Imbalances in Arnoux-Rauzy sequences
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.
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