BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Sinai Robins (University of Sao Paulo\, Brazil) DTSTART:20220525T183000Z DTEND:20220525T185500Z DTSTAMP:20260423T011437Z UID:CANT2022/25 DESCRIPTION:Title: The covariogram and an extension of Siegel's formula\nby Sinai Robin s (University of Sao Paulo\, Brazil) as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAbstract\nWe extend a formula of Carl Ludw ig Siegel in the geometry of numbers.\nSiegel's original formula assumed t hat there is exactly one lattice point in the interior of the body\, while here\nwe relax that condition\, so that the body may contain an arbitrary number of interior lattice points. Our extension involves a lattice sum of the covariogram for any compact set $\\mathcal K \\subset \\mathbb{R}^ d$\, where the covariogram of $\\mathcal K$ at $x \\in \\mathbb R^d$ is defined by $\\rm{vol}$$( \\mathcal K \\cap (\\mathcal K + x))$. \nThe pr oof hinges on a variation of the Poisson summation formula which we derive here\, and the Fourier methods herein also allow for more general admissi ble sets. One of the consequences of these results is a new characterizat ion of multi-tilings of Euclidean space by translations\, using the lower bound on lattice sums of such covariograms. The classical result known as Van der Corput's inequality\, also follows immediately from the main resu lt\, as well as a new spectral formula for the volume of a compact set. \ n\nThis is joint work with Michel Faleiros Martins.\n LOCATION:https://researchseminars.org/talk/CANT2022/25/ END:VEVENT END:VCALENDAR