Polarization and covering on sets of low smoothness

TL;DR

This paper investigates the asymptotic behavior of optimal point configurations for covering and polarization on non-smooth sets, including fractals and rectifiable sets, establishing their existence and properties.

Contribution

It provides the first rigorous proof of asymptotics for best covering and maximal polarization on low-smoothness sets like fractals.

Findings

Existence of asymptotics for best covering configurations.

Existence of asymptotics for maximal polarization.

Applicable to fractal and rectifiable sets.

Abstract

In this paper we study the asymptotic properties of point configurations that achieve optimal covering of sets lacking smoothness. Our results include the proofs of the existence of asymptotics of best covering and maximal polarization for $d$-rectifiable sets and classical fractal sets.