Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels

TL;DR

This paper studies the asymptotic behavior of greedy energy sequences on spheres, showing they achieve optimal growth for Green and Riesz energies within a specific parameter range.

Contribution

It establishes bounds on polarization and demonstrates the optimal second-order growth behavior of greedy sequences for Riesz and Green energies.

Findings

Greedy sequences attain optimal second-order energy growth.

Bounds on polarization are established using well-separation properties.

Results hold for d-2 ≤ s < d.

Abstract

We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_{n}\}_{n=1}^{\infty}$ on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1, x_2, ..., x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2 \leq s < d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.