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

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.

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.

@article{JTNB_2004__16_1_125_0,
     author = {Gazeau, Jean-Pierre and Verger-Gaugry, Jean-Louis},
     title = {Geometric study of the beta-integers for a Perron number and mathematical quasicrystals},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {1},
     year = {2004},
     pages = {125-149},
     doi = {10.5802/jtnb.437},
     mrnumber = {2145576},
     zbl = {1075.11007},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2004__16_1_125_0}
}
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, Volume 16 (2004) no. 1, pp. 125-149. doi : 10.5802/jtnb.437. http://www.numdam.org/item/JTNB_2004__16_1_125_0/

[Ak] S. Akiyama, Cubic Pisot units with finite β-expansions. Algebraic number theory and Diophantine analysis, (Graz, 1998), de Gruyter, Berlin, (2000), 11–26. | MR 1770451 | Zbl 1001.11038

[Ak1] S. Akiyama, Pisot numbers and greedy algorithm. Number Theory, (Eger, 1996), de Gruyter, Berlin, (1998), 9–21. | MR 1628829 | Zbl 0919.11063

[ABEI] 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. | MR 1888763 | Zbl 1017.68147

[AI] P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. 8 (2001), 181–207. | MR 1838930 | Zbl 1007.37001

[Be] A. Bertrand, Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sc. Paris, Série A, t. 285 (1977), 419–421. | MR 447134 | Zbl 0362.10040

[Be1] 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. | MR 1029085 | Zbl 0695.10005

[Be2] A. Bertrand - Mathis, Développement en base  θ , Répartition modulo un de la suite  (xθ n ) n0 . Langages codes et  θ - shift, Bull. Soc. Math. France, 114 (1986), 271–323. | Numdam | MR 878240 | Zbl 0628.58024

[Be3] 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. | Zbl 0766.11043

[Be4] A. Bertrand - Mathis, Questions diverses relatives aux systèmes codés: applications au θ-shift. Preprint.

[Bl] F. Blanchard, β-expansions and Symbolic Dynamics. Theor. Comp. Sci. 65 (1989), 131–141. | MR 1020481 | Zbl 0682.68081

[Bu] 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. | MR 1644115 | Zbl 0941.52019

[Fro] C. Frougny, Number representation and finite automata. London Math. Soc. Lecture Note Ser. 279 (2000), 207–228. | MR 1776760 | Zbl 0976.11003

[Fro1] Ch. Frougny and B. Solomyak, Finite beta-expansions. Ergod. Theor. Dynam. Syst. 12 (1992), 713–723. | MR 1200339 | Zbl 0814.68065

[Gaz] 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. | MR 1460024 | Zbl 0887.11043

[HS] M. W. Hirsch and S. Smale, Differential Equations. Dynamical Systems and Linear Algebra, Academic Press, New York, (1974). | MR 486784 | Zbl 0309.34001

[IK] S. Ito and M. Kimura, On the Rauzy fractal. Japan J. Indust. Appl. Math. 8 (1991), 461–486. | MR 1137652 | Zbl 0734.28010

[IS] S. Ito and Y. Sano, On periodic β - expansions of Pisot numbers and Rauzy fractals. Osaka J. Math. 38 (2001), 349–368. | MR 1833625 | Zbl 0991.11040

[La] J. Lagarias, Geometric Models for Quasicrystals I. Delone Sets of Finite Type. Discrete Comput. Geom. 21 no 2 (1999), 161–191. | MR 1668082 | Zbl 0924.68190

[La1] J.C. Lagarias, Geometrical models for quasicrystals. II. Local rules under isometry. Disc. and Comp. Geom., 21 no 3 (1999), 345–372. | MR 1672976 | Zbl 0930.51019

[La2] 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. | MR 1798989 | Zbl 1161.52312 | Zbl 01584913

[Li] D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. & Dynam. Sys. 4 (1984), 283–300. | MR 766106 | Zbl 0546.58035

[Me] A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy. J. Théor. Nombres Bordeaux, 10 (1998), 135–162. | Numdam | MR 1827290 | Zbl 0918.11048

[Me1] A. Messaoudi, Frontière du fractal de Rauzy et système de numérotation complexe Acta Arith. 95 (2000), 195–224. | MR 1793161 | Zbl 0968.28005

[Mey] Y. Meyer, Algebraic Numbers and Harmonic Analysis. North-Holland (1972). | MR 485769 | Zbl 0267.43001

[Mi] H. Minc, Nonnegative matrices. John Wiley and Sons, New York (1988). | MR 932967 | Zbl 0638.15008

[Mo] 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. | MR 1460032 | Zbl 0880.43008

[MVG] G. Muraz and J.-L. Verger-Gaugry, On lower bounds of the density of packings of equal spheres of   n . Institut Fourier, preprint n o 580 (2003).

[Pa] W. Parry, On the β - expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960), 401–416. | MR 142719 | Zbl 0099.28103

[PF] N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics. Lect. Notes Math. 1794, Springer-Verlag, (2003). | MR 1970385 | Zbl 1014.11015

[Ra] G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France, 110 (1982), 147–178. | Numdam | MR 667748 | Zbl 0522.10032

[Re] A. Renyi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957), 477–493. | MR 97374 | Zbl 0079.08901

[Ru] D. Ruelle, Statistical Mechanics; Rigorous results. Benjamin, New York, (1969). | MR 289084 | Zbl 0177.57301

[Sc] J. Schmeling, Symbolic dynamics for β- shifts and self-normal numbers. Ergod. Th. & Dynam. Sys. 17 (1997), 675–694. | MR 1452189 | Zbl 0908.58017

[Sch] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269–278. | MR 576976 | Zbl 0494.10040

[So] B. Solomyak, Dynamics of Self-Similar Tilings. Ergod. Th. & Dynam. Sys. 17 (1997), 695–738. | MR 1452190 | Zbl 0884.58062

[Th] W. Thurston Groups, tilings and finite state automata. Preprint (1989).