\(\beta\)-shift, numeration systems, and automata (Q1910684)

In this paper general numeration systems are considered from the point of view of formal languages. The author mainly considers enumeration systems defined via (not necessarily finite) linear recurrences as discussed by \textit{P. J. Grabner}, \textit{P. Liardet} and the reviewer [Acta Arith. 70, 103-123 (1995; Zbl 0822.11008)]. The language of the numeration system is described explicitly and the regularity is investigated. Moreover, the author gives a characterization of the arithmetico-geometric sequences and the mixed radix sequences such that the language of the corresponding numeration system is regular. Furthermore, Ostrowski systems of enumeration (related to continued fraction expansions) are studied and a new proof of a recent result of Shallit is given: the Ostrowski systems have a regular language if and only if they are associated to a quadratic irrational.