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Zbl 1004.37008
Cassaigne, Julien; Ferenczi, Sébastien; Zamboni, Luca Q.
Imbalances in Arnoux-Rauzy sequences. (English)
[J] Ann. Inst. Fourier 50, No.4, 1265-1276 (2000). ISSN 0373-0956; ISSN 1777-5310/e

Arnoux-Rauzy sequences were introduced in [{\it P. Arnoux} and {\it 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 {\it X. Droubay, J. Justin} and {\it 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. \par Note that several papers in the bibliography have appeared: \par [3] in Bull. Belg. Math. Soc., Simon Stevenin 8, 181-207 (2001; Zbl 1007.37001); \par [5] (with a slightly different title) in Discrete Math. 223, 27-53 (2002; Zbl 0970.68124); \par [6] (with a slightly different title) in Trans. Am. Math. Soc. 353, 5121-5144 (2001; Zbl 1142.37302); \par [11] in J. Théor. Nombrés Bordx. 13, 371-394 (2001; Zbl 1038.37010); \par [12] in Theor. Comput. Sci. 255, 539-553 (2001; Zbl 0981.68126); \par [15] in Ann. Inst. Fourier 51, 861-901 (2001; Zbl 1029.11036); \par [22] in Acta Arith. 95, 195-224 (2000; Zbl 0968.28005); \par [29] in Acta Arith. 95, 167-184 (2000; Zbl 0953.11007); \par [30] in Acta Arith. 96, 261-278 (2001; Zbl 0973.11030). \par Also note that [20] has appeared: Cambridge, 2002.
[J.-P.Allouche (Orsay)]

MSC 2000:: *37B10 Symbolic dynamics
68R15 Combinatorics on words
11B83 Special sequences of integers and polynomials

Keywords: infinite words; codings of rotations; return times; bounded reaminder sets; balanced sequences; Arnoux-Rauzy sequences; Sturmian sequences

Citations: Zbl 0789.28011; Zbl 0981.68126; Zbl 1007.37001; Zbl 0970.68124; Zbl 1142.37302; Zbl 1038.37010; Zbl 1029.11036; Zbl 0968.28005; Zbl 0953.11007; Zbl 0973.11030

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