Minimal Riesz energy on balanced fractal sets

TL;DR

This paper studies the asymptotic behavior of minimal Riesz energy configurations on fractal sets with non-integer dimensions, revealing explicit subsequence behaviors and the non-existence of a general asymptotic trend.

Contribution

It provides new insights into the asymptotic properties of Riesz energy on fractals with algebraically dependent contraction ratios, including explicit subsequence results.

Findings

Asymptotic behavior characterized along explicit subsequences

General asymptotic behavior does not exist for these fractals

Results apply to sets with non-integer, fractal dimensions

Abstract

We investigate the asymptotic behavior of minimal $N$-point Riesz $s$-energy on fractal sets of non-integer dimension, with algebraically dependent contraction ratios. For $s$ bigger than the dimension of the set $A$, we prove the asymptotic behavior of the minimal $N$-point Riesz $s$-energy of $A$ along explicit subsequences, but we show that the general asymptotic behavior does not exist.