# Discrepancy of minimal Riesz energy points

## TL;DR

This paper establishes improved upper bounds for the spherical cap discrepancy of minimal Riesz energy points on spheres, extending previous results and utilizing Sobolev discrepancy bounds inspired by Wolff's work.

## Contribution

It provides new upper bounds for the discrepancy of Riesz energy minimizers on spheres, improving upon prior bounds for various ranges of s and dimensions.

## Key findings

- Improved bounds for spherical cap discrepancy when 0 ≤ s < 2 in S^2
- Enhanced discrepancy bounds for d - t_0 < s < d in higher dimensions
- Utilization of Sobolev discrepancy estimates from Wolff's unpublished work

## Abstract

We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz $s$-energy on the sphere $\mathbb S^d.$ Our results are based in bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in $\mathbb S^2$ were obtained. Our result improves previously known bounds for $0\le s<2$ and $s\neq 1$ in $\mathbb S^2,$ where $s=0$ is Wolff's result, and for $d-t_0<s<d$ with $t_0\approx 2.5$ when $d\ge 3$ and $s\neq d-1.$