Approximation to uniform distribution in SO(3)

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 L2-norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interestCarlos Beltrán was supported by the Spanish “Ministerio de Economía y Competitividad” under projects MTM2017-83816-P and MTM2017-90682-REDT (Red ALAMA), as well as by the Banco Santander and Universidad de Cantabria under project 21.SI01.6465