Sharp inequalities for discrete and continuous multi-tiling, using the Bombieri-Siegel approach

Given a finite subset $F$ of integer points in $\mathbb Z^d$, it is of interest to seek conditions on $F$ that allow it to multi-tile $\mathbb Z^d$ by translations. To this end, we give a discretized version of the Bombieri-Siegel formula, which represents a finite sum of discrete covariograms in terms of Fourier transforms. As a consequence, we arrive at a new equivalent condition for multi-tiling $\mathbb Z^d$ by translating $F$ with a fixed integer sublattice. In the continuous case, we study lattice sums of the cross covariogram for any two bounded sets $A, B\subset \mathbb R^d$, and we prove a refined continuous version of the classical Bombieri-Siegel formula from the geometry of numbers. To achieve this goal, we use a variant of the Poisson Summation formula, adapted for continuous functions of compact support. As an application of this refined Bombieri-Siegel formula, a new characterization of multi-tilings of Euclidean space by translations of a compact set by using a lattice is given. One consequence is a novel spectral formula for the volume of any bounded measurable set. Another consequence is a novel spectral formula for the product of the volumes of any two bounded measurable sets.