no. 1
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. 
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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://archive.numdam.org/articles/10.5802/jtnb.491/

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