2022/04/19 by Michel Faleiros Martins, Martins, Michel Faleiros, Sinai Robins +1
Computer Science · Mathematics · #11H06 #11P21 #52C07 #52C22 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Point processes and geometric inequalities #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2204.08606
openalex publication_date 2022/04/19 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28
Given a finite subset F of integer points in \mathbb Zd, it is of interest to seek conditions on F that allow it to multi-tile \mathbb Zd 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 Zd 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⊂ \mathbb Rd, 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.