Ekström-Persson conjecture regarding random covering sets
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916141609254912 |
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| author | Järvenpää, Esa Järvenpää, Maarit Myllyoja, Markus Stenflo, Örjan |
| author_facet | Järvenpää, Esa Järvenpää, Maarit Myllyoja, Markus Stenflo, Örjan |
| contents | We consider the Hausdorff dimension of random covering sets generated by balls and general measures in Euclidean spaces. We prove, for a certain parameter range, a conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii $(n^{-α})_{n=1}^\infty$. For generating balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström-Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18289 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ekström-Persson conjecture regarding random covering sets Järvenpää, Esa Järvenpää, Maarit Myllyoja, Markus Stenflo, Örjan Classical Analysis and ODEs We consider the Hausdorff dimension of random covering sets generated by balls and general measures in Euclidean spaces. We prove, for a certain parameter range, a conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii $(n^{-α})_{n=1}^\infty$. For generating balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström-Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii. |
| title | Ekström-Persson conjecture regarding random covering sets |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2402.18289 |