Riesz energy on self-similar sets

A. Reznikov, O. Vlasiuk

We investigate properties of minimal N-point Riesz s-energy on fractal sets of non-integer dimension, as well as asymptotic behavior of N-point con gurations that minimize this energy. For s bigger than the dimension of the set A, we constructively prove a negative result concerning the asymptotic behavior (namely, its nonexistence) of the minimal N-point Riesz s-energy of A, but we show that the asymptotic exists over reasonable sub-sequences of N. Furthermore, we give a short proof of a result concerning asymptotic behavior of con gurations that minimize the discrete Riesz s-energy.