THE COVARIOGRAM AND AN EXTENSION OF SIEGEL’S FORMULA MICHEL FALEIROS MARTINS AND SINAI ROBINS 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 covariogram of any compact set K ⊂ R d . The Fourier methods herein also allow for more general admissible sets, due to the Poisson summation formula. As one of the consequences of these results, we obtain a new characterization of multi-tilings of Euclidean space by translations, using the lower bound on lattice sums of such covariograms. Contents 1. Introduction 1 2. Preliminaries 6 3. A nonconvex polygon that multi-tiles with multiplicity 2 7 4. A nonconvex polygon that does not multi-tile 9 5. Lemmas 10 6. Proofs of Theorem 1 and Theorem 2 12 7. Proof of Corollary 1 15 8. Proofs of Corollaries 2 and 4 16 9. More examples 17 10. Further remarks 22 References 22 1. Introduction We extend some of Siegel’s results from the Geometry of numbers by studying the covariogram of a compact set K, defined by (1) g K (x) := vol(K ∩ (K + x)), defined for all x ∈ R d . We define a body to be any compact subset of R d , following the standard convention of convex geometry. Given any full-rank lattice L⊂ R d , we study the following sum of the covariagram g K over the lattice L: (2)  n∈L g K (n) :=  n∈L vol(K ∩ (K + n)), 1 arXiv:2204.08606v1 [math.NT] 19 Apr 2022