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arXiv · 2607.08049

A set of points on the sphere with small Riesz energy

Stefan Steinerberger
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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.

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Stefan Steinerberger. 2026-07-09. A set of points on the sphere with small Riesz energy. https://arxiv.org/abs/2607.08049

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