# Polarization and Greedy Energy on the Sphere

## TL;DR

This paper studies a greedy algorithm for placing points on a sphere to minimize Riesz energy, showing it achieves near-optimal energy behavior and providing bounds on discrepancy, with numerical evidence of high uniformity.

## Contribution

It proves that greedy sequences on the sphere attain optimal second-order Riesz energy behavior for 0<s<d and relates polarization to energy and discrepancy.

## Key findings

- Greedy sequences achieve optimal second-order Riesz energy for 0<s<d.
- Second-order polarization term is of order N^{s/d} for 0<s<d.
- Upper bounds on L^2 spherical cap discrepancy are established, with numerical evidence of low discrepancy.

## Abstract

We investigate the behavior of a greedy sequence on the sphere $\mathbb{S}^d$ defined so that at each step the point that minimizes the Riesz $s$-energy is added to the existing set of points. We show that for $0<s<d$, the greedy sequence achieves optimal second-order behavior for the Riesz $s$-energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz $s$-kernels is of order $N^{s/d}$ in the same range $0<s<d$. Furthermore, using the Stolarsky principle relating the $L^2$-discrepancy of a point set with the pairwise sum of distances (Riesz energy with $s=-1$), we also obtain a simple upper bound on the $L^2$-spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.