Acta Mathematica

On the complexity of algebraic numbers, II. Continued fractions

Boris Adamczewski and Yann Bugeaud

Full-text: Open access

Note

The second author was supported by the Austrian Science Fund (FWF), Grant M822-N12.

Article information

Source
Acta Math., Volume 195, Number 1 (2005), 1-20.

Dates
Received: 16 May 2005
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891760

Digital Object Identifier
doi:10.1007/BF02588048

Mathematical Reviews number (MathSciNet)
MR2233683

Zentralblatt MATH identifier
1195.11093

Rights
2005 © Institut Mittag-Leffler

Citation

Adamczewski, Boris; Bugeaud, Yann. On the complexity of algebraic numbers, II. Continued fractions. Acta Math. 195 (2005), no. 1, 1--20. doi:10.1007/BF02588048. https://projecteuclid.org/euclid.acta/1485891760


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References

  • Adamczewski, B. & Bugeaud, Y., On the complexity of algebraic numbers I. Expansions in integer bases. To appear in Ann. of Math.
  • Adamczewski, B., Bugeaud, Y. & Davison, J. L., Continued fractions and transcendental numbers. Preprint, 2005.
  • Adamczewski, B., Bugeaud, Y. & Luca, F., Sur la complexité des nombres algébriques. C. R. Acad. Sci. Paris, 339 (2000), 19–34.
  • Allouche, J.-P., Nouveaux résultats de transcendance de réels à développement non aléatoire. Gaz. Math., 84 (2000), 19–34.
  • 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.
  • Allouche, J.-P. & Shallit, J., Automatic Sequences: Theory, Applications, Generalizations. Cambridge Univ. Press, Cambridge, 2003.
  • 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.
  • Baxa, C., Extremal values of continuants and transcendence of certain continued fractions. Adv. in Appl. Math., 32 (2004), 754–790.
  • 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.
  • Cassaigne, J., Sequences with grouped factors, in Developments in Language Theory, III (Thessaloniki, 1997), pp. 211–222. World Sci. Publishing, River Edge, NJ, 1998.
  • Cobham, A., Uniform tag sequences. Math. Systems Theory, 6 (1972), 164–192.
  • Davison, J. L., A class of transcendental numbers with bounded partial quotients, in Number Theory and Applications (Banff, AB, 1988), pp. 365–371, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 265. Kluwer, Dordrecht, 1989.
  • — Continued fractions with bounded partial quotients. Proc. Edinburgh Math. Soc., 45 (2002), 653–671.
  • 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 factors Ergodic Theory Dynam. Systems, 23 (2003), 663–669.
  • Hartmanis, J. & Stearns, R. E., On the computational complexity of algorithms. Trans. Amer. Math. Soc., 117 (1965), 285–306.
  • 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.
  • Lang, S., Introduction to Diophantine Approximations, 2nd edition. Springer, New York, 1995.
  • Mendès France, M., Principe de la symétrie perturbée in Séminaire de Théorie des Nombres (Paris, 1979–80), pp. 77–98. Progr. Math., 12. Birkhäuser, Boston, MA, 1981.
  • Morse, M. & Hedlung, G. A., Symbolic dynamics. Amer. J. Math., 60 (1938), 815–866.
  • Perron, O., Die Lehre von den Kettenbrüchen. Teubner, Leipzig, 1929.
  • Queffélec, M., Transcendance des fractions continues de Thue-Morse. J. Number Theory, 73 (1998), 201–211.
  • — Irrational numbers with automaton-generated continued fraction expansion, in Dynamical Systems (Luminy-Marseille, 1998), pp. 190–198. World Sci. Publishing, River Edge, NJ, 2000.
  • 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.
  • Schmidt, W. M., On simultaneous approximations of two algebraic numbers by rationals. Acta Math., 119 (1967), 27–50.
  • —, Norm form equations. Ann. of Math., 96 (1972), 526–551.
  • —, Diophantine Approximation. Lecture Notes in Math., 785. Springer, Berlin, 1980.
  • Shallit, J., Real numbers with bounded partial quotients: a survey. Enseign. Math., 38 (1992), 151–187.
  • Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc., 42 (1937), 230–265.
  • Waldschmidt, M., Un demi-siècle de transcendance, in Development of Mathematics 1950–2000, pp. 1121–1186. Birkhäuser, Basel, 2000.