Dimensions of random covering sets in Riemann manifolds
Feng, De-Jun; Järvenpää, Esa; Järvenpää, Maarit; Suomala, Ville (2018-04-12)
Feng, De-Jun; Järvenpää, Esa; Järvenpää, Maarit; Suomala, Ville. Dimensions of random covering sets in Riemann manifolds. Ann. Probab. 46 (2018), no. 3, 1542--1596. doi:10.1214/17-AOP1210. https://projecteuclid.org/euclid.aop/1523520024
© Institute of Mathematical Statistics, 2018. Published in this repository with the kind permission of the publisher.
https://rightsstatements.org/vocab/InC/1.0/
https://urn.fi/URN:NBN:fi-fe201804176586
Tiivistelmä
Abstract
Let \(\mathbf{M}\), \(\mathbf{N}\) and \(\mathbf{K}\) be \(d\)-dimensional Riemann manifolds. Assume that \(\mathbf{A}:=(A_{n})_{n\in{\mathbb{N}}}\) is a sequence of Lebesgue measurable subsets of \(\mathbf{M}\) satisfying a necessary density condition and \({\mathbf{x}}:=(x_{n})_{n\in{\mathbb{N}}}\) is a sequence of independent random variables, which are distributed on \(\mathbf{K}\) according to a measure, which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets \({\mathbf{E}}({\mathbf{x}},{\mathbf{A}}):=\limsup_{n\to\infty}A_{n}(x_{n})\subset{\mathbf{N}}\). Here, \(A_{n}(x_{n})\) is a diffeomorphic image of \(A_{n}\) depending on \(x_{n}\). We also verify that the packing dimensions of \({\mathbf{E}}({\mathbf{x}},{\mathbf{A}})\) equal \(d\) almost surely.
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