Abstract β-expansions and ultimately periodic representations
Rigo, Michel1 ; Steiner, Wolfgang2
1 Université de Liège, Institut de Mathématiques, Grande Traverse 12 (B 37), B-4000 Liège, Belgium.
2 TU Wien, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8-10/104, A-1040 Wien, Austria Universität Wien, Institut für Mathematik, Strudlhofgasse 4, A-1090 Wien, Austria.
Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 283-299.

Pour les systèmes de numération abstraits construits sur des langages réguliers exponentiels (comme par exemple, ceux provenant des substitutions), nous montrons que l’ensemble des nombres réels possédant une représentation ultimement périodique est ℚ(β) lorsque la valeur propre dominante β>1 de l’automate acceptant le langage est un nombre de Pisot. De plus, si β n’est ni un nombre de Pisot, ni un nombre de Salem, alors il existe des points de ℚ(β) n’ayant aucune représentation ultimement périodique.

For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is ℚ(β) if the dominating eigenvalue β>1 of the automaton accepting the language is a Pisot number. Moreover, if β is neither a Pisot nor a Salem number, then there exist points in ℚ(β) which do not have any ultimately periodic representation.

DOI : 10.5802/jtnb.491
Rigo, Michel 1 ; Steiner, Wolfgang 2

1 Université de Liège, Institut de Mathématiques, Grande Traverse 12 (B 37), B-4000 Liège, Belgium.
2 TU Wien, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8-10/104, A-1040 Wien, Austria Universität Wien, Institut für Mathematik, Strudlhofgasse 4, A-1090 Wien, Austria. 
@article{JTNB_2005__17_1_283_0,
     author = {Rigo, Michel and Steiner, Wolfgang},
     title = {Abstract $\beta $-expansions and ultimately periodic representations},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {283--299},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.491},
     zbl = {1084.11059},
     mrnumber = {2152225},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.491/}
}
TY  - JOUR
AU  - Rigo, Michel
AU  - Steiner, Wolfgang
TI  - Abstract $\beta $-expansions and ultimately periodic representations
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
SP  - 283
EP  - 299
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.491/
DO  - 10.5802/jtnb.491
LA  - en
ID  - JTNB_2005__17_1_283_0
ER  - 
%0 Journal Article
%A Rigo, Michel
%A Steiner, Wolfgang
%T Abstract $\beta $-expansions and ultimately periodic representations
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 283-299
%V 17
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.491/
%R 10.5802/jtnb.491
%G en
%F JTNB_2005__17_1_283_0
Rigo, Michel; Steiner, Wolfgang. Abstract $\beta $-expansions and ultimately periodic representations. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 283-299. doi : 10.5802/jtnb.491. http://www.numdam.org/articles/10.5802/jtnb.491/

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

[2] V. Bruyère, G. Hansel, Bertrand numeration systems and recognizability. Latin American Theoretical INformatics (Valparaíso, 1995). Theoret. Comput. Sci. 181 (1997), 17–43. | MR | Zbl

[3] J.-M. Dumont, A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions. J. Theoret. Comput. Sci. 65 (1989), 153–169. | MR | Zbl

[4] S. Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press , New York (1974). | MR | Zbl

[5] C. Frougny, B. Solomyak, On representation of integers in linear numeration systems. In Ergodic theory of Z d actions (Warwick, 1993–1994), 345–368, London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge (1996). | MR | Zbl

[6] C. Frougny, Numeration systems. In M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications 90. Cambridge University Press, Cambridge (2002).

[7] P. B. A. Lecomte, M. Rigo, Numeration systems on a regular language. Theory Comput. Syst. 34 (2001), 27–44. | MR | Zbl

[8] P. Lecomte, M. Rigo, On the representation of real numbers using regular languages. Theory Comput. Syst. 35 (2002), 13–38. | MR | Zbl

[9] P. Lecomte, M. Rigo, Real numbers having ultimately periodic representations in abstract numeration systems. Inform. and Comput. 192 (2004), 57–83. | MR | Zbl

[10] W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416. | Zbl

[11] A. Rényi, Representation for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. | MR | Zbl

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

Cité par Sources :