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On powers of words occurring in binary codings of rotations. (English) Zbl 1113.37002

The question of the repetition of finite words occuring in an infinite sequence has been related to various fields, including the transcendence of real numbers, Diophantine approximation and quasicrystals. In this direction, it has been for instance proved that the real number \(\alpha\) having the Fibonacci sequence as continued fraction expansion is transcendental. Such a result is true for any Sturmian sequence, that is, for any coding of rotation of angle \(\alpha\) with respect to the partition \([0,1-\alpha[ \cup [1-\alpha,1[\). The key point is to prove that Sturmian sequences begins in arbitrary long squares.
In this paper, the author investigates the generalization of these results to characteristic non-degenerate codings of rotations (of angle \(\alpha\) with respect to a partition \([0,\beta[\cup [\beta,1[\), such that \(\beta \not\in {\mathbb Z}+\alpha{\mathbb Z}\)). Sequences that contain only bounded powers of words are characterized. Contrary to Sturmian sequences, such codings of rotations can not begin with arbitrary long squares. This prevents one to generalize the proof of transcendence of numbers with Sturmian continued fraction expansions. However, when restricting to characteristic codings (that is, coding of the orbit of \(0\)) of non-periodic rotation or three interval exchanges transformation, the sequence always has non trivial asymptotic repetitions too far from the beginning, so that real numbers with such a sequence as a continued fraction expansion are transcendental.

MSC:

