Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number
Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2437-2461.

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.

On caractérise au moyen de la théorie des distributions la transformée de Fourier d’un peigne de Dirac avec poids, plus particulièrement la partie purement ponctuelle qui correspond aux pics de Bragg dans le spectre de diffraction. La fonction intensité de ces derniers est donnée d’une manière explicite. On en déduit le spectre de diffraction d’ensembles de Delaunay avec poids supportés par les beta-réseaux dans le cas où le poids est factorisable et où beta est le nombre d’or.

DOI: 10.5802/aif.2245
Classification: 52C23,  78A45,  42A99
Keywords: Delone set, Meyer set, beta-integer, beta-lattice, PV number, mathematical diffraction
Gazeau, Jean-Pierre 1; Verger-Gaugry, Jean-Louis 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)
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Gazeau, Jean-Pierre; 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, Volume 56 (2006) no. 7, pp. 2437-2461. doi : 10.5802/aif.2245. http://www.numdam.org/articles/10.5802/aif.2245/

[1] Argabright, L.; Gil de Lamadrid, J. Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups, Memoirs of the American Mathematical Society, 145, American Mathematical Society, Providence, RI, 1974 | MR | Zbl

[2] Baake, M.; Moody, R. V. Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math., Volume 573 (2004), pp. 61-94 | DOI | MR | Zbl

[3] Bell, J. P.; Hare, K. G. A Classification of (some) Pisot-Cyclotomic Numbers, J. Number Theory, Volume 115 (2005), pp. 215-229 | DOI | MR | Zbl

[4] Bertrandias, J.-P. Espaces de fonctions continues et bornées en moyenne asymptotique d’ordre p, Mémoire Soc. Math. France (1966) no. 5, pp. 3-106 | EuDML | Numdam | MR | Zbl

[5] Bertrandias, J.-P.; Couot, J.; Dhombres, J.; Mendès-France, M.; Phu Hien, P.; Vo Khac, Kh. Espaces de Marcinkiewicz, corrélations, mesures, systèmes dynamiques, Masson, Paris, 1987 | MR | Zbl

[6] Bombieri, E.; Taylor, J. E. Which distributions diffract? An initial investigation, J. Phys. Colloque, Volume 47 (1986) no. C3, pp. 19-28 | MR | Zbl

[7] Bombieri, E.; Taylor, J. E. Quasicrystal, tilings, and algebraic number theory: Some preliminary connections, The legacy of S. Kovalevskaya (Contemporary Mathematics), Volume 64, American Mathematical Society, Providence, RI, 1987, pp. 241-264 | MR | Zbl

[8] Burdík, Č.; Frougny, C.; Gazeau, J.-P.; Krejčar, R. Beta-integers as natural counting systems for quasicrystals, J. of Physics A: Math. Gen., Volume 31 (1998), pp. 6449-6472 | DOI | MR | Zbl

[9] Cordoba, A. Dirac combs, Lett. Math. Phys., Volume 17 (1989), pp. 191-196 | DOI | MR | Zbl

[10] Cowley, J.-M. Diffraction Physics, North-Holland, Amsterdam, 1986 (2nd edition)

[11] Denoyer, F.; Elkharrat, A.; Gazeau, J.-P. Beta-lattice multiresolution of quasicrystalline Bragg peaks (2006) (submitted)

[12] Elkharrat, A. Scale dependent partitioning of one-dimensional aperiodic set diffraction, Europ. Phys. J., Volume B39 (2004), pp. 287-294 and Thèse de l’Université Paris 7 - Denis Diderot (2004)

[13] Elkharrat, A.; Frougny, Ch.; Gazeau, J.-P.; Verger-Gaugry, J.-L. Symmetry groups for beta-lattices, Theor. Comp. Sci., Volume 319 (2004) no. 1-3, pp. 281-305 | DOI | MR | Zbl

[14] Fabre, S. Substitutions et β-systèmes de numération, Theor. Comp. Sci., Volume 137 (1995), pp. 219-236 | DOI | MR | Zbl

[15] Fraenkel, A. S. Systems of numeration, Amer. Math. Monthly, Volume 92 (1985) no. 2, pp. 105-114 | DOI | MR | Zbl

[16] Frougny, C. Number Representation and Finite Automata, London Math. Soc. Lecture Note Ser.;, Volume 279 (2000), pp. 207-228 | MR | Zbl

[17] Frougny, C.; Gazeau, J.-P.; Krejčar, R. Additive and multiplicative properties of point-sets based on beta-integers, Theor. Comp. Sci., Volume 303 (2003), pp. 491-516 | DOI | MR | Zbl

[18] Frougny, C.; Solomyak, B. Finite beta-expansions, Ergod. Theor. Dynam. Syst., Volume 12 (1992), pp. 713-723 | DOI | MR | Zbl

