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Transcendence of Sturmian or morphic continued fractions. (English) Zbl 0998.11036

The authors prove, using a theorem of W. Schmidt, that if the sequence of partial quotients of the continued fraction expansion of a positive irrational real number takes only two values and begins with arbitrary long blocks which are ‘almost squares’, then this number is either quadratic or transcendental. This result applies in particular to real numbers whose partial quotients form a Sturmian (or quasi-Sturmian) sequence, or are given by the sequence \((1 + (\lfloor n\alpha \rfloor \bmod 2))_{n\geq 0}\), or are a ‘repetitive’ fixed point of a binary morphism satisfying some technical conditions.
Reviewer: R.F.Tichy (Graz)

MSC:

11J70 Continued fractions and generalizations
11B83 Special sequences and polynomials
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References:

[1] Alessandri, P., Classification et représentation des suites de complexité \(n+2 (1995)\)
[2] Alessandri, P., Codages de rotations et basses complexités (1996)
[3] Allouche, J.-P.; Arnold, A.; Berstel, J.; Brlek, S.; Jockusch, W.; Plouffe, S.; Sagan, B. E., A relative of the Thue-Morse sequence, Discrete Math, 139, 455-461 (1995) · Zbl 0839.11007
[4] Allouche, J.-P.; Mendès France, M., Automata and automatic sequences, (Axel, F.; Gratias, D., Beyond Quasicrystals. Beyond Quasicrystals, Les Éditions de Physique (1995), Springer-Verlag: Springer-Verlag Berlin/New York), 293-367 · Zbl 0881.11026
[5] Allouche, J.-P.; Peyrière, J., Sur une formule de récurrence sur les traces de produits de matrices associés à certaines substitutions, C. R. Acad. Sci. Paris Sér. II, 302, 1135-1136 (1986) · Zbl 0587.65033
[6] Allouche, J.-P.; Zamboni, L. Q., Algebraic irrational binary numbers cannot be fixed points of non-trivial constant-length or primitive morphisms, J. Number Theory, 69, 119-124 (1998) · Zbl 0918.11016
[7] Arnoux, P.; Rauzy, G., Représentation géométrique de suites de complexité \(2n+1\), Bull. Soc. Math. France, 119, 199-215 (1991) · Zbl 0789.28011
[8] Axel, F.; Allouche, J.-P.; Kleman, M.; Mendès France, M.; Peyrière, J., Vibrational modes in a one dimensional “quasi-alloy”: the Morse case, J. Physique, 47, C3.181-C3.186 (1986) · Zbl 0696.10050
[9] Baker, A., Continued fractions of transcendental numbers, Mathematika, 9, 1-8 (1962) · Zbl 0105.03903
[10] J. Berstel, private communication, 1998.; J. Berstel, private communication, 1998.
[11] Berstel, J., On the index of Sturmian words, Jewels Are Forever (1999), Springer-Verlag: Springer-Verlag Berlin, p. 287-294 · Zbl 0982.11010
[12] Berthé, V., Fréquences des facteurs des suites sturmiennes, Theoret. Comput. Sci, 165, 295-309 (1996) · Zbl 0872.11018
[13] Bovier, A.; Ghez, J.-M., Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Comm. Math. Phys, 158, 45-66 (1993) · Zbl 0820.35099
[14] Brlek, S., Enumeration of factors in the Thue-Morse word, Discrete Appl. Math, 24, 83-96 (1989) · Zbl 0683.20045
[15] W.-T. Cao, and, Z.-Y. Wen, Some properties of the Sturmian sequences, preprint, 1999.; W.-T. Cao, and, Z.-Y. Wen, Some properties of the Sturmian sequences, preprint, 1999.
[16] J. Cassaigne, Sequences with grouped factors, in DLT’97, Developments in Language Theory III, Thessaloniki, Aristotle University of Thessaloniki, 1998, pp. 211-222. Also available at .; J. Cassaigne, Sequences with grouped factors, in DLT’97, Developments in Language Theory III, Thessaloniki, Aristotle University of Thessaloniki, 1998, pp. 211-222. Also available at .
