Abstract beta-expansions and ultimately periodic representations

Michel Rigo (1) , Wolfgang Steiner (2, 3)

1 Département de Mathématiques [Liège]
2 Institut für Geometrie
3 Fakultät für Mathematik [Wien]

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.

Mots clés: systèmes de numération; langage régulier; substitution; nombre de Pisot

Domaines: Théorie des nombres [math.NT]

Michel Rigo, Wolfgang Steiner. Abstract beta-expansions and ultimately periodic representations. Journal de Théorie des Nombres de Bordeaux, 2005, 17, pp.283-299. ⟨hal-00023235⟩