An Inequality for Gaussians on Lattices

We show that for any lattice L ⊆ Rn and vectors x, y ∈ Rn, ρ(L+ x)2ρ(L+ y)2 ≤ ρ(L)2ρ(L+ x + y)ρ(L+ x− y) , where ρ is the Gaussian mass function ρ(A) := ∑w∈A exp(−π‖w‖ ). 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.