An Inequality for Gaussians on Lattices

Abstract

$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that for any lattice $\lat \subseteq \R^n$ and vectors $\vec{x},\vec{y}\in {\mathbb{R}}^n$, \[ ρ(\lat + \vec{x})^2 ρ(\lat + \vec{y})^2 \leq ρ(\lat)^2 ρ(\lat + \vec{x} + \vec{y}) ρ(\lat + \vec{x} - \vec{y}) \; , \] where $\rho$ is the Gaussian measure $\rho (A):={\sum }_{\vec{w}\in A}\exp (-\pi \Vert \vec{w}{\Vert }^2)$. We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties of the heat kernel on flat tori, and a positive correlation inequality for Gaussian measures on lattices.

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