Point sets on the sphere $\mathbb{S}^2$ with small spherical cap discrepancy
Provides theoretical convergence rates for explicit point sets on the sphere, benefiting numerical integration and computational simulations.
The paper proves that spherical cap discrepancy of random point sets, spherical digital nets, and spherical Fibonacci lattices converges with order N^{-1/2}, making them useful for numerical integration.
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.