Abstract

Let 𝐌, 𝐍 and 𝐊 be d-dimensional Riemann manifolds. Assume that 𝐀 := (A_n)_n∈ℕ is a sequence of Lebesgue measurable subsets of 𝐌 satisfying a necessary density condition and 𝐱 := (x_n)_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 sup_n→∞A_n(x_n) ⊂ 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 𝐄(𝐱, 𝐀) equal d almost surely.