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 if the dominating eigenvalue of the automaton accepting the language is a Pisot number. Moreover, if is neither a Pisot nor a Salem number, then there exist points in which do not have any ultimately periodic representation.
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 lorsque la valeur propre dominante de l’automate acceptant le langage est un nombre de Pisot. De plus, si n’est ni un nombre de Pisot, ni un nombre de Salem, alors il existe des points de n’ayant aucune représentation ultimement périodique.
@article{JTNB_2005__17_1_283_0, author = {Michel Rigo and Wolfgang Steiner}, 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}, mrnumber = {2152225}, zbl = {1084.11059}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.491/} }
TY - JOUR AU - Michel Rigo AU - Wolfgang Steiner 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 - https://jtnb.centre-mersenne.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 Michel Rigo %A Wolfgang Steiner %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 https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.491/ %R 10.5802/jtnb.491 %G en %F JTNB_2005__17_1_283_0
Michel Rigo; Wolfgang Steiner. Abstract $\beta $-expansions and ultimately periodic representations. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 283-299. doi : 10.5802/jtnb.491. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.491/
[1] A. Bertrand, Développements en base de Pisot et répartition modulo . 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 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 -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
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