Asymptotics for Minimal Discrete Riesz Energy on Curves in ℝd

Abstract

AbstractWe consider the s-energy for point sets 𝒵 = {𝒵k,n: k = 0, …, n} on certain compact sets Γ in ℝd having finite one-dimensional Hausdorff measure,is the Riesz kernel. Asymptotics for the minimum s-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for s ≥ 1, the minimizing nodes for a rectifiable Jordan curve Γ distribute asymptotically uniformly with respect to arclength as n → ∞.