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

Dimensions of random covering sets in Riemann manifolds

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Author: Feng, De-Jun1; Järvenpää, Esa2; Järvenpää, Maarit2;
Organizations: 1Chinese University of Hong Kong
2University of Oulu
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
Language: English
Published: Institute of Mathematical Statistics, 2018
Publish Date: 2018-04-17
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Abstract

Let 𝐌, 𝐍 and 𝐊 be d-dimensional Riemann manifolds. Assume that 𝐀 := (An)n∈ℕ is a sequence of Lebesgue measurable subsets of 𝐌 satisfying a necessary density condition and 𝐱 := (xn)n∈ℕ is a sequence of independent random variables, which are distributed on 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 𝐄(𝐱, 𝐀) := lim supn→∞An(xn) ⊂ 𝐍. Here, An(xn) is a diffeomorphic image of An depending on xn. We also verify that the packing dimensions of 𝐄(𝐱, 𝐀) equal d almost surely.

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Series: Annals of probability
ISSN: 0091-1798
ISSN-E: 2168-894X
ISSN-L: 0091-1798
Volume: 46
Issue: 3
Pages: 1542 - 1596
DOI: 10.1214/17-AOP1210