On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

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

  • [1] Astroparticules et Cosmologie (APC, UMR 7164) Université Paris 7 Denis-Diderot Boite 7020 75251 Paris Cedex 05, France
  • [2] Université de Grenoble I Institut Fourier, CNRS UMR 5582 BP 74, Domaine Universitaire 38402 Saint-Martin d’Hères, France

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 3, page 673-705
  • ISSN: 1246-7405

Abstract

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The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of a new conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the p -rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyrière and M. Queffélec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.

How to cite

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Gazeau, Jean-Pierre, and Verger-Gaugry, Jean-Louis. "On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon." Journal de Théorie des Nombres de Bordeaux 20.3 (2008): 673-705. <http://eudml.org/doc/10855>.

@article{Gazeau2008,
abstract = {The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of a new conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyrière and M. Queffélec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.},
affiliation = {Astroparticules et Cosmologie (APC, UMR 7164) Université Paris 7 Denis-Diderot Boite 7020 75251 Paris Cedex 05, France; Université de Grenoble I Institut Fourier, CNRS UMR 5582 BP 74, Domaine Universitaire 38402 Saint-Martin d’Hères, France},
author = {Gazeau, Jean-Pierre, Verger-Gaugry, Jean-Louis},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Thue-Morse quasicrystal; spectrum; singular continuous component; rarefied sums; sum-of-digits fractal functions; approximation to distribution; Thue-Morse sequence; quasicrystal; singular continuous measure; sum of digits},
language = {eng},
number = {3},
pages = {673-705},
publisher = {Université Bordeaux 1},
title = {On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon},
url = {http://eudml.org/doc/10855},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Gazeau, Jean-Pierre
AU - Verger-Gaugry, Jean-Louis
TI - On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 3
SP - 673
EP - 705
AB - The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of a new conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyrière and M. Queffélec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.
LA - eng
KW - Thue-Morse quasicrystal; spectrum; singular continuous component; rarefied sums; sum-of-digits fractal functions; approximation to distribution; Thue-Morse sequence; quasicrystal; singular continuous measure; sum of digits
UR - http://eudml.org/doc/10855
ER -

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