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Lévy constants of transcendental numbers


Author: Christoph Baxa
Journal: Proc. Amer. Math. Soc. 137 (2009), 2243-2249
MSC (2000): Primary 11K50, 11J81
DOI: https://doi.org/10.1090/S0002-9939-09-09787-1
Published electronically: January 28, 2009
MathSciNet review: 2495257
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Abstract: We prove that every $ \gamma \ge \log \frac{1+\sqrt 5}{2}$ is the Lévy constant of a transcendental number; i.e., there exists a transcendental number $ \alpha $ such that $ \gamma =$ $ {\lim \limits _{m\to \infty }}\frac{1}{m}\log q_{m}(\alpha )$, where $ q_{m}(\alpha )$ denotes the denominator of the $ m$th convergent of $ \alpha $.


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  • 1. B. Adamczewski and Y. Bugeaud, On the Maillet-Baker continued fractions, J. Reine Angew. Math. 606 (2007), 105-121. MR 2337643 (2008d:11073)
  • 2. A. Baker, Continued fractions of transcendental numbers, Mathematika 9 (1962), 1-8. MR 0144853 (26:2394)
  • 3. A. Baker, On Mahler's classification of transcendental numbers, Acta Math. 111 (1964), 97-120. MR 0157943 (28:1171)
  • 4. J. L. Davison, Quasi-periodic continued fractions, J. Number Theory 127 (2007), 272-282. MR 2362436
  • 5. C. Faivre, The Lévy constant of an irrational number, Acta Math. Hungar. 74 (1997), 57-61. MR 1428047 (97j:11036)
  • 6. E. P. Golubeva, Estimates of the Lévy constant for $ \sqrt p$ and a class number one criterion for $ \mathbf{Q}(\sqrt p)$, J. Math. Sci. (N.Y.) 95 (1999), 2185-2191. MR 1691280 (2000d:11129)
  • 7. E. P. Golubeva, The spectrum of Lévy constants for quadratic irrationalities, J. Math. Sci. (N.Y.) 110 (2002), 3040-3047. MR 1756334 (2001b:11065)
  • 8. E. P. Golubeva, The spectrum of Lévy constants for quadratic irrationalities and class numbers of real quadratic fields, J. Math. Sci. (N.Y.) 118 (2003), 4740-4752. MR 1850361 (2002k:11199)
  • 9. H. Jager and P. Liardet, Distributions arithmétiques des dénominateurs de convergents de fractions continues, Indag. Math. 50 (1988), 181-197. MR 952514 (89i:11085)
  • 10. A. Ya. Khintchine, Zur metrischen Kettenbruchtheorie, Compositio Math. 3 (1936), 276-285. MR 1556944
  • 11. P. Lévy, Sur le dévelopment en fraction continue d'un nombre choisi au hasard, Comp. Math. 3 (1936), 286-303.
  • 12. E. Maillet, Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions, Gauthier-Villars, Paris, 1906.
  • 13. O. Perron, Die Lehre von den Kettenbrüchen, Band 1. Elementare Kettenbrüche, Teubner, Stuttgart, 1977. MR 0064172 (16:239e)
  • 14. W.M. Schmidt, On simultaneous approximations of two algebraic numbers by rationals, Acta Math. 119 (1967), 27-50. MR 0223309 (36:6357)
  • 15. W.M. Schmidt, Diophantine Approximation, Lecture Notes in Math. 785, Springer, Berlin, 1980. MR 568710 (81j:10038)
  • 16. J. Wu, On the Lévy constants for quadratic irrationals, Proc. Amer. Math. Soc. 134 (2006), 1631-1634. MR 2204273 (2007a:11112)

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Additional Information

Christoph Baxa
Affiliation: Department of Mathematics, University of Vienna, Nordbergstraße 15, A-1090, Wien, Austria
Email: christoph.baxa@univie.ac.at

DOI: https://doi.org/10.1090/S0002-9939-09-09787-1
Received by editor(s): August 26, 2008
Received by editor(s) in revised form: September 29, 2008
Published electronically: January 28, 2009
Communicated by: Ken Ono
Article copyright: © Copyright 2009 American Mathematical Society

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