Spherical cap discrepancy of perturbed lattices under the Lambert projection

TL;DR

This paper analyzes the spherical cap discrepancy of perturbed lattices mapped onto the sphere via Lambert projection, providing explicit bounds and demonstrating stability under perturbations, with implications for deterministic algorithms.

Contribution

It introduces explicit bounds for the spherical cap discrepancy of perturbed lattices under Lambert projection, including stability results and the smallest known constant for deterministic algorithms.

Findings

Discrepancy is at most of order N with explicit leading coefficient.

Bound remains stable under local lattice perturbations.

Achieves the smallest constant for cap discrepancy in deterministic algorithms.

Abstract

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.