In Beltrán and Etayo (J. Complexity 59, # 101471 (2020)) the authors presented a family of points on the sphere \(\mathbb {S}^{2}\) depending on many parameters, called the Diamond ensemble. In this paper we compute the spherical cap discrepancy of the Diamond ensemble as well as some other quantities. We also define an area regular partition on the sphere where each region contains exactly one point of the set. For a concrete choice of parameters, we prove that the Diamond ensemble provides the best spherical cap discrepancy, known until now for a deterministic family of points.

中文翻译：

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钻石组合的球冠差异

在Beltrán和Etayo（J. Complexity 59，＃101471（2020））中，作者根据称为戴蒙德系综的许多参数在球面上（\ mathbb {S} ^ {2} \）提出了一系列点。在本文中，我们计算了Diamond合奏的球冠差异以及其他一些数量。我们还在球体上定义了一个区域规则分区，其中每个区域恰好包含集合中的一个点。对于具体的参数选择，我们证明了Diamond集合提供了最佳的球冠差异，到目前为止，该球冠差异是确定性的点族。