A set of points on the sphere with small Riesz energy

Abstract

We construct a set of points $\left\{x_1,\dots ,x_n\right\}\subset {\mathbb{S}}^2$ such that $$\sum _{i\ne j}\frac{1}{\Vert x_i-x_j{\Vert }^2}\le \frac{n^2\log n}{4}+c{n}^2+O(n^{11/6}\log n),$$ where the constant $c\sim -0.085768\dots$ is given in closed form and matches the constant that was conjectured by Brauchart-Hardin-Saff to be optimal. The point set is motivated by the crystallization conjecture and consists of pieces of the hexagonal lattice projected onto the sphere in a tightly interlocked way.