We investigate in a geometrical way the point sets of obtained by the -numeration that are the -integers where is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the -numeration, allowing to lift up the -integers to some points of the lattice ( degree of ) lying about the dominant eigenspace of the companion matrix of . When is in particular a Pisot number, this framework gives another proof of the fact that is...
The Fourier transform of a weighted Dirac comb of beta-integers is characterized within the framework of the theory of Distributions, in particular its pure point part which corresponds to the Bragg part of the diffraction spectrum. The corresponding intensity function on this Bragg part is computed. We deduce the diffraction spectrum of weighted Delone sets on beta-lattices in the split case for the weight, when beta is the golden mean.
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 -rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner;...
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