Given any full rank lattice Λ in R^2 and a natural number N , we regard the point set Λ/N ∩ (0, 1)^2 under
the Lambert map to the unit sphere S^2 , 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
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