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

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

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 125-149.

Nous étudions géométriquement les ensembles de points de  $ℝ$  obtenus par la  $\beta$-numération que sont les  $\beta$-entiers  ${\mathbb{Z}}_{\beta }\subset \mathbb{Z}\left[\beta \right]$  où $\beta$  est un nombre de Perron. Nous montrons qu’il existe deux schémas de coupe-et-projection canoniques associés à la  $\beta$-numération, où les  $\beta$-entiers se relèvent en certains points du réseau  ${\mathbb{Z}}^m(m=$ degré de $\beta )$ , situés autour du sous-espace propre dominant de la matrice compagnon de  $\beta$ . Lorsque  $\beta$  est en particulier un nombre de Pisot, nous redonnons une preuve du fait que  ${\mathbb{Z}}_{\beta }$  est un ensemble de Meyer. Dans les espaces internes les fenêtres d’acceptation canoniques sont des fractals dont l’une est le fractal de Rauzy (à quasi-homothétie près). Nous le montrons sur un exemple. Nous montrons que  ${\mathbb{Z}}_{\beta }\cap {ℝ}^{+}$  est de type fini sur  $\mathbb{N}$, faisons le lien avec la classification de Lagarias des ensembles de Delaunay et donnons une borne supérieure effective de l’entier  $q$  dans la relation :  $x,y\in {\mathbb{Z}}_{\beta }\Rightarrow x+y(\mathrm{respectivement}x-y)\in {\beta }^{-q}{\mathbb{Z}}_{\beta }$ lorsque $x+y$ (respectivement $x-y$ ) a un $\beta$-développement de Rényi fini.

We investigate in a geometrical way the point sets of  $ℝ$  obtained by the  $\beta$-numeration that are the  $\beta$-integers  ${\mathbb{Z}}_{\beta }\subset \mathbb{Z}\left[\beta \right]$  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  ${\mathbb{Z}}^m$  ($m=$  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  ${\mathbb{Z}}_{\beta }$  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  ${\mathbb{Z}}_{\beta }\cap {ℝ}^{+}$  is finitely generated over  $\mathbb{N}$  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:  $x,y\in {\mathbb{Z}}_{\beta }\Rightarrow x+y(\mathrm{respectively}x-y)\in {\beta }^{-q}{\mathbb{Z}}_{\beta }$ if $x+y$ (respectively $x-y$ ) has a finite Rényi $\beta$-expansion.

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