Extremal Property of the Square Lattice

TL;DR

This paper proves that the square lattice $\

Contribution

It demonstrates that the square lattice shares the same local extremal property as the hexagonal lattice regarding distances from deep holes under perturbations.

Findings

Distances from deep holes increase under small lattice perturbations.

The growth of these distances is approximately preserved by convex functions.

The result extends the understanding of extremal properties of lattice structures.

Abstract

Motivated by a 2019 result of Faulhuber-Steinerberger, we demonstrate that the real square lattice $\mathbb{Z}^2$ exhibits the same local, extremal property as the hexagonal lattice $\Lambda$, where distances of lattice points from the `deep holes' of natural fundamental domains increase under perturbation. If $\Delta$ is a small perturbation of $\mathbb{Z}^2$ in the space of unimodular lattices, consider $C_r$, the set of points in $A_r$ shifted to $\Delta$. If $\Delta$ is a perturbation of the lattice $\mathbb{Z}^2$ with respect to the Euclidean metric, then for a fixed deep hole $p$, the summed total distance of lattice points to $p$ strictly increases, and is bounded below by a function of the distance between the lattice and its perturbation. Additionally, we show this growth is approximately preserved by convex functions.