Asymptotics for minimal discrete riesz energy on curves in ℝ ^d

We consider the s-energy E(Z_n ; s) = ∑_≠j K(∥z_i,n - z_j,n∥ ; s) for point sets Z_n = {z_k,n: k = 0, ...,n} on certain compact sets Γ in ℝ ^d having finite one-dimensional Hausdorff measure, where K(t;s) = {t^-s, if s> 0, -lnt, if s = 0, is the Riesz kernel. Asyrnptotics 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 → ∞.