Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets

Given a closed d-rectifiable set A embedded in Euclidean space, we investigate minimal weighted Riesz energy points on A; that is, N points constrained to A and interacting via the weighted power law potential V = w(x,y)|x-y| -s ,where s > 0 is a fixed parameter and w is an admissible weight. (In the unweighted case (w ≡ 1) such points for N fixed tend to the solution of the best-packing problem on A as the parameter s →∞.) Our main results concern the asymptotic behavior as N → ∞ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, our results provide a method for generating N-point configurations on A that are "well-separated" and have asymptotic distribution p(x) as N → ∞.

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