Energy-minimizing measures supported near fractal 1-sets

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.