Discrepancy of Minimal Riesz Energy Points

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the sphere \(\mathbb S^d.\) Our results are based on bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in \(\mathbb S^2\) were obtained.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 \(s\ne 1\) 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 \(s\ne d-1.\)