Spherical cap discrepancy of perturbed lattices under the Lambert projection

Given any full rank lattice and a natural number N , we regard the point set given by the scaled lattice intersected with the unit square 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 the lattice 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 the lattice 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.