The Ekström–Persson conjecture regarding random covering sets
Järvenpää, Esa; Järvenpää, Maarit; Myllyoja, Markus; Stenflo, Örjan (2024-12-20)
London mathematical society
20.12.2024
Järvenpää, E., Järvenpää, M., Myllyoja, M. and Stenflo, Ö. (2025), The Ekström–Persson conjecture regarding random covering sets. J. London Math. Soc., 111: e70058. https://doi.org/10.1112/jlms.70058
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© 2024 The Author(s). Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
https://creativecommons.org/licenses/by/4.0/
© 2024 The Author(s). Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
https://creativecommons.org/licenses/by/4.0/
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202501151173
https://urn.fi/URN:NBN:fi:oulu-202501151173
Tiivistelmä
Abstract
We consider the Hausdorff dimension of random covering sets formed by balls with centres chosen independently at random according to an arbitrary Borel probability measure on ℝ𝑑 and radii given by a deterministic sequence tending to zero. We prove, for a certain parameter range, the conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii (𝑛−𝛼)∞𝑛=1. For 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.
We consider the Hausdorff dimension of random covering sets formed by balls with centres chosen independently at random according to an arbitrary Borel probability measure on ℝ𝑑 and radii given by a deterministic sequence tending to zero. We prove, for a certain parameter range, the conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii (𝑛−𝛼)∞𝑛=1. For 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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