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

top
We investigate in a geometrical way the point sets of   ℝ   obtained by the   β -numeration that are the   β -integers   ℤ β ⊂ ℤ [ 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   ℤ 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   ℤ 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   ℤ β ∩ ℝ +   is finitely generated over   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 ∈ ℤ β ⇒ x + y ( respectively x - y ) ∈ β - q ℤ β if x + y (respectively x - y ) has a finite Rényi β -expansion.

How to cite

top

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 -

References

top
  1. S. Akiyama, Cubic Pisot units with finite β -expansions. Algebraic number theory and Diophantine analysis, (Graz, 1998), de Gruyter, Berlin, (2000), 11–26. Zbl1001.11038MR1770451
  2. S. Akiyama, Pisot numbers and greedy algorithm. Number Theory, (Eger, 1996), de Gruyter, Berlin, (1998), 9–21. Zbl0919.11063MR1628829
  3. P. Arnoux, V. Berthé, H. Ei and S. Ito, Tilings, quasicrystals, discrete planes, generalized substitutions and multidimensional continued fractions. Disc. Math. and Theor. Comp. Sci. Proc. AA (DM-CCG), (2001), 59–78. Zbl1017.68147MR1888763
  4. P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. 8 (2001), 181–207. Zbl1007.37001MR1838930
  5. A. Bertrand, Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sc. Paris, Série A, t. 285 (1977), 419–421. Zbl0362.10040MR447134
  6. A. Bertrand - Mathis, Comment écrire les nombres entiers dans une base qui n’est pas entière. Acta Math. Hung., 54, (3-4) (1989), 237–241. Zbl0695.10005MR1029085
  7. A. Bertrand - Mathis, Développement en base   θ  , Répartition modulo un de la suite   ( x θ n ) n ≥ 0 . Langages codes et   θ - shift, Bull. Soc. Math. France, 114 (1986), 271–323. Zbl0628.58024MR878240
  8. A. Bertrand - Mathis, Nombres de Perron et questions de rationnalité. Séminaire d’Analyse, Université de Clermont II, Année 1988/89, Exposé 08; A. Bertrand - Mathis, Nombres de Perron et problèmes de rationnalité. Astérisque, 198-199-200 (1991), 67–76. Zbl0766.11043
  9. A. Bertrand - Mathis, Questions diverses relatives aux systèmes codés: applications au θ -shift. Preprint. 
  10. F. Blanchard, β -expansions and Symbolic Dynamics. Theor. Comp. Sci. 65 (1989), 131–141. Zbl0682.68081MR1020481
  11. Č. Burdík, Ch. Frougny, J.-P. Gazeau and R. Krejcar, Beta-integers as natural counting systems for quasicrystals. J. Phys. A: Math. Gen. 31 (1998), 6449–6472. Zbl0941.52019MR1644115
  12. C. Frougny, Number representation and finite automata. London Math. Soc. Lecture Note Ser. 279 (2000), 207–228. Zbl0976.11003MR1776760
  13. Ch. Frougny and B. Solomyak, Finite beta-expansions. Ergod. Theor. Dynam. Syst. 12 (1992), 713–723. Zbl0814.68065MR1200339
  14. J. P. Gazeau, Pisot-Cyclotomic Integers for Quasicrystals. The Mathematics of Long-Range Aperiodic Order, Ed. By R.V. Moody, Kluwer Academic Publishers, (1997), 175–198. Zbl0887.11043MR1460024
  15. M. W. Hirsch and S. Smale, Differential Equations. Dynamical Systems and Linear Algebra, Academic Press, New York, (1974). Zbl0309.34001MR486784
  16. S. Ito and M. Kimura, On the Rauzy fractal. Japan J. Indust. Appl. Math. 8 (1991), 461–486. Zbl0734.28010MR1137652
  17. S. Ito and Y. Sano, On periodic β - expansions of Pisot numbers and Rauzy fractals. Osaka J. Math. 38 (2001), 349–368. Zbl0991.11040MR1833625
  18. J. Lagarias, Geometric Models for Quasicrystals I. Delone Sets of Finite Type. Discrete Comput. Geom. 21 no 2 (1999), 161–191. Zbl0924.68190MR1668082
  19. J.C. Lagarias, Geometrical models for quasicrystals. II. Local rules under isometry. Disc. and Comp. Geom., 21 no 3 (1999), 345–372. Zbl0930.51019MR1672976
  20. J.C. Lagarias, Mathematical Quasicrystals and the problem of diffraction. Directions in Mathematical Physics, CRM Monograph Series, Vol 13, Ed. M. Baake and R.V. Moody, (2000), 61–94. Zbl1161.52312MR1798989
  21. D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. & Dynam. Sys. 4 (1984), 283–300. Zbl0546.58035MR766106
  22. A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy. J. Théor. Nombres Bordeaux, 10 (1998), 135–162. Zbl0918.11048MR1827290
  23. A. Messaoudi, Frontière du fractal de Rauzy et système de numérotation complexe Acta Arith. 95 (2000), 195–224. Zbl0968.28005MR1793161
  24. Y. Meyer, Algebraic Numbers and Harmonic Analysis. North-Holland (1972). Zbl0267.43001MR485769
  25. H. Minc, Nonnegative matrices. John Wiley and Sons, New York (1988). Zbl0638.15008MR932967
  26. R.V. Moody, Meyer sets and their duals The Mathematics of Long-Range Aperiodic Order, Ed. By R.V. Moody, Kluwer Academic Publishers, (1997), 403–441. Zbl0880.43008MR1460032
  27. G. Muraz and J.-L. Verger-Gaugry, On lower bounds of the density of packings of equal spheres of   ℝ n . Institut Fourier, preprint n o 580 (2003). 
  28. W. Parry, On the β - expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960), 401–416. Zbl0099.28103MR142719
  29. N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics. Lect. Notes Math. 1794, Springer-Verlag, (2003). Zbl1014.11015MR1970385
  30. G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France, 110 (1982), 147–178. Zbl0522.10032MR667748
  31. A. Renyi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957), 477–493. Zbl0079.08901MR97374
  32. D. Ruelle, Statistical Mechanics; Rigorous results. Benjamin, New York, (1969). Zbl0177.57301MR289084
  33. J. Schmeling, Symbolic dynamics for β - shifts and self-normal numbers. Ergod. Th. & Dynam. Sys. 17 (1997), 675–694. Zbl0908.58017MR1452189
  34. K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269–278. Zbl0494.10040MR576976
  35. B. Solomyak, Dynamics of Self-Similar Tilings. Ergod. Th. & Dynam. Sys. 17 (1997), 695–738. Zbl0884.58062MR1452190
  36. W. ThurstonGroups, tilings and finite state automata. Preprint (1989). 

Citations in EuDML Documents

top
  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 (−β)-integers
  4. Gilbert Muraz, Jean-Louis Verger-Gaugry, On a generalization of the Selection Theorem of Mahler
  5. Wolfgang Steiner, On the structure of (−)-integers
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.