Hausdorff measures of sets in Exact Diophantine approximation
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arXiv
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| Format: | Preprint |
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2025
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| author | Tan, Bo Tian, Chen Wang, Baowei Wu, Jun |
| author_facet | Tan, Bo Tian, Chen Wang, Baowei Wu, Jun |
| contents | Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $ϕ: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, ϕ)$ of points $x \in X$ that ``hit'' the shrinking balls $B(ξ,{ϕ(R(ξ))})$ for infinitely many $ξ\in Q$, yet, for every $ε\in (0,1)$, are eventually ``cleared out'' from the slightly smaller neighborhoods $B(ξ,{(1-ε)ϕ(R(ξ))})$, that is, they lie outside all but finitely many of these smaller balls.
We give sufficient conditions (also necessary under mild assumptions) for $E(Q, R, ϕ)$ to have infinite Hausdorff $f$-measure. This setting generalizes both the classical set $\mathrm{Exact}(ψ)$ of exactly $ψ$-approximable points (with $ψ$ non-increasing) and certain types of restricted Diophantine approximation sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02492 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff measures of sets in Exact Diophantine approximation Tan, Bo Tian, Chen Wang, Baowei Wu, Jun Number Theory Dynamical Systems 28A80, 11K55, 11J83 Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $ϕ: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, ϕ)$ of points $x \in X$ that ``hit'' the shrinking balls $B(ξ,{ϕ(R(ξ))})$ for infinitely many $ξ\in Q$, yet, for every $ε\in (0,1)$, are eventually ``cleared out'' from the slightly smaller neighborhoods $B(ξ,{(1-ε)ϕ(R(ξ))})$, that is, they lie outside all but finitely many of these smaller balls. We give sufficient conditions (also necessary under mild assumptions) for $E(Q, R, ϕ)$ to have infinite Hausdorff $f$-measure. This setting generalizes both the classical set $\mathrm{Exact}(ψ)$ of exactly $ψ$-approximable points (with $ψ$ non-increasing) and certain types of restricted Diophantine approximation sets. |
| title | Hausdorff measures of sets in Exact Diophantine approximation |
| topic | Number Theory Dynamical Systems 28A80, 11K55, 11J83 |
| url | https://arxiv.org/abs/2511.02492 |