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

J.-P. Gazeau 1 Jean-Louis Verger-Gaugry 2
1 APC - THEORIE
APC - UMR 7164 - AstroParticule et Cosmologie, Institut für theoretische Physik
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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Pré-publication, Document de travail
IF_PREPUB. 35 pages In honor of the $60$-th birthday of Henri Cohen. 2007
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Dernière modification le : jeudi 23 novembre 2017 - 13:32:03
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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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