Imbalances in Arnoux-Rauzy sequences
type: Item
instance of: scholarly article
title: Imbalances in Arnoux-Rauzy sequences
zbMATH Open document ID: 1004.37008
author: Julien Cassaigne
author: Sébastien Ferenczi
author: Luca Q. Zamboni
publication date: 2000-07-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75456
review text: Arnoux-Rauzy sequences were introduced in [\textit{P. Arnoux} and \textit{G. Rauzy}, Représentation géométrique de suites de complexité \(2n+1\), Bull. Soc. Math. Fr. 119, No. 2, 199-215 (1991; Zbl 0789.28011)]. The authors formulated at this time the conjecture that these sequences can be obtained as codings of rotations on the two-dimensional torus. This conjecture is disproved in the paper under review, where the authors actually disprove a conjecture of \textit{X. Droubay, J. Justin} and \textit{G. Pirillo} [Theor. Comput. Sci. 255, No. 1-2, 539-553 (2001; Zbl 0981.68126)] by showing that these sequences are not necessarily \(N\)-balanced: recall that a sequence is \(N\)-balanced for some integer \(N\) if for each letter the numbers of occurrences of this letter in any two words in the sequence having the same length differ by at most \(N\). Sturmian sequences are known to be 1-balanced. Note that several papers in the bibliography have appeared: [3] in Bull. Belg. Math. Soc., Simon Stevenin 8, 181-207 (2001; Zbl 1007.37001); [5] (with a slightly different title) in Discrete Math. 223, 27-53 (2002; Zbl 0970.68124); [6] (with a slightly different title) in Trans. Am. Math. Soc. 353, 5121-5144 (2001; Zbl 1142.37302); [11] in J. Théor. Nombrés Bordx. 13, 371-394 (2001; Zbl 1038.37010); [12] in Theor. Comput. Sci. 255, 539-553 (2001; Zbl 0981.68126); [15] in Ann. Inst. Fourier 51, 861-901 (2001; Zbl 1029.11036); [22] in Acta Arith. 95, 195-224 (2000; Zbl 0968.28005); [29] in Acta Arith. 95, 167-184 (2000; Zbl 0953.11007); [30] in Acta Arith. 96, 261-278 (2001; Zbl 0973.11030). Also note that [20] has appeared: Cambridge, 2002.
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 68R15
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 1478822
zbMATH Keywords: infinite words
zbMATH Keywords: codings of rotations
zbMATH Keywords: return times
zbMATH Keywords: bounded reaminder sets
zbMATH Keywords: balanced sequences
zbMATH Keywords: Arnoux-Rauzy sequences
zbMATH Keywords: Sturmian sequences
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
OpenAlex ID: W1984528804
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DOI: 10.5802/AIF.1792
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1478822