The covariogram and extensions of the Bombieri-Siegel formula

Abstract: We extend a formula of C. L. Siegel in the geometry of numbers, allowing the body to contain an arbitrary number of interior lattice points. Our extension involves a lattice sum of the cross covariogram for any two bounded sets $A, B\subseteq \mathbb R^d$, and turns out to also extend a result of E. Bombieri. We begin with a new variation of the Poisson summation formula, which may be of independent interest. One of the consequences of these results is a new characterization of multitilings of Euclidean space by translations, which is an application of Bombieri’s identity and of our extension of it. Some classical results, such as Van der Corput’s inequality, and Hardy’s identity for the Gauss circle problem, also follow as corollaries. This is joint work with Michel Faleiros Martins.

classical analysis and ODEsfunctional analysisrepresentation theoryspectral theory

Audience: researchers in the topic

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