Harmonic Analysis and Applications

June 4-8, 2018


"Bounds for Minimal Discrete Riesz and Gaussian Energy"

Saff, Edward Barry

Minimizing the potential energy of particles with a repulsive force between them that are confined to a manifold has significant applications to physics (determining ground states), coding theory (maximizing the separation between particles), and numerical methods (sampling for the purpose of quadrature or interpolation). 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 \emph{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.

« back