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Acta Mathematica

, Volume 195, Issue 1, pp 1–20 | Cite as

On the complexity of algebraic numbers, II. Continued fractions

  • Boris Adamczewski
  • Yann Bugeaud
Article

Keywords

Turing Machine Algebraic Number Finite Automaton Continue Fraction Expansion Proper Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Adamczewski, B. & Bugeaud, Y., On the complexity of algebraic numbers I. Expansions in integer bases. To appear inAnn. of Math.Google Scholar
  2. [2]
    Adamczewski, B., Bugeaud, Y. & Davison, J. L., Continued fractions and transcendental numbers. Preprint, 2005.Google Scholar
  3. [3]
    Adamczewski, B., Bugeaud, Y. &Luca, F., Sur la complexité des nombres algébriques.C. R. Acad. Sci. Paris, 339 (2000), 19–34.Google Scholar
  4. [4]
    Allouche, J.-P., Nouveaux résultats de transcendance de réels à développement non aléatoire.Gaz. Math., 84 (2000), 19–34.MathSciNetGoogle Scholar
  5. [5]
    Allouche, J.-P., Davison, J. L., Queffélec, M. &Zamboni, L. Q., Transcendence of Sturmian or morphic continued fractions.J. Number Theory, 91 (2001), 39–66.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Allouche, J.-P. &Shallit, J.,Automatic Sequences: Theory, Applications, Generalizations. Cambridge Univ. Press, Cambridge, 2003.MATHGoogle Scholar
  7. [7]
    Bailey, D. H., Borwein, J. M., Crandall, R. E. &Pomerance, C., On the binary expansions of algebraic numbers.J. Théor. Nombres Bordeaux, 16 (2004), 487–518.MATHMathSciNetGoogle Scholar
  8. [8]
    Baxa, C., Extremal values of continuants and transcendence of certain continued fractions.Adv. in Appl. Math., 32 (2004), 754–790.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Borel, É., Sur les chiffres décimaux de\(\sqrt 2 \) et divers problèmes de probabilités en chaîne.C. R. Acad. Sci. Paris, 230 (1950), 591–593.MATHMathSciNetGoogle Scholar
  10. [10]
    Cassaigne, J., Sequences with grouped factors, inDevelopments in Language Theory, III (Thessaloniki, 1997), pp. 211–222. World Sci. Publishing, River Edge, NJ, 1998.Google Scholar
  11. [11]
    Cobham, A., Uniform tag sequences.Math. Systems Theory, 6 (1972), 164–192.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Davison, J. L., A class of transcendental numbers with bounded partial quotients, inNumber Theory and Applications (Banff, AB, 1988), pp. 365–371, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 265. Kluwer, Dordrecht, 1989.Google Scholar
  13. [13]
    — Continued fractions with bounded partial quotients.Proc. Edinburgh Math. Soc., 45 (2002), 653–671.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Durand, F., Linearly recurrent subshifts have a finite number of non-periodic subshift factors.Ergodic Theory Dynam. Systems, 20 (2000), 1061–1078; Corrigendum and addendum.Ibid.Durand, F., Linearly recurrent subshifts have a finite number of non-periodic subshift factorsErgodic Theory Dynam. Systems, 23 (2003), 663–669.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Hartmanis, J. &Stearns, R. E., On the computational complexity of algorithms.Trans. Amer. Math. Soc., 117 (1965), 285–306.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Khinchin, A. Ya.,Continued Fractions, 2nd edition. Gosudarstv. Izdat. Techn.-Teor. Lit., Moscow-Leningrad, 1949 (Russian); English translation: The University of Chicago Press, Chicago-London, 1964.Google Scholar
  17. [17]
    Lang, S.,Introduction to Diophantine Approximations, 2nd edition. Springer, New York, 1995.MATHGoogle Scholar
  18. [18]
    Mendès France, M., Principe de la symétrie perturbée inSéminaire de Théorie des Nombres (Paris, 1979–80), pp. 77–98. Progr. Math., 12. Birkhäuser, Boston, MA, 1981.Google Scholar
  19. [19]
    Morse, M. &Hedlung, G. A., Symbolic dynamics.Amer. J. Math., 60 (1938), 815–866.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Perron, O.,Die Lehre von den Kettenbrüchen. Teubner, Leipzig, 1929.MATHGoogle Scholar
  21. [21]
    Queffélec, M., Transcendance des fractions continues de Thue-Morse.J. Number Theory, 73 (1998), 201–211.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    — Irrational numbers with automaton-generated continued fraction expansion, inDynamical Systems (Luminy-Marseille, 1998), pp. 190–198. World Sci. Publishing, River Edge, NJ, 2000.Google Scholar
  23. [23]
    Roth, K. F., Rational approximations to algebraic numbers.Mathematika, 2 (1955), 1–20; Corrigendum.Ibid.Roth, K. F., Rational approximations to algebraic numbers.Mathematika, 2 (1955), 168.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Schmidt, W. M., On simultaneous approximations of two algebraic numbers by rationals.Acta Math., 119 (1967), 27–50.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    —, Norm form equations.Ann. of Math., 96 (1972), 526–551.CrossRefMathSciNetGoogle Scholar
  26. [26]
    —,Diophantine Approximation. Lecture Notes in Math., 785. Springer, Berlin, 1980.MATHGoogle Scholar
  27. [27]
    Shallit, J., Real numbers with bounded partial quotients: a survey.Enseign. Math., 38 (1992), 151–187.MathSciNetMATHGoogle Scholar
  28. [28]
    Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem.Proc. London Math. Soc., 42 (1937), 230–265.CrossRefGoogle Scholar
  29. [29]
    Waldschmidt, M., Un demi-siècle de transcendance, inDevelopment of Mathematics 1950–2000, pp. 1121–1186. Birkhäuser, Basel, 2000.Google Scholar

Copyright information

© Institut Mittag-Leffler 2005

Authors and Affiliations

  • Boris Adamczewski
    • 1
  • Yann Bugeaud
    • 2
  1. 1.CNRS, Institute Camille JordanUniversité Claude Bernard Lyon IVilleurbanne CedexFrance
  2. 2.Université Louis PasteurU.F.R. de mathématiqueStrasbourg CedexFrance

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