[17] Cassaigne, J., Limit values of the recurrence quotient of Sturmian sequences, Theoret. Comput. Sci, 218, 3-12 (1999) · Zbl 0916.68115
[18] Coven, E. M., Sequences with minimal block growth II, Math. Systems Theory, 8, 376-382 (1975) · Zbl 0299.54032
[19] Coven, E. M.; Hedlund, G. A., Sequences with minimal block growth, Math. Systems Theory, 7, 138-153 (1973) · Zbl 0256.54028
[20] Damanik, D.; Killip, R.; Lenz, D., Uniform spectral properties of one-dimensional quasicrystals, III. \(α\)-continuity, Comm. Math. Phys, 212, 191-204 (2000) · Zbl 1045.81024
[21] D. Damanik, and, D. Lenz, The index of Sturmian sequences, European J. Combin, in press.; D. Damanik, and, D. Lenz, The index of Sturmian sequences, European J. Combin, in press. · Zbl 1002.11020
[22] J. L. Davison, Exponential growth in a class of difference equations, presented at the “Ontario Mathematics Meeting, Carleton University,”, 1977.; J. L. Davison, Exponential growth in a class of difference equations, presented at the “Ontario Mathematics Meeting, Carleton University,”, 1977.
[23] Davison, J. L., A class of transcendental numbers with bounded partial quotients, Number Theory and Applications (Banff, AB, 1988). Number Theory and Applications (Banff, AB, 1988), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci, 265 (1989), Kluwer Academic: Kluwer Academic Dordrecht, p. 365-371 · Zbl 0693.10028
[24] Didier, G., Combinatoire des codages de rotations, Acta Arith, 85, 157-177 (1998) · Zbl 0910.11007
[25] Didier, G., Codages de rotations et fractions continues, J. Number Theory, 71, 275-306 (1998) · Zbl 0921.11015
[26] Didier, G., Caractérisation des \(N\)-écritures et application à l’étude des suites de complexité ultimement \(n+c^{ ste } \), Theoret. Comput. Sci, 215, 31-49 (1999) · Zbl 0913.68163
[27] Ferenczi, S., Les transformations de Chacon: combinatoire, structure géométrique, lien avec les systèmes de complexité \(2n+1\), Bull. Soc. Math. France, 123, 271-292 (1995) · Zbl 0855.28008
[28] Ferenczi, S.; Mauduit, C., Transcendence of numbers with a low complexity expansion, J. Number Theory, 67, 146-161 (1997) · Zbl 0895.11029
[29] Flor, P., Ein Verteilungsproblem für arithmetische Folgen, Abh. Math. Sem. Univ. Hamburg, 25, 62-70 (1961) · Zbl 0118.28904
[30] Gottschalk, W. H., Substitution minimal sets, Trans. Amer. Math. Soc, 109, 467-491 (1963) · Zbl 0121.18002
[31] Harman, G.; Wong, K. C., A note on the metrical theory of continued fractions, Amer. Math. Monthly, 107, 834-837 (2000) · Zbl 0982.11043
[32] A. Heinis, On low complexity \(Z\)-words and their factors, J. Théor. Nombres Bordeaux, in press.; A. Heinis, On low complexity \(Z\)-words and their factors, J. Théor. Nombres Bordeaux, in press. · Zbl 1013.68155
[33] C. Holton, and, L. Q. Zamboni, Overlaps in Sturmian words, in preparation.; C. Holton, and, L. Q. Zamboni, Overlaps in Sturmian words, in preparation.
