# An Inequality for Gaussians on Lattices

@article{Regev2017AnIF, title={An Inequality for Gaussians on Lattices}, author={Oded Regev and Noah Stephens-Davidowitz}, journal={SIAM J. Discret. Math.}, year={2017}, volume={31}, pages={749-757} }

$ \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 $\vec{x}, \vec{y} \in \R^n$, \[ \rho(\lat + \vec{x})^2 \rho(\lat + \vec{y})^2 \leq \rho(\lat)^2 \rho(\lat + \vec{x} + \vec{y}) \rho(\lat + \vec{x} - \vec{y}) \; , \] where $\rho$ is the Gaussian measure $\rho(A) := \sum_{\vec{w} \in A} \exp(-\pi \| \vec{w} \|^2)$. We show a number of applications, including bounds on the…

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