Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number

Jean-Pierre Gazeau[1]; Jean-Louis Verger-Gaugry[2]

  • [1] Université Paris 7-Denis Diderot APC - UMR CNRS 7164 Boite 7020 75251 Paris cedex 05 (France)
  • [2] Université Grenoble I Institut Fourier - UMR CNRS 5582 BP 74 38402 Saint-Martin d’Hères (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 7, page 2437-2461
  • ISSN: 0373-0956

Abstract

top
The Fourier transform of a weighted Dirac comb of beta-integers is characterized within the framework of the theory of Distributions, in particular its pure point part which corresponds to the Bragg part of the diffraction spectrum. The corresponding intensity function on this Bragg part is computed. We deduce the diffraction spectrum of weighted Delone sets on beta-lattices in the split case for the weight, when beta is the golden mean.

How to cite

top

Gazeau, Jean-Pierre, and Verger-Gaugry, Jean-Louis. "Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number." Annales de l’institut Fourier 56.7 (2006): 2437-2461. <http://eudml.org/doc/10209>.

@article{Gazeau2006,
abstract = {The Fourier transform of a weighted Dirac comb of beta-integers is characterized within the framework of the theory of Distributions, in particular its pure point part which corresponds to the Bragg part of the diffraction spectrum. The corresponding intensity function on this Bragg part is computed. We deduce the diffraction spectrum of weighted Delone sets on beta-lattices in the split case for the weight, when beta is the golden mean.},
affiliation = {Université Paris 7-Denis Diderot APC - UMR CNRS 7164 Boite 7020 75251 Paris cedex 05 (France); Université Grenoble I Institut Fourier - UMR CNRS 5582 BP 74 38402 Saint-Martin d’Hères (France)},
author = {Gazeau, Jean-Pierre, Verger-Gaugry, Jean-Louis},
journal = {Annales de l’institut Fourier},
keywords = {Delone set; Meyer set; beta-integer; beta-lattice; PV number; mathematical diffraction},
language = {eng},
number = {7},
pages = {2437-2461},
publisher = {Association des Annales de l’institut Fourier},
title = {Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number},
url = {http://eudml.org/doc/10209},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Gazeau, Jean-Pierre
AU - Verger-Gaugry, Jean-Louis
TI - Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2437
EP - 2461
AB - The Fourier transform of a weighted Dirac comb of beta-integers is characterized within the framework of the theory of Distributions, in particular its pure point part which corresponds to the Bragg part of the diffraction spectrum. The corresponding intensity function on this Bragg part is computed. We deduce the diffraction spectrum of weighted Delone sets on beta-lattices in the split case for the weight, when beta is the golden mean.
LA - eng
KW - Delone set; Meyer set; beta-integer; beta-lattice; PV number; mathematical diffraction
UR - http://eudml.org/doc/10209
ER -

