Dimensions of random covering sets in Riemann manifolds

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>Let 𝐌, 𝐍 and 𝐊 be <em>d</em>-dimensional Riemann manifolds. Assume that 𝐀 := (<em>A_n</em>)_{<em>n</em>∈ℕ} is a sequence of Lebesgue measurable subsets of 𝐌 satisfying a necessary density condition and 𝐱 := (<em>x_n</em>)_{<em>n</em>∈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 sup_{<em>n</em>→∞}<em>A_n</em>(<em>x_n</em>) ⊂ 𝐍. Here, <em>A_n</em>(<em>x_n</em>) is a diffeomorphic image of <em>A_n</em> depending on <em>x_n</em>. We also verify that the packing dimensions of 𝐄(𝐱, 𝐀) equal <em>d</em> almost surely.</p></div>