Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates

TL;DR

This paper investigates the distribution of weighted Fekete sets in the plane, establishing their maximal spread relative to Beurling--Landau density and connecting this to spectral properties and correlation kernel estimates of random matrix ensembles.

Contribution

It introduces a novel approach combining spectral analysis and kernel estimates to analyze the equidistribution of weighted Fekete sets in the droplet.

Findings

Fekete sets are maximally spread out relative to Beurling--Landau density.

The method links spectral properties of the concentration operator to set distribution.

Correlation kernel estimates support the equidistribution results.

Abstract

In this paper we discuss equidistribution results for weighted Fekete sets in subsets of the plane. More precisely, we show that Fekete sets are maximally spread out relative to a rescaled version of the Beurling--Landau density, in the "droplet" corresponding to the given weight. Our method combines Landau's idea to relate the density of a family of discrete sets to properties of the spectrum of the concentration operator, with estimates for the correlation kernel of the corresponding random normal matrix ensemble.