Asymptotics of Discrete Riesz d-Polarization on Subsets of d-Dimensional Manifolds

Abstract

We prove a conjecture of T. Erdélyi and E.B. Saff, concerning the form of the dominant term (as N → ∞) of the N-point Riesz d-polarization constant for an infinite compact subset A of a d-dimensional C 1-manifold embedded in ℝ m (d ≤ m). Moreover, if we assume further that the d-dimensional Hausdorff measure of A is positive, we show that any asymptotically optimal sequence of N-point configurations for the N-point d-polarization problem on A is asymptotically uniformly distributed with respect to $\mathcal H_d|_A$ . These results also hold for finite unions of such sets A provided that their pairwise intersections have $\mathcal H_d$ -measure zero.