Comparison of probabilistic and deterministic point sets

Abstract

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (spherical $t$-designs) are better or as good as probabilistic ones. We find asymptotic equalities for the discrete Riesz $s$-energy of sequences of well separated $t$-designs on the unit sphere ${\mathbb{S}}^d\subset {\mathbb{R}}^{d+1}$, $d\ge 2$. The case $d=2$ was studied Hesse and Leopardi. Bondarenko, Radchenko, and Viazovska established, that for $d\ge 2$, there exists a constant $c_d$, such that for every $N>c_d{t}^d$ there exists a well-separated spherical $t$-design on ${\mathbb{S}}^d$ with $N$ points. For this reason, in our paper we assume, that the sequence of well separated spherical $t$-designs is such that $t$ and $N$ are related by $N\asymp t^d$.