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

Abstract

We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, 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 \[ 0\leq \frac{2^{α+1}}{α+2}N^2-\sum_{x,y\in\cP_N}|x-y|^α\le C_αN^{1-α/2}. \] This result shows that $\cP_N\subset\mathbb S^2$ 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 $\cP_N\subset\mathbb S^2$ has $L^2$ spherical cap discrepancy of asymptotically optimal order \[ D_{L^2}^C(\mathcal P_N)\asymp N^{-3/4}, \] making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set $\cP_N$ has Sobolev $H^{s}(\Sph)$ worst-case error of optimal order for all $1<s<2$.

19 pages, 1 figure