Applications of a novel Bombieri-Siegel covariogram identity in the Geometry of Numbers
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-covari...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | Brasil |
| Institución: | Universidade de São Paulo (USP) |
| Repositorio: | Biblioteca Digital de Teses e Dissertações da USP |
| Idioma: | inglés |
| OAI Identifier: | oai:teses.usp.br:tde-23082024-105838 |
| Acceso en línea: | https://www.teses.usp.br/teses/disponiveis/45/45131/tde-23082024-105838/ |
| Access Level: | acceso abierto |
| Palabra clave: | Covariogram Covariograma Fourier Transform Lattice Multi-ladrilhamento Multi-tiling Poisson Summation Reticulado Somatório de Poisson Transformada de Fourier |
| Sumario: | 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. |
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