An Extremal Property of the Square Lattice

Abstract

\nI{Motivated} by a 2019 result of Faulhuber-Steinerberger \cite{extremal} on the hexagonal lattice $\Lambda$, we demonstrate that the square lattice $\Z^2$ exhibits the same local extremal property as $\Lambda$, where distances of lattice points from the barycenters of natural fundamental domains increase under perturbation. These two lattices are very special lattices in $\R^2$, as they have nontrivial symmetries. Precisely, we show the following: let $p = (1/2, 1/2)$ denote the center of the standard square fundamental domain $[0, 1]^2$ for $\Z^2$ acting on $\R^2$, and let $A_r$ denote the set of lattice points that are at distance exactly $r$ from $p$. If $\Delta$ is a small perturbation of $\Z^2$ in the space of unimodular lattices, consider $C_r$, the set of points in $A_r$ shifted to $\Delta$. Then,\begin{equation} \sum_{\delta \in C_r}{ \| p - \delta\|} - \sum_{\lambda \in A_r}{ \| p - z\|} \geq r \, |A_r| \, d(\Delta, \Z^2)^2, \end{equation} where $d(\Delta, \Z^2)$ denotes the distance between the lattices, measured by, for example, the distances between basis vectors of $\Delta$ and those of $\Z^2$. As mentioned above, this says that the distances of lattice points from the barycenter of the fundamental domain strictly increase under perturbation, and we give an explicit bound for the minimum increase. Further, we conjecture that many higher-dimensional symmetric lattices will exhibit similar extremal properties.