Discrete & Computational Geometry

, Volume 58, Issue 3, pp 596–613 | Cite as

A Minkowski Theorem for Quasicrystals



The aim of this paper is to generalize Minkowski’s theorem. This theorem is usually stated for a centrally symmetric convex body and a lattice both included in \(\mathbf {R}^n\). In some situations, one may replace the lattice by a more general set for which a notion of density exists. In this paper, we prove a Minkowski theorem for quasicrystals, which bounds from below the frequency of differences appearing in the quasicrystal and belonging to a centrally symmetric convex body. The last part of the paper is devoted to quite natural applications of this theorem to Diophantine approximation and to discretization of linear maps.


Minkowski theorem Quasicrystals Diophantine approximation Discretization 

Mathematics Subject Classification

05A20 11B05 52C23 11H06 


  1. 1.
    Berger, M.: Geometry Revealed. A Jacob’s Ladder to Modern Higher Geometry. Springer, Heidelberg (2010)MATHGoogle Scholar
  2. 2.
    Cassels, J.W.S.: An Introduction to the Geometry of Numbers. Classics in Mathematics. Springer, Berlin (1997)MATHGoogle Scholar
  3. 3.
    de Bruijn, N.G.: Algebraic theory of Penrose’s nonperiodic tilings of the plane. I, II. Nederl. Akad. Wetensch. Indag. Math. 43(1), 39–52, 53–66 (1981)Google Scholar
  4. 4.
    Gruber, P.M., Lekkerkerker, C.G.: Geometry of Numbers. North-Holland Mathematical Library, vol. 37, 2nd edn. North-Holland, Amsterdam (1987)MATHGoogle Scholar
  5. 5.
    Guihéneuf, P.-A.: Model sets, almost periodic patterns, uniform density and linear maps. Rev. Mat. Iberoam. arXiv:1512.00650 (2015, to appear)
  6. 6.
    Guihéneuf, P.-A.: Discretizations of isometries. In: Normand, N., Guédon, J., Autrusseau, F. (eds.) Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 9647, pp. 71–92. Springer, Cham (2016)CrossRefGoogle Scholar
  7. 7.
    Henk, M.: Successive minima and lattice points. Rend. Circ. Mat. Palermo (2) Suppl. 70(part I), 377–384 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Lagarias, J.C.: Meyer’s concept of quasicrystal and quasiregular sets. Commun. Math. Phys. 179(2), 365–376 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Marklof, J., Strömbergsson, A.: Visibility and directions in quasicrystals. Int. Math. Res. Not. IMRN 2015(15), 6588–6617 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Meyer, Y.: Algebraic Numbers and Harmonic Analysis. North-Holland Mathematical Library, vol. 2. North-Holland, New York (1972)Google Scholar
  11. 11.
    Meyer, Y.: Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling. Afr. Diaspora J. Math. 13(1), 1–45 (2012)MathSciNetMATHGoogle Scholar
  12. 12.
    Minkowski, H.: Geometrie der Zahlen, vol. 1. B.G. Teubner, Leipzig (1910)MATHGoogle Scholar
  13. 13.
    Moody, R.V.: Model sets: A survey. In: Axel, F., Dénoyer, F., Gazeau, J.-P. (eds.) From Quasicrystals to More Complex Systems. Centre de Physique des Houches, vol. 13, pp. 145–166. Springer, Heidelberg (2000)Google Scholar
  14. 14.
    Moody, R.V.: Uniform distribution in model sets. Can. Math. Bull. 45(1), 123–130 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Siegel, C.L.: Lectures on the Geometry of Numbers. Springer, Berlin (1989)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade Federal FluminenseNiteroiBrazil
  2. 2.IMJ-PRG, 4 place Jussieu, case 247Paris Cedex 05France
  3. 3.Modal’X, Bureau E08, Bâtiment GUniversité Paris OuestNanterreFrance

Personalised recommendations