Point sets on the sphere $\mathbb{S}^2$ with small spherical cap discrepancy

TL;DR

This paper analyzes the discrepancy of various explicit point sets on the sphere, showing they achieve an order of convergence suitable for numerical integration, using an area-preserving Lambert map for analysis.

Contribution

It provides a detailed analysis of the spherical cap discrepancy for random, digital net, and Fibonacci lattice point sets on the sphere, with convergence order established.

Findings

Discrepancy converges at order N^{-1/2} for the studied point sets.

The Lambert map is used to analyze the discrepancy via pre-image level sets.

Results support the use of these point sets in numerical integration on the sphere.

Abstract

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order $N^{-1/2}$. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given.