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arXiv · 2402.18289

Ekström-Persson conjecture regarding random covering sets

Esa Järvenpää·Maarit Järvenpää·Markus Myllyoja·Örjan Stenflo
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Abstract

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.

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Esa Järvenpää, Maarit Järvenpää, Markus Myllyoja, Örjan Stenflo. 2024-02-28. Ekström-Persson conjecture regarding random covering sets. https://arxiv.org/abs/2402.18289

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