Comparison of probabilistic and deterministic point sets on the sphere

Abstract

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t-designs) are better or as good as probabilistic ones like the jittered sampling model.We find asymptotic equalities for the discrete Riesz s-energy of sequences of well separated t-designs on the unit sphere Sd⊂Rd+1, d≥2. The case d=2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d≥2, there exists a constant cd, such that for every N>cdtd there exists a well-separated spherical t-design on Sd with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t-designs such that t and N are related by N≍td.