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On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
Gazeau J.-P., Verger-Gaugry J.-L.
Mathématiques/Théorie des nombres
On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
J.-P. Gazeau ()1, Jean-Louis Verger-Gaugry ()2
1 :  APC - UMR 7164 - AstroParticule et Cosmologie
CNRS : UMR7164 – IN2P3 – Observatoire de Paris – Université Paris VII - Paris Diderot – CEA : DSM/IRFU
APC - UMR 7164, Université Paris Diderot, 10 rue Alice Domon et Léonie Duquet, case postale 7020, F-75205 Paris Cedex 13
2 :  IF - Institut Fourier
CNRS : UMR5582 – Université Joseph Fourier - Grenoble I
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.

Thue-Morse quasicrystal – spectrum – singular continuous component – rarefied sums – sum-of-digits fractal functions – approximation to distribution
MSC 11A63, 11B85, 42A38, 42A55, 43A30, 52C23, 62E17
35 pages In honor of the $60$-th birthday of Henri Cohen

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