Equidistribution of points via energy

TL;DR

This paper investigates how points distribute evenly on a plane set by analyzing their energy and potential theory, providing quantitative measures and applications to polynomial zero distributions.

Contribution

It introduces new energy estimates and quantitative methods for understanding equidistribution of points and polynomial zero behavior related to Robin's constant.

Findings

Quantitative bounds on equidistribution rates

Estimates for growth of Fekete and Leja polynomials

Convergence rates of discrete energy to Robin's constant

Abstract

We study the asymptotic equidistribution of points with discrete energy close to Robin's constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin's constant, and problems on the means of zeros of polynomials with integer coefficients.