An Inequality for Gaussians on Lattices

Source abstract

We show that for any lattice $\mathcal{L} \subseteq \mathbb{R}^n$ and vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$, $\rho(\mathcal{L} + \mathbf{x})^2 \rho(\mathcal{L} + \mathbf{y})^2 \leq \rho(\mathcal{L})^2 \rho(\mathcal{L} + \mathbf{x} + \mathbf{y}) \rho(\mathcal{L} + \mathbf{x} - \mathbf{y}),$ where $\rho$ is the Gaussian mass function $\rho(A) := \sum_{\mathbf{w} \in A} \exp(-\pi \lVert\mathbf{w}\rVert^2)$. 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.