Asymptotics for the minimum Riesz energy and best-packing on sets of finite packing premeasure

Abstract

We show that for every compact set $A\subset {\mathbb R}^m$ of finite $\alpha$-dimensional packing premeasure $0<\alpha\leq m$, the lower limit of the normalized discrete minimum Riesz $s$-energy ($s>\alpha$) coincides with the outer measure of $A$ constructed from this limit by method I. The asymptotic behavior of the discrete minimum energy on compact subsets of a self-similar set $K$ satisfying the open set condition is also studied for $s$ greater than the Hausdorff dimension of $K$. In addition, similar problems are studied for the best-packing radius.