[en] 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 $\mathbb{Q}(\beta)$ if the dominating eigenvalue $\beta>1$ of the automaton accepting the language is a Pisot number. Moreover, if $\beta$ is neither a Pisot nor a Salem number, then there exist points in $\mathbb{Q}(\beta)$ which do not have any ultimately periodic representation.
Disciplines :
Mathematics
Author, co-author :
Rigo, Michel ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Steiner, Wolfgang
Language :
English
Title :
Abstract beta-expansion and ultimately periodic representations
A. Bertrand, Dèveloppements en base de Pisot et répartition modulo 1. C. R. Acad. Sc. Paris 285 (1977), 419-421
V. Bruyère, G. Hansel, Bertrand numeration systems and recognizability. Latin American Theoretical INformatics (Valparaíso, 1995). Theoret. Comput. Sci. 181 (1997), 17-43
J.-M. Dumont, A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions. J. Theoret. Comput. Sci. 65 (1989), 153-169
S. Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press, New York (1974)
C. Frougny, B. Solomyak, On representation of integers in linear numeration systems. In Ergodic theory of Zd actions (Warwick, 1993-1994), 345-368, London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge (1996)
C. Frougny, Numeration systems. In M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications 90. Cambridge University Press, Cambridge (2002)
P. B. A. Lecomte, M. Rigo, Numeration systems on a regular language. Theory Comput. Syst. 34 (2001), 27-44
P. Lecomte, M. Rigo, On the representation of real numbers using regular languages. Theory Comput. Syst. 35 (2002), 13-38
P. Lecomte, M. Rigo, Real numbers having ultimately periodic representations in abstract numeration systems. Inform. and Comput. 192 (2004), 57-83
W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416
A. Rényi, Representation for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493
K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269-278.
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