,
title = {Metric results of the intersection of sets in Diophantine approximation},
author = {Chen Tian and Liuqing Peng},
journal= {arXiv preprint arXiv:2502.14513},
year = {2025}
}