\(\beta\)-shift, numeration systems, and automata
type: Item
instance of: scholarly article
title: \(\beta\)-shift, numeration systems, and automata
zbMATH Open document ID: 0843.11013
author: Nathalie Loraud
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-08-13T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/226997
full work available at URL: jtnb7-2.html
review text: 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.
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 68Q45
Mathematics Subject Classification ID: 11A63
zbMATH DE Number: 858576
zbMATH Keywords: \(\beta\)-shift
zbMATH Keywords: automata
zbMATH Keywords: numeration systems
zbMATH Keywords: formal languages
zbMATH Keywords: linear recurrences
zbMATH Keywords: regularity
zbMATH Keywords: arithmetico-geometric sequences
zbMATH Keywords: mixed radix sequences
zbMATH Keywords: Ostrowski systems
zbMATH Keywords: continued fraction expansions
zbMATH Keywords: regular language
zbMATH Keywords: quadratic irrational
zbMATH Keywords: numeration system
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
OpenAlex ID: W2328530075
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DOI: 10.5802/JTNB.153
description: scientific article; zbMATH DE number 858576