References

top
  1. L. Argabright, J. Gil de Lamadrid, Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups, 145 (1974), American Mathematical Society, Providence, RI Zbl0294.43002MR621876
  2. M. Baake, R. V. Moody, Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math. 573 (2004), 61-94 Zbl1188.43008MR2084582
  3. J. P. Bell, K. G. Hare, A Classification of (some) Pisot-Cyclotomic Numbers, J. Number Theory 115 (2005), 215-229 Zbl1084.11058MR2180499
  4. J.-P. Bertrandias, Espaces de fonctions continues et bornées en moyenne asymptotique d’ordre p , Mémoire Soc. Math. France (1966), 3-106 Zbl0148.11701MR196411
  5. J.-P. Bertrandias, J. Couot, J. Dhombres, M. Mendès-France, P. Phu Hien, Kh. Vo Khac, Espaces de Marcinkiewicz, corrélations, mesures, systèmes dynamiques, (1987), Masson, Paris Zbl0617.46034MR878355
  6. E. Bombieri, J. E. Taylor, Which distributions diffract? An initial investigation, J. Phys. Colloque 47 (1986), 19-28 Zbl0693.52002
  7. E. Bombieri, J. E. Taylor, Quasicrystal, tilings, and algebraic number theory: Some preliminary connections, The legacy of S. Kovalevskaya 64 (1987), 241-264, American Mathematical Society, Providence, RI Zbl0617.43002MR881466
  8. Č. Burdík, C. Frougny, J.-P. Gazeau, R. Krejčar, Beta-integers as natural counting systems for quasicrystals, J. of Physics A: Math. Gen. 31 (1998), 6449-6472 Zbl0941.52019MR1644115
  9. A. Cordoba, Dirac combs, Lett. Math. Phys. 17 (1989), 191-196 Zbl0681.42013MR995797
  10. J.-M. Cowley, Diffraction Physics, (1986), North-Holland, Amsterdam 
  11. F. Denoyer, A. Elkharrat, J.-P. Gazeau, Beta-lattice multiresolution of quasicrystalline Bragg peaks, (2006) 
  12. A. Elkharrat, Scale dependent partitioning of one-dimensional aperiodic set diffraction, Europ. Phys. J. B39 (2004), 287-294 
  13. A. Elkharrat, Ch. Frougny, J.-P. Gazeau, J.-L. Verger-Gaugry, Symmetry groups for beta-lattices, Theor. Comp. Sci. 319 (2004), 281-305 Zbl1068.52028MR2074957
  14. S. Fabre, Substitutions et β -systèmes de numération, Theor. Comp. Sci. 137 (1995), 219-236 Zbl0872.11017MR1311222
  15. A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly 92 (1985), 105-114 Zbl0568.10005MR777556
  16. C. Frougny, Number Representation and Finite Automata, London Math. Soc. Lecture Note Ser.; 279 (2000), 207-228 Zbl0976.11003MR1776760
  17. C. Frougny, J.-P. Gazeau, R. Krejčar, Additive and multiplicative properties of point-sets based on beta-integers, Theor. Comp. Sci. 303 (2003), 491-516 Zbl1036.11034MR1990778
  18. C. Frougny, B. Solomyak, Finite beta-expansions, Ergod. Theor. Dynam. Syst. 12 (1992), 713-723 Zbl0814.68065MR1200339
  19. J.-P. Gazeau, Pisot-cyclotomic integers for quasilattices, The Mathematics of Long-Range Aperiodic Order (1997), 175-198, MoodyR.V.R., Dordrecht Zbl0887.11043MR1460024
  20. J.-P. Gazeau, J.-L. Verger-Gaugry, Geometric study of the beta-integers for a Perron number and mathematical quasicrystals, J. Théorie Nombres Bordeaux 16 (2004), 125-149 Zbl1075.11007MR2145576
  21. J. Gil de Lamadrid, L. Argabright, Almost Periodic Measures, Memoirs of the American Mathematical Society, American Mathematical Society, Providence, RI 85 (1990) Zbl0719.43006MR979431
  22. A. Guinier, Theory and Techniques for X-Ray Crystallography, (1964), Dunod, Paris 
  23. A. Hof, On diffraction by aperiodic structures, Commun. Math. Phys. 169 (1995), 25-43 Zbl0821.60099MR1328260
  24. J. C. Lagarias, Geometric Models for Quasicrystals I. Delone Sets of Finite Type, Discr. Comput. Geom. 21 (1999), 161-191 Zbl0924.68190MR1668082
  25. J. C. Lagarias, Mathematical Quasicrystals and the problem of diffraction, Directions in Mathematical Quasicrystals (2000), 61-93, BaakeM.M., Providence, RI Zbl1161.52312MR1798989
  26. M. Lothaire, Algebraic Combinatorics on Words, (2002), Cambridge University Press Zbl1001.68093MR1905123
  27. Y. Meyer, Nombres de Pisot, Nombres de Salem et Analyse Harmonique, Lect. Notes Math. 117 (1969), Springer Zbl0189.14301MR568288
  28. Y. Meyer, Algebraic Numbers and Harmonic Analysis, (1972), North-Holland Zbl0267.43001MR485769
  29. Y. Meyer, Quasicrystals, Diophantine approximation and algebraic numbers, Beyond Quasicrystals (1995), 3-16, AxelF.F. Zbl0881.11059MR1420415
  30. R. V. Moody, Meyer sets and their duals, The Mathematics of Long-Range Aperiodic Order (1997), 403-442, MoodyR. V.R. V. Zbl0880.43008MR1460032
  31. R. V. Moody, From quasicrystals to more complex systems, Model Sets: A Survey (2000), 145-166, AxelF.F. 
  32. G. Muraz, J.-L. Verger-Gaugry, On lower bounds of the density of Delone sets and holes in sequences of sphere packings, Exp. Math. 14 (2005), 47-57 Zbl1108.52021MR2146518
  33. W. Parry, On the β -expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416 Zbl0099.28103MR142719
  34. N. Pythéas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Math. 1794 (2003), Springer Zbl1014.11015MR1970385
  35. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957), 477-493 Zbl0079.08901MR97374
  36. M. Schlottmann, Cut-and-Project sets in locally compact Abelian groups, Quasicrystals and Discrete Geometry 10 (1998), 247-264, PateraJ.J., Providence, RI Zbl0912.22002MR1636782
  37. L. Schwartz, Théorie des distributions, (1973), Hermann, Paris Zbl0962.46025MR209834
  38. D. Shechtman, I. Blech, D. Gratias, J. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951-1953 
  39. N. Strungaru, Almost Periodic Measures and Long-Range Order in Meyer Sets, Discr. Comput. Geom. 33 (2005), 483-505 Zbl1062.43008MR2121992
  40. W. P. Thurston, Groups, tilings, and finite state automata, (Summer 1989) 
  41. J.-L. Verger-Gaugry, On gaps in Rényi β -expansions of unity for β &gt; 1 an algebraic number, (2006) Zbl1177.11013
  42. J.-L. Verger-Gaugry, On self-similar finitely generated uniformly discrete (SFU-) sets and sphere packings, Number Theory and Physics (2006), NyssenL.L. Zbl1170.52303
  43. K. Vo Khac, Fonctions et distributions stationnaires. Application à l’étude des solutions stationnaires d’équations aux dérivées partielles, Espaces de Marcinkiewicz, corrélations, mesures, systèmes dynamiques (1987), 11-57, Masson, Paris 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.