The covariogram and an extension of Siegel's formula

Abstract: We extend a formula of Carl Ludwig Siegel in the geometry of numbers. Siegel's original formula assumed that there is exactly one lattice point in the interior of the body, while here we relax that condition, so that the body may contain an arbitrary number of interior lattice points. Our extension involves a lattice sum of the covariogram for any compact set $\mathcal K \subset \mathbb{R}^d$, where the covariogram of $\mathcal K$ at $x \in \mathbb R^d$ is defined by $\rm{vol}$$( \mathcal K \cap (\mathcal K + x))$. The proof hinges on a variation of the Poisson summation formula which we derive here, and the Fourier methods herein also allow for more general admissible sets. One of the consequences of these results is a new characterization of multi-tilings of Euclidean space by translations, using the lower bound on lattice sums of such covariograms. The classical result known as Van der Corput's inequality, also follows immediately from the main result, as well as a new spectral formula for the volume of a compact set.

This is joint work with Michel Faleiros Martins.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)