Spherical cap discrepancy of perturbed lattices under the Lambert projection

Abstract

Given any full rank lattice Λ and a natural number N , we regard the point set Λ/N ∩ (0, 1) 2 under the Lambert map to the unit sphere, and show that its spherical cap discrepancy is at most of order N , with leading coefficient given explicitly and depending on Λ only. The proof is established using a lemma that bounds the amount of intersections of certain curves with fundamental domains that tile R 2 , and even allows for local perturbations of Λ without affecting the bound, proving to be stable for numerical applications. A special case yields the smallest constant for the leading term of the cap discrepancy for deterministic algorithms up to date.