Applications of a novel Bombieri-Siegel covariogram identity in the Geometry of Numbers

• Ano de defesa: 2024
• Autor(a) principal: Martins, Michel Faleiros
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Covariogram; Covariograma; Fourier Transform; Lattice; Multi-ladrilhamento; Multi-tiling; Poisson Summation; Reticulado; Somatório de Poisson; Transformada de Fourier
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-23082024-105838/

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 conditions under which a union of sublattices translates can multi-tile \\R^d, and how to relate the Minkowski Conjecture about linear forms to Fourier transforms of cones.