Open Access
May 2018 Dimensions of random covering sets in Riemann manifolds
De-Jun Feng, Esa Järvenpää, Maarit Järvenpää, Ville Suomala
Ann. Probab. 46(3): 1542-1596 (May 2018). DOI: 10.1214/17-AOP1210


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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De-Jun Feng. Esa Järvenpää. Maarit Järvenpää. Ville Suomala. "Dimensions of random covering sets in Riemann manifolds." Ann. Probab. 46 (3) 1542 - 1596, May 2018.


Received: 1 February 2016; Revised: 1 February 2017; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894781
MathSciNet: MR3785595
Digital Object Identifier: 10.1214/17-AOP1210

Primary: 28A80 , 60D05

Keywords: Hausdorff dimension , Packing dimension , Random covering set

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 3 • May 2018
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