Asymptotics of discrete Riesz $d$-polarization on subsets of $d$-dimensional manifolds

TL;DR

This paper proves a conjecture about the asymptotic behavior of Riesz $d$-polarization constants on subsets of $d$-dimensional manifolds, showing the dominant term and distribution of optimal configurations.

Contribution

It establishes the asymptotic form of the Riesz $d$-polarization constant and the uniform distribution of optimal point configurations on manifolds.

Findings

Proved the conjecture on the dominant term of the Riesz $d$-polarization constant.

Showed asymptotic uniform distribution of optimal configurations.

Identified conditions under which the results hold, including positive Hausdorff measure.

Abstract

We prove a conjecture of T. Erd\'{e}lyi and E.B. Saff, concerning the form of the dominant term (as $N\to \infty$) of the $N$-point Riesz $d$-polarization constant for an infinite compact subset $A$ of a $d$-dimensional $C^{1}$-manifold embedded in $\mathbb{R}^{m}$ ($d\leq m$). 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 $\mathcal H_d|_A$.