Hausdorff dimension and exact approximation order in $\mathbb{R}^n$
| Autor: | Bandi, Prasuna, de Saxcé, Nicolas |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Given a non-increasing function $\psi\colon\mathbb{N}\to\mathbb{R}^+$ such that $s^{\frac{n+1}{n}}\psi(s)$ tends to zero as $s$ goes to infinity, we show that the set of points in $\mathbb{R}^n$ that are exactly $\psi$-approximable is non-empty, and we compute its Hausdorff dimension. For $n\geq 2$, this answers questions of Jarn\'{i}k and of Beresnevich, Dickinson, and Velani. Comment: 20 pages, 4 figures |
| Databáze: | arXiv |
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