[19] Gazeau, J.-P.; Moody, R.V. Pisot-cyclotomic integers for quasilattices, The Mathematics of Long-Range Aperiodic Order (NATO advances Science Institutes, Series C: Mathematical and Physical Sciences 489), Kluwer Academic Publishers, Dordrecht, 1997, pp. 175-198 | MR | Zbl

[20] Gazeau, J.-P.; Verger-Gaugry, J.-L. Geometric study of the beta-integers for a Perron number and mathematical quasicrystals, J. Théorie Nombres Bordeaux, Volume 16 (2004), pp. 125-149 | DOI | Numdam | MR | Zbl

[21] Gil de Lamadrid, J.; Argabright, L. Almost Periodic Measures, Memoirs of the American Mathematical Society, American Mathematical Society, Providence, RI, Volume 85 (1990) no. 428, pp. vi+219 | MR | Zbl

[22] Guinier, A. Theory and Techniques for X-Ray Crystallography, Dunod, Paris, 1964

[23] Hof, A. On diffraction by aperiodic structures, Commun. Math. Phys., Volume 169 (1995), pp. 25-43 | DOI | MR | Zbl

[24] Lagarias, J. C. Geometric Models for Quasicrystals I. Delone Sets of Finite Type, Discr. Comput. Geom., Volume 21 (1999), pp. 161-191 | DOI | MR | Zbl

[25] Lagarias, J. C.; Baake, M.; Moody, R. V. Mathematical Quasicrystals and the problem of diffraction, Directions in Mathematical Quasicrystals (CRM Monograph Series), Amer. Math. Soc., Providence, RI, 2000, pp. 61-93 | MR | Zbl

[26] Lothaire, M. Algebraic Combinatorics on Words, Cambridge University Press, 2002 | MR | Zbl

[27] Meyer, Y. Nombres de Pisot, Nombres de Salem et Analyse Harmonique, Lect. Notes Math., Volume 117, Springer, 1969, pp. 63 | MR | Zbl

[28] Meyer, Y. Algebraic Numbers and Harmonic Analysis, North-Holland, 1972 | MR | Zbl

[29] Meyer, Y.; Axel, F.; Gratias, D. Quasicrystals, Diophantine approximation and algebraic numbers, Beyond Quasicrystals, Springer-Verlag & Les Editions de Physique, 1995, pp. 3-16 | MR | Zbl

[30] Moody, R. V.; Moody, R. V. Meyer sets and their duals, The Mathematics of Long-Range Aperiodic Order, Kluwer, 1997, pp. 403-442 | MR | Zbl

[31] Moody, R. V.; Axel, F.; Denoyer, F.; Gazeau, J.-P. From quasicrystals to more complex systems, Model Sets: A Survey, Springer & Les Editions de Physique, 2000, pp. 145-166

[32] Muraz, G.; Verger-Gaugry, J.-L. On lower bounds of the density of Delone sets and holes in sequences of sphere packings, Exp. Math., Volume 14 (2005) no. 1, pp. 47-57 | DOI | MR | Zbl

[33] Parry, W. On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar., Volume 11 (1960), pp. 401-416 | DOI | MR | Zbl

[34] Pythéas Fogg, N. Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Math., Volume 1794, Springer, 2003 | MR | Zbl

[35] Rényi, A. Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., Volume 8 (1957), pp. 477-493 | DOI | MR | Zbl

[36] Schlottmann, M.; Patera, J. Cut-and-Project sets in locally compact Abelian groups, Quasicrystals and Discrete Geometry (Fields Institute Monograph Series), Volume 10, Amer. Math. Soc., Providence, RI, 1998, pp. 247-264 | MR | Zbl

[37] Schwartz, L. Théorie des distributions, Hermann, Paris, 1973 | MR | Zbl

[38] Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett., Volume 53 (1984), pp. 1951-1953 (1951) | DOI

[39] Strungaru, N. Almost Periodic Measures and Long-Range Order in Meyer Sets, Discr. Comput. Geom., Volume 33 (2005), pp. 483-505 | DOI | MR | Zbl

[40] Thurston, W. P. Groups, tilings, and finite state automata (Summer 1989) (A.M.S. Colloquium Lectures, Boulder)

[41] Verger-Gaugry, J.-L. On gaps in Rényi β -expansions of unity for β > 1 an algebraic number (2006) (Annales Institut Fourier) | Numdam

[42] Verger-Gaugry, J.-L.; Nyssen, L. On self-similar finitely generated uniformly discrete (SFU-) sets and sphere packings, Number Theory and Physics (IRMA Lectures in Mathematics and Theoretical Physics), E.M.S. Publishing House, 2006 | Zbl

[43] Vo Khac, K. Fonctions et distributions stationnaires. Application à l’étude des solutions stationnaires d’équations aux dérivées partielles, Espaces de Marcinkiewicz, corrélations, mesures, systèmes dynamiques, Masson, Paris, 1987, pp. 11-57

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