Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets

Abstract

Given a compact \(d\)-rectifiable set \(A\) embedded in Euclidean space and a distribution \(\rho(x)\) with respect to \(d\)-dimensional Hausdorff measure on \(A\), we address the following question: how can one generate optimal configurations of \(N\) points on \(A\) that are "well-separated" and have asymptotic distribution \(\rho (x)\) as \(N\to \infty\)? For this purpose we investigate minimal weighted Riesz energy points, that is, points interacting via the weighted power law potential \(V=w(x,y)|x-y|^{-s}\), where \(s>0\) is a fixed parameter and \(w\) is suitably chosen. In the unweighted case (\(w\equiv 1\)) such points for \(N\) fixed tend to the solution of the best-packing problem on \(A\) as the parameter \(s\to \infty\).