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 $$ \sum_{i \neq j} \frac{1}{\|x_i - x_j\|^2} \leq \frac{n^2 \log{n}}{4} + cn^2 + O(n^{11/6} \log{n}),$$ 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.