An Inequality for Gaussians on Lattices

Oded Regev, Noah Stephens-Davidowitz

Published 2015-02-17Version 1

We show that for any lattice $L \subseteq R^n$ and vectors $x, y \in R^n$, \[ \rho(L+ x)^2 \rho(L + y)^2 \leq \rho(L)^2 \rho(L + x+y) \rho(L + x - y) \;, \] where $\rho$ is the Gaussian measure $\rho(A) := \sum_{w \in A} e^{-\pi ||w||^2}$. We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties, and a positive correlation inequality for Gaussian measures on lattices.