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Jean-Pierre Gazeau; Jean-Louis Verger-Gaugry
On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
Journal de théorie des nombres de Bordeaux, 20 no. 3 (2008), p. 673-705, doi: 10.5802/jtnb.645
Article PDF | Analyses MR 2523312 | Zbl pre05572696
Mots clés: Thue-Morse quasicrystal, spectrum, singular continuous component, rarefied sums, sum-of-digits fractal functions, approximation to distribution

Résumé - Abstract

On explore le spectre d’un peigne de Dirac pondéré supporté par le quasicristal de Thue-Morse au moyen de la Conjecture de Bombieri-Taylor, pour les pics de Bragg, et d’une nouvelle conjecture que l’on appelle Conjecture de Aubry-Godrèche-Luck, pour la composante singulière continue. La décomposition de la transformée de Fourier du peigne de Dirac pondéré est obtenue dans le cadre de la théorie des distributions tempérées. Nous montrons que l’asymptotique de l’arithmétique des sommes $p$-raréfiées de Thue-Morse (Dumont ; Goldstein, Kelly and Speer ; Grabner ; Drmota and Skalba,...), précisément les fonctions fractales des sommes de chiffres, jouent un rôle fondamental dans la description de la composante singulière continue du spectre, combinées à des résultats classiques sur les produits de Riesz de Peyrière et de M. Queffélec. Les lois d’échelle dominantes des suites de mesures approximantes sont contrôlées sur une partie de la composante singulière continue par certaines inégalités dans lesquelles le nombre de classes de diviseurs et le régulateur de corps quadratiques réels interviennent.

Bibliographie

[AMF] J.-P. Allouche and M. Mendès-France, Automata and automatic sequences, in Beyond Quasicrystals, Ed. F. Axel and D. Gratias, Course 11, Les Editions de Physique, Springer (1995), 293–367.  MR 1420422 |  Zbl 0881.11026
[AGL] S. Aubry, C. Godrèche and J.-M. Luck, Scaling Properties of a Structure Intermediate between Quasiperiodic and Random, J. Stat. Phys. 51 (1988), 1033–1075.  MR 971045 |  Zbl 1086.37522
[AT] F. Axel and H. Terauchi, High-resolution X-ray-diffraction spectra of Thue-Morse GaAs-AlAs heterostructures: Towards a novel description of disorder, Phys. Rev. Lett. 66 (1991), 2223–2226.
[B] Zai-Qiao Bai, Multifractal analysis of the spectral measure of the Thue-Morse sequence: a periodic orbit approach, J. Phys. A: Math. Gen. 39 (2006) 10959–10973.  MR 2277352 |  Zbl 1097.81032
[Bs1] J.-P. Bertrandias, Espaces de fonctions continues et bornées en moyenne asymptotique d’ordre $p$, Mémoire Soc. Math. france (1966), no. 5, 3–106. Numdam |  MR 196411 |  Zbl 0148.11701
[B-VK] J.-P. Bertrandias, J. Couot, J. Dhombres, M. Mendès-France, P. Phu Hien and K. Vo Khac, Espaces de Marcinkiewicz, corrélations, mesures, systèmes dynamiques, Masson, Paris (1987).  MR 878355 |  Zbl 0617.46034
[BT1] E. Bombieri and J.E. Taylor, “Which distributions of matter diffract ? An initial investigation”, J. Phys. Colloque 47 (1986), C3, 19–28.  MR 866320 |  Zbl 0693.52002
[BT2] E. Bombieri and J.E. Taylor, Quasicrystals, tilings, and algebraic number theory: some preliminary connections, Contemp. Math. 64 (1987), 241–264.  MR 881466 |  Zbl 0617.43002
[BS] Z.I. Borevitch and I.R. Chafarevitch, Théorie des Nombres, Gauthiers-Villars, Paris (1967).  MR 205908 |  Zbl 0145.04901
[CSM] Z. Cheng, R. Savit and R. Merlin, Structure and electronic properties of Thue-Morse lattices, Phys. Rev B 37 (1988), 4375–4382.
