Distribution of point charges with small discrete energy

TL;DR

This paper investigates how points distribute themselves near compact sets in multi-dimensional space, using energy estimates for Riesz potentials, and provides quantitative convergence rates for discrete measures to equilibrium measures.

Contribution

It introduces new asymptotic and quantitative methods for analyzing the distribution of points with small discrete energy near compact sets.

Findings

Quantifies weak convergence of discrete measures to equilibrium measures.

Provides estimates of convergence rates for discrete potentials.

Analyzes distribution behavior in classical Newtonian case.

Abstract

We study the asymptotic equidistribution of points near arbitrary compact sets of positive capacity in $\R^d,\ d\ge 2$. Our main tools are the energy estimates for Riesz potentials. We also consider the quantitative aspects of this equidistribution in the classical Newtonian case. In particular, we quantify the weak convergence of discrete measures to the equilibrium measure, and give the estimates of convergence rates for discrete potentials to the equilibrium potential.