37B10 Symbolic dynamics
68R15 Combinatorics on words
11J70 Continued fractions and generalizations
11B85 Automata sequences
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[1] Adamczewski, B., Répartitions des suites et substitutions, Acta Arith., 112, 1-22 (2004) · Zbl 1060.11043
[2] Adamczewski, B., Codages de rotations et phénomènes d’autosimilarité, J. Théor. Nombres Bordeaux, 14, 351-386 (2002) · Zbl 1113.37003
[3] Adamczewski, B.; Cassaigne, J., On the transcendence of real numbers with a regular expansion, J. Number Theory, 103, 27-37 (2003) · Zbl 1052.11052
[4] Adamczewski, B.; Damanik, D., Linearly recurrent circle map subshifts and an application to Schrödinger operators, Ann. Henri Poincaré, 3, 1019-1047 (2002) · Zbl 1023.47019
[5] Allouche, J.-P.; Davison, J. L.; Queffélec, M.; Zamboni, L. Q., Transcendence of Sturmian or morphic continued fractions, J. Number Theory, 91, 39-66 (2001) · Zbl 0998.11036
[6] Arnoux, P.; Ferenczi, S.; Hubert, P., Trajectories of rotations, Acta Arith., 87, 209-217 (1999) · Zbl 0921.11033
[7] Arnoux, P.; Rauzy, G., Représentation géométrique de suites de complexité \(2 n + 1\), Bull. Soc. Math. France, 119, 199-215 (1991) · Zbl 0789.28011
[8] C. Baxa, Extremal values of continuants and transcendence of certain continued fractions, Adv. in Appl. Math., in press; C. Baxa, Extremal values of continuants and transcendence of certain continued fractions, Adv. in Appl. Math., in press · Zbl 1063.11019
[9] Berstel, J., On the index of Sturmian words, (Jewels Are Forever (1999), Springer-Verlag: Springer-Verlag Berlin), 287-294 · Zbl 0982.11010
[10] V. Berthé, C. Holton, L.Q. Zamboni, Initial powers of Sturmian words, Preprint, 2003; V. Berthé, C. Holton, L.Q. Zamboni, Initial powers of Sturmian words, Preprint, 2003
[11] Bovier, A.; Ghez, J.-M., Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Comm. Math. Phys., 158, 1, 45-66 (1993) · Zbl 0820.35099
[12] Y. Bugeaud, M. Laurent, Exposants d’approximation de fractions continues sturmiennes, Preprint, 2003; Y. Bugeaud, M. Laurent, Exposants d’approximation de fractions continues sturmiennes, Preprint, 2003
[13] Damanik, D.; Killip, R.; Lenz, D., Uniform spectral properties of one-dimensional quasicrystals. III. \(α\)-continuity, Comm. Math. Phys., 212, 1, 191-204 (2000) · Zbl 1045.81024
[14] Damanik, D.; Lenz, D., Powers in Sturmian sequences, European J. Combin., 24, 4, 377-390 (2003) · Zbl 1030.68068
[15] Davison, J. L., A class of transcendental numbers with bounded partial quotients, (Number Theory and Applications. Number Theory and Applications, Banff, AB, 1988 (1989), Kluwer Academic: Kluwer Academic Dordrecht), 365-371 · Zbl 0693.10028
[16] Davison, J. L., Continued fractions with bounded partial quotients, Proc. Edinb. Math. Soc. (2), 45, 3, 653-671 (2002) · Zbl 1107.11303
[17] Durand, F., A characterization of substitutive sequences using return words, Discrete Math., 179, 89-101 (1998) · Zbl 0895.68087
[18] Ferenczi, S.; Mauduit, C., Transcendence of numbers with a low complexity expansion, J. Number Theory, 67, 146-161 (1997) · Zbl 0895.11029
[19] Justin, J.; Pirillo, G., Fractional powers in Sturmian words, Theoret. Comput. Sci., 255, 1-2, 363-376 (2001) · Zbl 0974.68159
[20] Keane, M., Interval exchange transformations, Math. Z., 141, 25-31 (1975) · Zbl 0278.28010
[21] Lenz, D., Uniform ergodic theorems on subshifts over a finite alphabet, Ergodic Theory Dynam. Systems, 22, 1, 245-255 (2002) · Zbl 1004.37005
[22] Mignosi, F., Infinite words with linear subword complexity, Theoret. Comput. Sci., 65, 2, 221-242 (1989) · Zbl 0682.68083
[23] Mignosi, F.; Pirillo, G., Repetitions in the Fibonacci infinite word, RAIRO Inform. Théor. Appl., 26, 3, 199-204 (1992) · Zbl 0761.68078
[24] Morse, M.; Hedlund, G. A., Symbolic dynamics, Amer. J. Math., 60, 815-866 (1938) · JFM 64.0798.04
[25] Morse, M.; Hedlund, G. A., Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62, 1-42 (1940) · JFM 66.0188.03
[26] Queffélec, M., Transcendance des fractions continues de Thue-Morse, J. Number Theory, 73, 201-211 (1998) · Zbl 0920.11045
[27] Queffélec, M., Irrational numbers with automaton-generated continued fraction expansion, (Dynamical Systems. Dynamical Systems, Luminy-Marseille, 1998 (2000), World Scientific: World Scientific River Edge, NJ), 190-198 · Zbl 1196.11015
[28] Rote, G., Sequences with subword complexity \(2n\), J. Number Theory, 46, 196-213 (1994) · Zbl 0804.11023
[29] Roy, D., Approximation simultanée d’un nombre et de son carré, C. R. Math. Acad. Sci. Paris, 336, 1, 1-6 (2003) · Zbl 1038.11042
[30] Roy, D., Approximation to real numbers by cubic algebraic integers II, Ann. of Math., 158, 1-7 (2003)
[31] Roy, D., Approximation to real numbers by cubic algebraic integers I, Proc. London Math. Soc. (3), 88, 1, 42-62 (2004) · Zbl 1035.11028
[32] Schmidt, W. M., On simultaneous approximations of two algebraic numbers by rationals, Acta Math., 119, 27-50 (1967) · Zbl 0173.04801
[33] Thue, A., Über unendliche Zeichenreihen, Norske Vid. Selsk. Skr. Mat. Nat. Kl.. (Nagell, T., Selected Mathematical Papers of Axel Thue (1977), Universitetsforlaget: Universitetsforlaget Oslo), 7, 139-158 (1906), Reprinted in · JFM 39.0283.01
[34] Vandeth, D., Sturmian words and words with a critical exponent, Theoret. Comput. Sci., 242, 1-2, 283-300 (2000) · Zbl 0944.68148
[35] Veech, W. A., Interval exchange transformations, J. Anal. Math., 33, 222-272 (1978) · Zbl 0455.28006
[36] Veech, W. A., Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115, 201-242 (1982) · Zbl 0486.28014
[37] Weyl, H., Über die Gleichverteilung mod. Eins, Math. Ann., 77, 313-352 (1916) · JFM 46.0278.06
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