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 Sd. Our results are based on bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in S2 were obtained. Our result improves previously known bounds for 0 ≤ s< 2 and s≠ 1 in S2, where s= 0 is Wolff’s result, and for d- t< s< d with t≈ 2.5 when d≥ 3 and s≠ d- 1.