The Ekström–Persson conjecture regarding random covering sets

Abstract

We consider the Hausdorff dimension of random covering sets formed by balls with centres chosen independently at random according to an arbitrary Borel probability measure on $R^d$ and radii given by a deterministic sequence tending to zero. We prove, for a certain parameter range, the conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii ${\left(n^{-\alpha }\right)}_{n=1}^{\infty }$. For balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström–Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.