Ekström-Persson conjecture regarding random covering sets

2024/02/28 by Esa Järvenpää, Maarit Järvenpää, Järvenpää, Esa +5 · 1 citation
Computer Science · #Rough Sets and Fuzzy Logic #Computational Geometry and Mesh Generation

paper · pdf · doi:10.48550/arxiv.2402.18289

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^∞. 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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• Dynamical covering sets in self-similar sets

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