Dimensions of random covering sets in Riemann manifolds

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.