Geometric study of the beta-integers for a Perron number and mathematical quasicrystals

Abstract : We investigate in a geometrical way the point sets of ~$\rb$~ obtained by the ~$\beta$-numeration that are the ~$\beta$-integers ~$\zb_{\beta} \subset \zb[\beta]$~ where ~$\beta$~ is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the ~$\beta$-numeration, allowing to lift up the ~$\beta$-integers to some points of the lattice ~$\zb^{m}$~ ($m = $~ degree of ~$\beta$) lying about the dominant eigenspace of the companion matrix of ~$\beta$~. When ~$\beta$~ is in particular a Pisot number, this framework gives another proof of the fact that ~$\zb_{\beta}$~ is a Meyer set. In the internal spaces, the canonical acceptance windows are fractals and one of them is the Rauzy fractal (up to quasi-dilation). We show it on an example. We show that ~$\zb_{\beta} \cap \rb^{+}$~ is finitely generated over ~$\nb$~ and make a link with the classification of Delone sets proposed by Lagarias. Finally we give an effective upper bound for the integer ~$q$~ taking place in the relation: ~$x, y \in \zb_{\beta} ~ \Longrightarrow x+y ~(\mbox{{\rm respectively}} ~x-y~~) \in \beta^{-q} \zb_{\beta}$ if ~$x+y$~ (respectively ~$x-y$~~) has a finite Rényi ~$\beta$- expansion.
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Jean-Louis Verger-Gaugry, Jean-Pierre Gazeau. Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, 2004, 16, pp.125--149. ⟨hal-00353156⟩

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