158 articles – 1990 Notices  [english version]
 HAL : hal-00341927, version 1
 arXiv : 0811.4361
 On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
 (23/12/2007)
 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.
 équipe(s) de recherche : APC - THEORIE
 Domaine : Mathématiques/Théorie des nombres
 Mots Clés : Thue-Morse quasicrystal – spectrum – singular continuous component – rarefied sums – sum-of-digits fractal functions – approximation to distribution
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 hal-00341927, version 1 http://hal.archives-ouvertes.fr/hal-00341927 oai:hal.archives-ouvertes.fr:hal-00341927 Contributeur : Jean-Louis Verger-Gaugry <> Soumis le : Mercredi 26 Novembre 2008, 13:03:39 Dernière modification le : Jeudi 12 Avril 2012, 17:29:10