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

1 APC - THEORIE
Institut für theoretische Physik, APC - UMR 7164 - AstroParticule et Cosmologie
Abstract : The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated, and characterized up to a measure zero set, by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of another 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.
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Type de document :
Pré-publication, Document de travail
IF_PREPUB. 35 pages In honor of the $60$-th birthday of Henri Cohen. 2007

https://hal.archives-ouvertes.fr/hal-00341927
Contributeur : Jean-Louis Verger-Gaugry <>
Soumis le : mercredi 26 novembre 2008 - 13:03:39
Dernière modification le : mardi 11 octobre 2016 - 14:55:46
Document(s) archivé(s) le : lundi 7 juin 2010 - 22:02:03

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• HAL Id : hal-00341927, version 1
• ARXIV : 0811.4361

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J.-P. Gazeau, Jean-Louis Verger-Gaugry. On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon. IF_PREPUB. 35 pages In honor of the $60$-th birthday of Henri Cohen. 2007. <hal-00341927>

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