Inventiones mathematicae

, Volume 200, Issue 2, pp 585–606 | Cite as

Quasicrystals and Poisson’s summation formula



We characterize the measures on \(\mathbb {R}\) which have both their support and spectrum uniformly discrete. A similar result is obtained in \(\mathbb {R}^n\) for positive measures.


  1. 1.
    Allouche, J.-P., Meyer, Y.: Quasicrystals, model sets, and automatic sequences. C. R. Phys. 15, 6–11 (2014)CrossRefGoogle Scholar
  2. 2.
    Baake, M., Lenz, D., Moody, R.: Characterization of model sets by dynamical systems. Ergod. Theory Dyn. Syst. 27, 341–382 (2007)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bombieri, E., Taylor, J. E.: Quasicrystals, tilings, and algebraic number theory: some preliminary connections. The legacy of Sonya Kovalevskaya. Contemp. Math. 64, 241–264 (1987) (American Mathematical Society, Providence, RI)Google Scholar
  4. 4.
    Cahn, J. W., Taylor, J. E.: An introduction to quasicrystals. The legacy of Sonya Kovalevskaya. Contemp. Math. 64, 265–286 (1987) (American Mathematical Society, Providence, RI)Google Scholar
  5. 5.
    Córdoba, A.: La formule sommatoire de Poisson. C. R. Acad. Sci. Paris Sér. I Math. 306, 373–376 (1988)Google Scholar
  6. 6.
    Córdoba, A.: Dirac combs. Lett. Math. Phys. 17, 191–196 (1989)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Dyson, F.: Birds and frogs. Notices Am. Math. Soc. 56, 212–223 (2009)MATHMathSciNetGoogle Scholar
  8. 8.
    Guinand, A.P.: Concordance and the harmonic analysis of sequences. Acta Math. 101, 235–271 (1959)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Gurarii, V. P.: Group methods of commutative harmonic analysis. Current problems in mathematics. In: Fundamental Directions, Vol. 25 (Russian). Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (English translation in Commutative harmonic analysis II, edited by V. P. Havin and N. K. Nikolski, Springer-Verlag, Berlin, 1998)Google Scholar
  10. 10.
    Kahane, J.-P., Mandelbrojt, S.: Sur l’équation fonctionnelle de Riemann et la formule sommatoire de Poisson. Ann. Sci. École Norm. Sup. 75, 57–80 (1958)MATHMathSciNetGoogle Scholar
  11. 11.
    Kolountzakis, M.N., Lagarias, J.C.: Structure of tilings of the line by a function. Duke Math. J. 82, 653–678 (1996)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Lagarias, J.C.: Meyer’s concept of quasicrystal and quasiregular sets. Commun. Math. Phys. 179, 365–376 (1996)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Lagarias, J. C.: Mathematical quasicrystals and the problem of diffraction. Directions in mathematical quasicrystals. In: CRM Monograph Series, vol. 13, pp. 61–93. American Mathematical Society, Providence (2000)Google Scholar
  14. 14.
    Landau, H.J.: Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117, 37–52 (1967)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Lev, N., Olevskii, A.: Measures with uniformly discrete support and spectrum. C. R. Math. Acad. Sci. Paris 351, 613–617 (2013)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Matei, B., Meyer, Y.: Simple quasicrystals are sets of stable sampling. Complex Var. Elliptic Equ. 55, 947–964 (2010)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Meyer, Y.: Nombres de Pisot, nombres de Salem et analyse harmonique. In: Lecture Notes in Mathematics, vol. 117. Springer-Verlag, New York (1970)Google Scholar
  18. 18.
    Meyer, Y.: Algebraic Numbers and Harmonic Analysis. North-Holland, Amsterdam (1972)MATHGoogle Scholar
  19. 19.
    Meyer, Y.: Quasicrystals, diophantine approximation and algebraic numbers. In: Beyond Quasicrystals (Les Houches, 1994), pp. 3–16. Springer, Berlin (1995)Google Scholar
  20. 20.
    Mitkovski, M., Poltoratski, A.: Pólya sequences, Toeplitz kernels and gap theorems. Adv. Math. 224, 1057–1070 (2010)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Moody, R. V.: Meyer sets and their duals. The mathematics of long-range aperiodic order (Waterloo, ON, 1995). In: NATO Advanced Science Institute Series C: Mathematical and Physical Sciences, vol. 489, pp. 403–441. Kluwer Academic Publishers, Dordrecht (1997)Google Scholar
  22. 22.
    Nitzan, S., Olevskii, A.: Revisiting Landau’s density theorems for Paley–Wiener spaces. C. R. Math. Acad. Sci. Paris 350, 509–512 (2012)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Olevskii, A., Ulanovskii, A.: Universal sampling and interpolation of band-limited signals. Geom. Funct. Anal. 18, 1029–1052 (2008)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Olevskii, A., Ulanovskii, A.: On multi-dimensional sampling and interpolation. Anal. Math. Phys. 2, 149–170 (2012)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)MATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael
  2. 2.School of Mathematical SciencesTel-Aviv UniversityTel AvivIsrael

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