Equality Case in van der Corput’s Inequality and Collisions in Multiple Lattice Tilings

  • Gennadiy AverkovEmail author


Van der Corput’s provides the sharp bound \(\mathop {\mathrm {vol}}\nolimits (C) \le m 2^d\) on the volume of a d-dimensional origin-symmetric convex body C that has \(2m-1\) points of the integer lattice in its interior. For \(m=1\), a characterization of the equality case \(\mathop {\mathrm {vol}}\nolimits (C)= m 2^d\) is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for \(m \ge 2\), no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit characterization of the equality case for all \(m \ge 2\). Our result reveals that, the equality case for \(m \ge 2\) is more restrictive than for \(m=1\). We also present consequences of our characterization in the context of multiple lattice tilings.


Lattice Multiple tiling Tiling Van der Corput’s inequality 

Mathematics Subject Classification

05B45 11H06 52C22 



Research supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—314838170, GRK 2297 MathCoRe.


  1. 1.
    Averkov, G.: Local optimality of Zaks–Perles–Wills simplices (2018). arXiv:1803.04852
  2. 2.
    Averkov, G., Krümpelmann, J., Nill, B.: Largest integral simplices with one interior integral point: solution of Hensley’s conjecture and related results. Adv. Math. 274, 118–166 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Averkov, G., Krümpelmann, J., Nill, B.: Lattice simplices with a fixed positive number of interior lattice points: A nearly optimal volume bound. Int. Math. Res. Not.
  4. 4.
    Balletti, G., Kasprzyk, A.M.: Three-dimensional lattice polytopes with two interior lattice points (2016). arXiv:1612.08918
  5. 5.
    Bolle, U.: On multiple tiles in \(E^2\). In: Böröczky, K., Fejes Tóth, G. (eds.) Intuitive Geometry (Szeged, 1991). Colloquia Mathematica Societatis János Bolyai, vol. 63, pp. 39–43. North-Holland, Amsterdam (1994)Google Scholar
  6. 6.
    Cassels, J.W.S.: An Introduction to the Geometry of Numbers. Classics in Mathematics. Springer, Berlin (1997). Corrected reprint of the 1971 editionzbMATHGoogle Scholar
  7. 7.
    Draisma, J., McAllister, T.B., Nill, B.: Lattice-width directions and Minkowski’s \(3^d\)-theorem. SIAM J. Discrete Math. 26(3), 1104–1107 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Freiman, G., Heppes, A., Uhrin, B.: A lower estimation for the cardinality of finite difference sets in \({\bf R}^n\). In: Györy, K., Halász, G. (eds.) Number Theory, Vol. I (Budapest, 1987). Colloquia Mathematica Societatis János Bolyai, pp. 125–139. North-Holland, Amsterdam (1990)Google Scholar
  9. 9.
    González Merino, B., Henze, M.: A generalization of the discrete version of Minkowski’s fundamental theorem. Mathematika 62(3), 637–652 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gravin, N., Robins, S., Shiryaev, D.: Translational tilings by a polytope, with multiplicity. Combinatorica 32(6), 629–649 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gruber, P.M.: Convex and Discrete Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 336. Springer, Berlin (2007)Google Scholar
  12. 12.
    Gruber, P.M., Lekkerkerker, C.G.: Geometry of Numbers. North-Holland Mathematical Library, vol. 37, 2nd edn. North-Holland, Amsterdam (1987)zbMATHGoogle Scholar
  13. 13.
    Hensley, D.: Lattice vertex polytopes with interior lattice points. Pac. J. Math. 105(1), 183–191 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lagarias, J.C., Ziegler, G.M.: Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Can. J. Math. 43(5), 1022–1035 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pikhurko, O.: Lattice points in lattice polytopes. Mathematika 48(1–2), 15–24 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151, expanded edn. Cambridge University Press, Cambridge (2014)Google Scholar
  17. 17.
    Uhrin, B.: On a generalization of Minkowski’s convex body theorem. J. Number Theory 13(2), 192–209 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yang, Q., Zong, C.: Multiple lattice tilings in Euclidean spaces (2017). arXiv:1710.05506
  19. 19.
    Yang, Q., Zong, C.: Multiple translative tilings in Euclidean spaces (2017). arXiv:1711.02514
  20. 20.
    Zong, C.: Characterization of the two-dimensional five-fold lattice tiles (2017). arXiv:1712.01122

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

Personalised recommendations