Abstract β-expansions and ultimately periodic representations
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, p. 283-299

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 (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 (β) 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 (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 (β) n’ayant aucune représentation ultimement périodique.

@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},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     pages = {283-299},
     doi = {10.5802/jtnb.491},
     mrnumber = {2152225},
     zbl = {1084.11059},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2005__17_1_283_0}
}
Rigo, Michel; Steiner, Wolfgang. 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. http://www.numdam.org/item/JTNB_2005__17_1_283_0/

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