Energy-minimizing measures supported near fractal 1-sets

TL;DR

This paper investigates energy-minimizing measures near fractal 1-sets, revealing their distribution properties and limitations of energy methods in analyzing Favard length, using physical analogies and computational techniques.

Contribution

It introduces a novel approach to studying measures supported near fractal sets without requiring self-similarity, using a physical analogy and a variant of the fast multipole method.

Findings

Strong equidistribution results for energy-minimizing measures.

Sets with mild geometric constraints can be analyzed without self-similarity.

Demonstrates limitations of energy techniques for Favard length studies.

Abstract

Energy techniques can be used to study the structure of fractal sets; the existence of a measure with finite Riesz energy supported on a set gives information about its dimension, distribution, and density. In this paper, we study energy-minimizing measures supported near fractal $1$-sets. Using physical analogy and a variant of the fast multipole method, we show a strong equidistribution result for these measures. We impose only mild geometric constraints on our sets, assuming only a generational structure of the approximations. This allows us to consider sets which do not exhibit self-similarity or other algebraic constraints. As a corollary, we demonstrate a fundamental limitation in the use of energy techniques for studying Favard length.