Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO(3), with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on SO(3). The variance of rotation matrices sampled by a certain determinantal point process is estimated, and formulas for the L 2 -norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interest.

Gegenbauer, also known as ultra-spherical, polynomials appear often in numerical analysis or interpolation. In the present text we find a recursive formula for and compute the asymptotic behavior of their $$L^2$$ L 2 -norm.

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.