Title: Applications of a novel Bombieri-Siegel covariogram identity, to the Geometry of Numbers
Data: 25 de julho de 2024 – 5a feira – 11h - Auditório Jacy Monteiro, Bloco B do IME
Student: Michel Faleiros Martins

Abstract: We explore adaptations of the classical well-established conditions for applying the Poisson Summation Formula to derive a variant suitable for continuous functions of compact support. This culminates in a refined Bombieri-Siegel formula, which we leverage to develop lattice sums of the cross-covariogram for any two bounded sets $A,B \subset \R^d$. As an application of this refined Bombieri-Siegel formula, we present a new characterization of multi-tilings of Euclidean space by translations of a compact set using a lattice. A further consequence is a spectral formula for the volume of any bounded measurable set. We also apply the newly derived identity for cross-covariograms and Fourier transforms to problems related to counting lattice points inside a body and problems in continuous and discrete multi-tiling. For instance, given a finite subset $F$ of integer points in $\Z^d$, it is of interest to identify conditions on $F$ that enable it to multi-tile $\Z^d$ by translations. Similar questions pertaining to convex bodies have been extensively investigated. Specifically, we provide a discretized version of the Bombieri-Siegel formula, which entails a finite sum of discrete covariograms taken over any finite set of integer points in $\R^d$. As a result, we establish a new equivalent condition for multi-tiling $\Z^d$ by translating $F$ with a fixed integer sublattice. Additionally, we explore two additional topics. Namely, what are conditions under which a union of sublattice translates can multi-tile $\R^d$? And finally, we give an equivalence between the Minkowski Conjecture concerning linear forms, and the Fourier-Laplace transforms of cones.