Universally Optimal Periodic Configurations in the Plane

TL;DR

This paper establishes universal optimality of certain lattice configurations in the plane by developing lower bounds for energy and employing polynomial interpolation, demonstrating optimality among specific periodic configurations.

Contribution

It introduces a framework for deriving sharp energy bounds for periodic configurations and proves universal optimality of scaled and rotated $A_2$ lattices with specific point additions.

Findings

Scaling of $A_2$ is universally optimal among certain 4-point configurations.

A scaled and rotated $A_2$ is universally optimal among certain 6-point configurations.

Framework uses polynomial interpolation to establish sharp bounds.

Abstract

We develop lower bounds for the energy of configurations in $\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial interpolation problem. We use this framework to show a scaling of the equitriangular lattice $A_2$ is universally optimal among all configurations of the form $\omega_4+ A_2$ where $\omega_4$ is a 4-point configuration in $\mathbb{R}^2$. Likewise, we show a scaling and rotation of $A_2$ is universally optimal among all configurations of the form $\omega_6+L$ where $\omega_6$ is a 6-point configuration in $\mathbb{R}^2$ and $L=\mathbb{Z} \times \sqrt{3} \mathbb{Z}$.