[34] Justin, J.; Pirillo, G., Fractional powers in Sturmian words, Theoret. Comput. Sci, 255, 363-376 (2001) · Zbl 0974.68159
[35] Khintchine, A. Y., Continued Fractions (1949), Gosudarstv. Izdat. Tehn.-Teor. Lit: Gosudarstv. Izdat. Tehn.-Teor. Lit Moscow/Leningrad
[36] Knuth, D. E.; Morris, J.; Pratt, V., Fast pattern matching in strings, Siam J. Comput, 6, 323-350 (1977) · Zbl 0372.68005
[37] Lévy, P., Sur le développement en fraction continue d’un nombre choisi au hasard, Compositio Math, 3, 286-303 (1936) · JFM 62.0246.01
[38] Liardet, P.; Stambul, P., Séries de Engel et fractions continuées, J. Théor. Nombres Bordeaux, 12, 37-68 (2000) · Zbl 1007.11045
[39] M. Lothaire, in, Algebraic Combinatorics on Words, (, J. Berstel and P. Séébold, Eds.), Chap. 3, Cambridge University Press, in press.; M. Lothaire, in, Algebraic Combinatorics on Words, (, J. Berstel and P. Séébold, Eds.), Chap. 3, Cambridge University Press, in press. · Zbl 1001.68093
[40] Maillet, E., Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions (1906), Gauthier-Villars: Gauthier-Villars Paris · JFM 37.0237.02
[41] Michel, P., Stricte ergodicité d’ensembles minimaux de substitution, C. R. Acad. Sci. Paris, Sér. A, 278, 811-813 (1974) · Zbl 0274.60028
[42] Michel, P., Stricte ergodicité d’ensembles minimaux de substitution, Théorie ergodique, Actes Journées Ergodiques, Rennes, 1973/1974. Théorie ergodique, Actes Journées Ergodiques, Rennes, 1973/1974, Lecture Notes in Mathematics, 532 (1976), Springer-Verlag: Springer-Verlag Berlin, p. 189-201 · Zbl 0331.54036
[43] Mignotte, M., Quelques remarques sur l’approximation rationnelle des nombres algébriques, J. Reine Angew. Math, 268/269, 341-347 (1974) · Zbl 0284.10011
[44] Morse, M.; Hedlund, G. A., Symbolic dynamics II: Sturmian trajectories, Amer. J. Math, 62, 1-42 (1940) · JFM 66.0188.03
[45] Oxtoby, J. C., Ergodic sets, Bull. Amer. Math. Soc, 58, 116-136 (1952) · Zbl 0046.11504
[46] Paul, M. E., Minimal symbolic flows having minimal block growth, Math. Systems Theory, 8, 309-315 (1975) · Zbl 0306.54056
[47] Peyrière, J., Trace maps, (Axel, F.; Gratias, D., Beyond Quasicrystals (1995), Springer/Les Éditions de Physique), 465-480 · Zbl 0882.20017
[48] Prodinger, H.; Urbanek, F. J., Infinite 0-1-sequences without long adjacent identical blocks, Discrete Math, 28, 277-289 (1979) · Zbl 0421.05007
[49] Queffélec, M., Transcendance des fractions continues de Thue-Morse, J. Number Theory, 73, 201-211 (1998) · Zbl 0920.11045
[50] Queffélec, M., Irrational numbers with automaton-generated continued fraction expansion, (Gambaudo, J.-M.; Hubert, P.; Vaienti, S., Dynamical systems. From crystal to chaos. Proceedings of the conference in honor of G. Rauzy on his 60th birthday, held in Luminy-Marseille, France, July 6-10, 1998 (2000), World Scientific: World Scientific Singapore), 190-198 · Zbl 1196.11015
[51] Risley, R. N.; Zamboni, L. Q., A generalization of Sturmian flows; combinatorial structure and transcendence, Acta Arith, 95, 167-184 (2000) · Zbl 0953.11007
[52] Rote, G., Sequences with subword complexity \(2n\), J. Number Theory, 46, 196-213 (1994) · Zbl 0804.11023
[53] Schmidt, W., On simultaneous approximations of two algebraic numbers by rationals, Acta Math, 119, 27-50 (1967) · Zbl 0173.04801
[54] Shallit, J., Real numbers with bounded partial quotients: a survey, Enseign. Math, 38, 151-187 (1992) · Zbl 0753.11006
[55] Shallit, J., Automaticity IV: sequences, sets, and diversity, J. Théor. Nombres Bordeaux, 8, 347-367 (1996) · Zbl 0876.11010
[56] Smith, H. J.S., Note on continued fractions, Messenger Math, 6, 1-14 (1876)
[57] Vandeth, D., Sturmian words and words with a critical exponent, Theoret. Comput. Sci, 242, 283-300 (2000) · Zbl 0944.68148
[58] Wen, Z.-X.; Wen, Z.-Y., Remarques sur la suite engendrée par des substitutions composées, Ann. Fac. Sci. Toulouse, 9, 55-63 (1988) · Zbl 0657.10060
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