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Jean-Pierre Gazeau; Jean-Louis Verger-Gaugry
On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
Journal de théorie des nombres de Bordeaux, 20 no. 3 (2008), p. 673-705, doi: 10.5802/jtnb.645
Article PDF | Reviews MR 2523312 | Zbl pre05572696
Keywords: Thue-Morse quasicrystal, spectrum, singular continuous component, rarefied sums, sum-of-digits fractal functions, approximation to distribution

Résumé - 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.

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