Asymptotics for Minimal Discrete Riesz Energy on Curves in ℝ<i>^d</i>

Abstract

Abstract We consider the s -energy for point sets 𝒵 = {𝒵 k,n : k = 0, …, n } on certain compact sets Γ in ℝ d 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 → ∞.