[CL1] H. Cohen and H.W. Lenstra, Jr., Heuristics on Class groups, Lect. Notes Math. 1052 (1984), 26–36.  MR 750661 |  Zbl 0532.12008
[CL2] H. Cohen and H.W. Lenstra, Jr., Heuristics on Class groups of number fields, Number Theory, Proc. Journ. Arithm., Noodwijkerhout/Neth. 1983, Lect. Notes Math. 1068 (1984), 33–62.  MR 756082 |  Zbl 0558.12002
[CM] H. Cohen and J. Martinet, Class Groups of Number Fields: Numerical Heuristics, Math. Comp. 48 (1987), 123–137.  MR 866103 |  Zbl 0627.12006
[Ct] J. Coquet, A summation formula related to the binary digits, Inv. Math. 73 (1983), 107–115.  MR 707350 |  Zbl 0528.10006
[Cy] J.-M. Cowley, Diffraction physics, North-Holland, Amsterdam (1986), 2nd edition.
[DS1] M. Drmota and M. Skalba, Sign-changes of the Thue-Morse fractal fonction and Dirichlet $L$-series, Manuscripta Math. 86 (1995), 519–541. Article |  MR 1324686 |  Zbl 0828.11013
[DS2] M. Drmota and M. Skalba, Rarefied sums of the Thue-Morse sequence, Trans. Amer. Math. Soc. 352 (2000), 609–642.  MR 1491859 |  Zbl 0995.11017
[D] J.M. Dumont, Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Série I 297 (1983), 145–148.  MR 725391 |  Zbl 0533.10005
[GVG] J. P. Gazeau and J.-L. Verger-Gaugry, Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number, Ann. Inst. Fourier 56 (2006), 2437–2461. Cedram |  MR 2290786 |  Zbl 1119.52015
[Gd] A.O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. 13 (1968), 259–265. Article |  MR 220693 |  Zbl 0155.09003
[GL1] C. Godrèche and J.-M. Luck, Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures, J. Phys. A: Math. Gen. 23 (1990), 3769–3797.  MR 1069478 |  Zbl 0713.11021
[GL2] C. Godrèche and J.-M. Luck, Indexing the diffraction spectrum of a non-Pisot self-structure, Phys. Rev. B 45 (1992), 176–185.
[GKS] S. Goldstein, K.A. Kelly and E.R. Speer, The fractal structure of rarefied sums of the Thue-Morse sequence, J. Number Th. 42 (1992), 1–19.  MR 1176416 |  Zbl 0788.11010
[Gr1] P.J. Grabner, A note on the parity of the sum-of-digits function, Actes 30ième Séminaire Lotharingien de Combinatoire (Gerolfingen, 1993), 35–42.  MR 1312627 |  Zbl 1060.11506
[Gr2] P.J. Grabner, Completely $q$-Multiplicative Functions: the Mellin Transform Approach, Acta Arith. 65 (1993), 85–96. Article |  MR 1239244 |  Zbl 0783.11035
[GHT] P.J. Grabner, T. Herendi and R.F. Tichy, Fractal digital sums and Codes, AAECC 8 (1997), 33-39.  MR 1465087 |  Zbl 0874.11012
[G] A. Guinier, Theory and Technics for X-Ray Crystallography, Dunod, Paris (1964).
