Dimensions of random covering sets in Riemann manifolds

Abstract

Let $M$, $N$ and $K$ be $d$-dimensional Riemann manifolds. Assume that $\mathbf{A}:=(A_n{)}_{n\in \mathbb{N}}$ is a sequence of Lebesgue measurable subsets of $M$ 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 $K$ 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 $\mathbf{E}(\mathbf{x},\mathbf{A}):={\mathrm{limsup}}_{n\to \infty }{A}_n(x_n)\subset N$. 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.