  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.
