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

Gazeau, Jean-Pierre; Verger-Gaugry, Jean-Louis

Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 673-705.

On explore le spectre d’un peigne de Dirac pondéré supporté par le quasicristal de Thue-Morse au moyen de la Conjecture de Bombieri-Taylor, pour les pics de Bragg, et d’une nouvelle conjecture que l’on appelle Conjecture de Aubry-Godrèche-Luck, pour la composante singulière continue. La décomposition de la transformée de Fourier du peigne de Dirac pondéré est obtenue dans le cadre de la théorie des distributions tempérées. Nous montrons que l’asymptotique de l’arithmétique des sommes $p$-raréfiées de Thue-Morse (Dumont ; Goldstein, Kelly and Speer ; Grabner ; Drmota and Skalba,...), précisément les fonctions fractales des sommes de chiffres, jouent un rôle fondamental dans la description de la composante singulière continue du spectre, combinées à des résultats classiques sur les produits de Riesz de Peyrière et de M. Queffélec. Les lois d’échelle dominantes des suites de mesures approximantes sont contrôlées sur une partie de la composante singulière continue par certaines inégalités dans lesquelles le nombre de classes de diviseurs et le régulateur de corps quadratiques réels interviennent.

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.

Mots clés : Thue-Morse quasicrystal, spectrum, singular continuous component, rarefied sums, sum-of-digits fractal functions, approximation to distribution

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