Geometric study of the beta-integers for a Perron number and mathematical quasicrystals
Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 125-149.

Nous étudions géométriquement les ensembles de points de    obtenus par la  β-numération que sont les  β-entiers   β [β]  où β  est un nombre de Perron. Nous montrons qu’il existe deux schémas de coupe-et-projection canoniques associés à la  β-numération, où les  β-entiers se relèvent en certains points du réseau   m (m= degré de β) , situés autour du sous-espace propre dominant de la matrice compagnon de  β . Lorsque  β  est en particulier un nombre de Pisot, nous redonnons une preuve du fait que   β   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   β +   est de type fini sur  , 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 β x+y( respectivement x-y)β -q β lorsque x+y (respectivement x-y ) a un β-développement de Rényi fini.

We investigate in a geometrical way the point sets of    obtained by the  β-numeration that are the  β-integers   β [β]  where  β  is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the  β-numeration, allowing to lift up the  β-integers to some points of the lattice   m   (m=  degree of  β) lying about the dominant eigenspace of the companion matrix of  β . When  β  is in particular a Pisot number, this framework gives another proof of the fact that   β   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   β +   is finitely generated over    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 β x+y( respectively x-y)β -q β if x+y (respectively x-y ) has a finite Rényi β-expansion.

DOI : 10.5802/jtnb.437
Gazeau, Jean-Pierre 1 ; Verger-Gaugry, Jean-Louis 2

1 LPTMC Université Paris 7 Denis Diderot mailbox 7020 2 place Jussieu 75251 Paris Cedex 05, France
2 Institut Fourier UJF Grenoble UFR de Mathématiques CNRS UMR 5582 BP 74 - Domaine Universitaire 38402 Saint Martin d’Hères, France
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Gazeau, Jean-Pierre; Verger-Gaugry, Jean-Louis. Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 125-149. doi : 10.5802/jtnb.437. http://www.numdam.org/articles/10.5802/jtnb.437/

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