[H] A. Hof, On diffraction by aperiodic structures, Commun. Math. Phys. 169 (1995), 25–43. Article |  MR 1328260 |  Zbl 0821.60099
[Ha] L.-K. Hua, Introduction to Number Theory, Springer-Verlag, Berlin-New York (1982).  MR 665428 |  Zbl 0483.10001
[K] M. Kac, On the distribution of values of sums of the type $\sum f(2^k t)$, Ann. Math. 47 (1946), 33–49.  MR 15548 |  Zbl 0063.03091
[KIR] M. Kolár, B. Iochum and L. Raymond, Structure factor of 1D systems (superlattices) based on two-letter substitution rules: I. $\delta $ (Bragg) peaks, J. Phys. A: Math. Gen. 26 (1993), 7343–7366.  MR 1257767
[La1] J.C. Lagarias, Meyer’s concept of quasicrystal and quasiregular sets, Comm. Math. Phys. 179 (1995), 365–376. Article |  MR 1400744 |  Zbl 0858.52010
[La2] J.C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in Directions in Mathematical Quasicrystals, ed. M. Baake & R.V. Moody, CRM Monograph Series, Amer. Math. Soc. Providence, RI, (2000), 61–93.  MR 1798989 |  Zbl pre01584913
[Le] H.W. Lenstra Jr., On Artin’s conjecture and Euclid’s algorithm in global fields, Invent. Math. 42 (1977), 201–224.  MR 480413 |  Zbl 0362.12012
[Lz] D. Lenz, Continuity of Eigenfonctions of Uniquely Ergodic Dynamical Systems and Intensity of Bragg peaks, preprint (2006). arXiv |  MR 2480747
[Lu] J.-M. Luck, Cantor spectra and scaling of gap widths in deterministic aperiodic systems, Phys. Rev. B 39 (1989), 5834–5849.
[M] R.V. Moody, Meyer sets and their duals, in The Mathematics of Long-Range Aperiodic Order, Ed. R.V. Moody, Kluwer (1997), 403–442.  MR 1460032 |  Zbl 0880.43008
[N] D.J. Newman, On the number of binary digits in a multiple of three, Proc. Am. Math. Soc. 21 (1969), 719–721.  MR 244149 |  Zbl 0194.35004
[Oa] C.R. de Oliveira, A proof of the dynamical version of the Bombieri-Taylor Conjecture, J. Math. Phys. 39 (1998), 4335–4342.  MR 1643245 |  Zbl 0940.81017
[P] J. Peyrière, Etude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier 25 (1975), 127–169. Cedram |  MR 404973 |  Zbl 0302.43003
[PCA] J. Peyrière, E. Cockayne and F. Axel, Line-Shape Analysis of High Resolution X-Ray Diffraction Spectra of Finite Size Thue-Morse GaAs-AlAs Multilayer Heterostructures, J. Phys. I France 5 (1995), 111–127.
[Q1] M. Queffélec, Dynamical systems - Spectral Analysis, Lect. Notes Math. 1294 (1987).  Zbl 0642.28013
[Q2] M. Queffélec, Spectral study of automatic and substitutive sequences, in Beyond Quasicrystals, Ed. F. Axel and D. Gratias, Course 12, Les Editions de Physique - Springer (1995), 369–414.  MR 1420423 |  Zbl 0881.11027
[R] D. Raikov, On some arithmetical properties of summable functions, Rec. Math. de Moscou 1 (43;3) (1936) 377–383.  Zbl 0014.39701 |  JFM 62.0254.02
[Sz] L. Schwartz, Théorie des distributions, Hermann, Paris (1973).  MR 209834 |  Zbl 0962.46025
[Su] N. Strungaru, Almost Periodic Measures and Long-Range Order in Meyer Sets, Discr. Comput. Geom. 33 (2005), 483–505.  MR 2121992 |  Zbl 1062.43008
[VG] J.-L. Verger-Gaugry, On self-similar finitely generated uniformly discrete (SFU-) sets and sphere packings, in IRMA Lect. in Math. and Theor. Phys. 10, Ed. L. Nyssen, E.M.S. (2006), 39–78.  MR 2277756 |  Zbl pre05263002
[VK] K. Vo Khac, Fonctions et distributions stationnaires. Application à l’étude des solutions stationnaires d’équations aux dérivées partielles, in [B-VK], pp 11–57.
[WWVG] J. Wolny, A. Wnek and J.-L. Verger-Gaugry, Fractal behaviour of diffraction patterns of Thue-Morse sequence, J. Comput. Phys. 163 (2000), 313.  MR 1783556 |  Zbl 1073.82617