Asymptotic Linear Programming Lower Bounds for the Energy of Minimizing Riesz and Gauss Configurations

TL;DR

This paper develops asymptotic linear programming bounds for the minimal energy of large point configurations on spheres and Euclidean spaces, extending to Riesz and Gaussian potentials, with implications for energy minimization problems.

Contribution

It introduces new asymptotic linear programming bounds for Riesz and Gaussian energies, extending existing frameworks to the hypersingular case and infinite configurations.

Findings

Derived lower bounds for Riesz energy as N approaches infinity.

Established lower estimates for minimal energy on compact sets.

Provided bounds for Gaussian potential energy with prescribed density.

Abstract

Utilizing frameworks developed by Delsarte, Yudin and Levenshtein, we deduce linear programming lower bounds (as $N\to \infty$) for the Riesz energy of $N$-point configurations on the $d$-dimensional unit sphere in the so-called hypersingular case; i.e, for non-integrable Riesz kernels of the form $|x-y|^{-s}$ with $s>d.$ As a consequence, we immediately get (thanks to the Poppy-seed bagel theorem) lower estimates for the large $N$ limits of minimal hypersingular Riesz energy on compact $d$-rectifiable sets. Furthermore, for the Gaussian potential $\exp(-\alpha|x-y|^2)$ on $\mathbb{R}^p,$ we obtain lower bounds for the energy of infinite configurations having a prescribed density.