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the sphere \(\mathbb S^d.\) Our results are based on bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in \(\mathbb S^2\) were obtained.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

我们找到球面上（\ mathbb S ^ d。\）的Riesz s-能量极小子集的球冠差异的上限。我们的结果基于托马斯·沃尔夫（Thomas Wolff）在未出版的Sobolev差异的界限在手稿中获得了对数能量极小值在\（\ mathbb S ^ 2 \）中的球形盖差异的估计值。我们的结果改善了\（\ mathbb S ^ 2，\）中\（0 \ le s <2 \）和\（s \ ne 1 \）的已知边界，其中\（s = 0 \）是Wolff的结果，并且为\（d-T_0 <S <d \）与\（T_0 \约2.5 \）时\（d \ GE 3 \）和\（s \ ne d-1。\）