Hitting Probabilities and the Ekstr{ö}m-Persson conjecture
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911001988825088 |
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| author | Järvenpää, Esa Myllyoja, Markus Seuret, Stéphane |
| author_facet | Järvenpää, Esa Myllyoja, Markus Seuret, Stéphane |
| contents | We consider the Ekst\''om-Persson conjecture concerning the value of the Hausdorff dimension of random covering sets formed by balls with radii $(k^{-α})_{k=1}^\infty$ and centres chosen independently at random according to an arbitrary Borel probability measure $μ$ on $\mathbb{R}^d$. The conjecture has been solved positively in the case $\frac 1α\le \overline{\dim}_H μ$, where $\overline{\dim}_H μ$ stands for the upper Hausdorff dimension of $μ$. In this paper, we develop a new approach in order to answer the full conjecture, proving in particular that the conjectured value is only a lower bound for the dimension. Our approach opens the way to study more general limsup sets, and has consequences on the so-called hitting probability questions. For instance, we are able to determine whether and what part of a deterministic analytic set can be hit by random covering sets formed by open sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10448 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hitting Probabilities and the Ekstr{ö}m-Persson conjecture Järvenpää, Esa Myllyoja, Markus Seuret, Stéphane Probability We consider the Ekst\''om-Persson conjecture concerning the value of the Hausdorff dimension of random covering sets formed by balls with radii $(k^{-α})_{k=1}^\infty$ and centres chosen independently at random according to an arbitrary Borel probability measure $μ$ on $\mathbb{R}^d$. The conjecture has been solved positively in the case $\frac 1α\le \overline{\dim}_H μ$, where $\overline{\dim}_H μ$ stands for the upper Hausdorff dimension of $μ$. In this paper, we develop a new approach in order to answer the full conjecture, proving in particular that the conjectured value is only a lower bound for the dimension. Our approach opens the way to study more general limsup sets, and has consequences on the so-called hitting probability questions. For instance, we are able to determine whether and what part of a deterministic analytic set can be hit by random covering sets formed by open sets. |
| title | Hitting Probabilities and the Ekstr{ö}m-Persson conjecture |
| topic | Probability |
| url | https://arxiv.org/abs/2506.10448 |