Metric results of the intersection of sets in Diophantine approximation
Chen Tian, Liuqing Peng
TL;DR
This work determines the metric properties of the ψ-intersection set E(ψ) in one-dimensional Diophantine approximation and derives a sharp divergence-convergence dichotomy for its s-dimensional Hausdorff measure under the condition that x^2ψ^s(x) is non-increasing and ψ^s(x)→0, namely H^s(E(ψ)) equals 0 when ∑_{n} nψ^s(n) < ∞ and equals ∞ (indeed, (0,1) in measure) when ∑_{n} nψ^s(n) = ∞. It further establishes a precise product-dimension formula dim_H E(ψ_1)×···×E(ψ_n) = min_{1≤i≤n} { n−1 + 2/λ_i }, with λ_i = liminf_{x→∞} (−log ψ_i(x))/log x, extending Erdős-type ideas to higher-dimensional products. The methodology combines Cantor subset constructions, a robust mass distribution framework, and local dimension analysis to obtain both lower and upper bounds, yielding a complete picture of when E(ψ) has maximal or diminished fractal size and how this size behaves under Cartesian products. These results advance the understanding of exact-type sets in Diophantine approximation and illuminate how fractal geometry interacts with number-theoretic approximation across dimensions.
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 $1\le i \le n.$