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

Jean-Pierre Gazeau[1]; Jean-Louis Verger-Gaugry[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

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 1, page 125-149
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Gazeau, Jean-Pierre, and Verger-Gaugry, Jean-Louis. "Geometric study of the beta-integers for a Perron number and mathematical quasicrystals." Journal de Théorie des Nombres de Bordeaux 16.1 (2004): 125-149. <http://eudml.org/doc/249261>.

@article{Gazeau2004,
abstract = {We investigate in a geometrical way the point sets of  $\mathop \{\mathbb\{R\}\}\nolimits $  obtained by the  $\beta $-numeration that are the  $\beta $-integers  $\mathop \{\mathbb\{Z\}\}\nolimits _\{\beta \} \subset \mathop \{\mathbb\{Z\}\}\nolimits [\beta ]$  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  $\mathop \{\mathbb\{Z\}\}\nolimits ^\{m\}$  ($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  $\mathop \{\mathbb\{Z\}\}\nolimits _\{\beta \}$  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  $\mathop \{\mathbb\{Z\}\}\nolimits _\{\beta \} \cap \mathop \{\mathbb\{R\}\}\nolimits ^\{+\}$  is finitely generated over  $\mathop \{\mathbb\{N\}\}\nolimits $  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:  $x, y \in \mathop \{\mathbb\{Z\}\}\nolimits _\{\beta \} ~ \Rightarrow x+y ~(\mbox \{\{\rm respectively\}\} ~x-y~~) \in \beta ^\{-q\} \mathop \{\mathbb\{Z\}\}\nolimits _\{\beta \}$ if $x+y$ (respectively $x-y$ ) has a finite Rényi $\beta $-expansion.},
affiliation = {LPTMC Université Paris 7 Denis Diderot mailbox 7020 2 place Jussieu 75251 Paris Cedex 05, France; Institut Fourier UJF Grenoble UFR de Mathématiques CNRS UMR 5582 BP 74 - Domaine Universitaire 38402 Saint Martin d’Hères, France},
author = {Gazeau, Jean-Pierre, Verger-Gaugry, Jean-Louis},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {beta-integers; Perron numbers; Pisot numbers; quasi-crystals; Meyer sets; Delaunay sets; -expansions},
language = {eng},
number = {1},
pages = {125-149},
publisher = {Université Bordeaux 1},
title = {Geometric study of the beta-integers for a Perron number and mathematical quasicrystals},
url = {http://eudml.org/doc/249261},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Gazeau, Jean-Pierre
AU - Verger-Gaugry, Jean-Louis
TI - Geometric study of the beta-integers for a Perron number and mathematical quasicrystals
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 1
SP - 125
EP - 149
AB - We investigate in a geometrical way the point sets of  $\mathop {\mathbb{R}}\nolimits $  obtained by the  $\beta $-numeration that are the  $\beta $-integers  $\mathop {\mathbb{Z}}\nolimits _{\beta } \subset \mathop {\mathbb{Z}}\nolimits [\beta ]$  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  $\mathop {\mathbb{Z}}\nolimits ^{m}$  ($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  $\mathop {\mathbb{Z}}\nolimits _{\beta }$  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  $\mathop {\mathbb{Z}}\nolimits _{\beta } \cap \mathop {\mathbb{R}}\nolimits ^{+}$  is finitely generated over  $\mathop {\mathbb{N}}\nolimits $  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:  $x, y \in \mathop {\mathbb{Z}}\nolimits _{\beta } ~ \Rightarrow x+y ~(\mbox {{\rm respectively}} ~x-y~~) \in \beta ^{-q} \mathop {\mathbb{Z}}\nolimits _{\beta }$ if $x+y$ (respectively $x-y$ ) has a finite Rényi $\beta $-expansion.
LA - eng
KW - beta-integers; Perron numbers; Pisot numbers; quasi-crystals; Meyer sets; Delaunay sets; -expansions
UR - http://eudml.org/doc/249261
ER -

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Citations in EuDML Documents

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  1. Jean-Pierre Gazeau, Jean-Louis Verger-Gaugry, Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number
  2. Jean-Louis Verger-Gaugry, On gaps in Rényi β -expansions of unity for β &gt; 1 an algebraic number
  3. Wolfgang Steiner, On the structure of (−)-integers
  4. Wolfgang Steiner, On the structure of (−β)-integers
  5. Gilbert Muraz, Jean-Louis Verger-Gaugry, On a generalization of the Selection Theorem of Mahler
  6. Valérie Berthé, Timo Jolivet, Anne Siegel, Connectedness of fractals associated with Arnoux–Rauzy substitutions
  7. Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner, Dynamical directions in numeration

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