AN INEQUALITY FOR GAUSSIANS ON LATTICES

We show that for any lattice L subset of R-n and vectors x, y is an element of R-n, rho(L + x)(2)rho(L + x)(3) <= p(r)2 rho(L + x + y)rho(L + x - y), where rho is the Gaussian mass function rho(A) := Sigma(w is an element of A) eXP(-pi parallel to W parallel to(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.