Spherical cap discrepancy of the Diamond ensemble

TL;DR

This paper analyzes the spherical cap discrepancy of the Diamond ensemble, a deterministic set of evenly distributed points on the sphere, demonstrating it achieves the best known discrepancy for such configurations.

Contribution

It introduces the Diamond ensemble and proves it has optimal spherical cap discrepancy among deterministic point sets for certain parameters.

Findings

Diamond ensemble has the lowest known spherical cap discrepancy among deterministic points.

Defined an area regular partition on the sphere aligned with the Diamond ensemble.

Proved optimal discrepancy results for specific parameter choices.

Abstract

We compute the spherical cap discrepancy of the Diamond ensemble (a set of evenly distributed spherical points) 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 Diamond ensemble. For a concrete choice of parameters, we prove that the Diamond ensemble provides the best spherical cap discrepancy known until date for a deterministic family of points.