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- W4293794230 abstract "We extend the Bombieri-Siegel formula from the geometry of numbers, by studying a lattice sum of the cross covariogram for any two bounded sets $A,B subset mathbb R^d$. Using a variation of the Poisson summation formula, our extension also refines the summation index of the Bombieri-Siegel formula; this refinement turns out to be useful in applications. One of the consequences of these results is a new characterization of multi-tilings of Euclidean space by translations, as an application of Bombieri-Siegel formula, and of our extension. Another consequence is a spectral formula for the volume of any bounded measurable set. Finally, we give an application of the main results to arithmetic combinatorics, namely a formula for finite sums of discrete covariograms over any finite set of integer points in $mathbb R^d$. This is a discretized version of the Bombieri-Siegel formula. As a consequence, we arrive at a new equivalent condition for the multi-tiling $mathbb Z^d$ by a finite set of integer points." @default.
- W4293794230 created "2022-08-31" @default.
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- W4293794230 creator A5046774469 @default.
- W4293794230 date "2022-04-18" @default.
- W4293794230 modified "2023-09-26" @default.
- W4293794230 title "The covariogram and extensions of the Bombieri-Siegel formula" @default.
- W4293794230 doi "https://doi.org/10.48550/arxiv.2204.08606" @default.
- W4293794230 hasPublicationYear "2022" @default.
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