A set of points with optimal $L^2$ spherical cap discrepancy

We introduce an explicit deterministic collection of $N$ spherical points, ${\text{cP}}_N\subset {\mathbb{S}}^2$, that we call the {\em deterministic Diamond points}. For every $0<\alpha <2$ we prove that there exists a constant $C_{\alpha }>0$ such that $$0\le \frac{2^{\alpha +1}}{\alpha +2}{N}^2-\sum _{x,y\in {\text{cP}}_N}\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }\le C_{\alpha }{N}^{1-\alpha \mathrm{/}2}\mathrm{.}$$ This result shows that ${\text{cP}}_N\subset {\mathbb{S}}^2$ has a Riesz energy deficit of optimal order in this range of the parameter. In particular, for $\alpha =1$, Stolarsky’s invariance principle implies that ${\text{cP}}_N\subset {\mathbb{S}}^2$ has $L^2$ spherical cap discrepancy of asymptotically optimal order $$D_{L^2}^C({\mathcal{P}}_N)\asymp N^{-3\mathrm{/}4},$$ making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set ${\text{cP}}_N$ has Sobolev $H^s(\text{Sph})$ worst-case error of optimal order for all $1<s<2$.