Riesz energy on self-similar sets

TL;DR

This paper studies the behavior of minimal Riesz energy configurations on fractal sets, revealing nonexistence of certain asymptotics for high s and providing new insights into their asymptotic properties.

Contribution

It proves the nonexistence of asymptotic minimal Riesz energy for fractals when s exceeds the set dimension and offers a simplified proof for the asymptotic behavior of minimal energy configurations.

Findings

Minimal Riesz energy asymptotics do not exist for s > set dimension.

Asymptotics exist over subsequences of N.

Simplified proof of minimal energy configuration behavior.

Abstract

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 configurations 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 configurations that minimize the discrete Riesz $s$-energy.