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Divergence points with fast growth orders of the partial quotients in continued fractions

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Abstract

This paper is concerned with the divergence points with fast growth orders of the partial quotients in continued fractions. Let S be a nonempty interval. We are interested in the size of the set of divergence points

$$ E_\varphi (S) = \left\{ {x \in [0,1):{\rm A}\left( {\frac{1} {{\varphi (n)}}\sum\limits_{k = 1}^n {\log a_k (x)} } \right)_{n = 1}^\infty = S} \right\}, $$

where A denotes the collection of accumulation points of a sequence and φ: ℕ → ℕ with φ(n)/n → ∞ as n → ∞. Mainly, it is shown, in the case φ being polynomial or exponential function, that the Hausdorff dimension of E φ (S) is a constant. Examples are also given to indicate that the above results cannot be expected for the general case.

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References

  1. I. S. Baek, L. Olsen and N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math., 214 (2007), 267–287.

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Barreira and J. Schmeling, Sets of non-typical points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29–70.

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Cajar, Billingsley Dimension in Probability Space, Lecture Notes in Mathematics, 892, Springer-Verlag (Berlin-New York, 1981).

    Google Scholar 

  4. R. Cawley and R.D. Mauldin, Multifractal decomposition of Moran fractals, Adv. Math., 92 (1992), 196–236.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Ercai, T. Küpper and S. Lin, Topological entropy for divergence points, Ergod. Th. Dynam. Sys., 25 (2005), 1173–1208.

    Article  MATH  Google Scholar 

  6. E. Cesaratto and B. Vallée, Hausdorff dimension of real numbers with bounded digits averages, Acta Arith., 125 (2) (2006), 115–162.

    Article  MATH  MathSciNet  Google Scholar 

  7. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Application, Wiley (1990).

  8. A. H. Fan, D. J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229–244.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. H. Fan, L. M. Liao, B. W. Wang and J. Wu, On Kintchine exponents and Lyapunov exponents of continued fractions, Ergod. Th. Dynam. Sys., 29 (2009), 73–109.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. J. Feng, J. Wu, J. C. Liang and S. Tseng, Appendix to the paper by T. Lúczak- a simple proof of the lower bound: “On the fractional dimension of sets of continued fractions”, Mathematika, 44 (1) (1997), 54–55.

    MATH  MathSciNet  Google Scholar 

  11. P. Hanus, R. D. Mauldin and M. Urbański, Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Acta Math. Hungar., 96 (2002), 27–98.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and its Applications, 547, Kluwer Academic Publishers (Dordrecht, 2002).

    Google Scholar 

  13. S. Jaffard, Multifractal formalism for functions. I. Results valid for all functions, SIAM J. Math. Anal., 28 (1997), 944–970.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Ya. Khintchine, Continued Fractions, P. Noordhoff, Groningen, The Netherlands (1963).

    MATH  Google Scholar 

  15. T. Lúczak, On the fractional dimension of sets of continued fractions, Mathematika, 44 (1997), 50–53.

    Article  MATH  MathSciNet  Google Scholar 

  16. E. Oliver, Dimension de Billingsley d’ensembles satures, C.R. Acad. Sci. Paris Sr. I Math., 328 (1999), 13–16.

    Google Scholar 

  17. L. Olsen, Divergence points of deformed empirical measures, Math. Res. Lett., 9 (2002), 701–713.

    MATH  MathSciNet  Google Scholar 

  18. L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591–1649.

    MATH  MathSciNet  Google Scholar 

  19. L. Olsen, Applications of multifractal divergence points to some sets of numbers defined by their N-adic expansion, Math. Proc. Camb. Phil. Soc., 140 (2003), 335–350.

    MATH  MathSciNet  Google Scholar 

  20. L. Olsen, Mixed divergence points for self-similar measures, Indiana Univ. Math. J., 52 (2003), 1343–1372.

    Article  MATH  MathSciNet  Google Scholar 

  21. L. Olsen, Applications of multifractal divergence points to some sets of d-tuples of numbers defined by their N-adic expansion, Bull. Sci. Math., 128 (2004), 265–289.

    Article  MATH  MathSciNet  Google Scholar 

  22. L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. III, Aequationes Math., 71 (2006), 29–53.

    Article  MATH  MathSciNet  Google Scholar 

  23. L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension, Bull. Sci. Math., 132 (2008), 650–678.

    Article  MATH  MathSciNet  Google Scholar 

  24. L. Olsen and S. Winter, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. London Math. Soc., 67 (2003), 103–122.

    Article  MATH  MathSciNet  Google Scholar 

  25. L. Olsen and S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra, Bull. Sci. Math., 131 (2007), 518–558.

    Article  MATH  MathSciNet  Google Scholar 

  26. C.-E. Pfister and W. G. Sullivan, Billingsley dimension on shift space, Nonlinearity, 16 (2003), 661–682.

    Article  MATH  MathSciNet  Google Scholar 

  27. C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Sys., 27 (2007), 929–956.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation, Comm. Math. Phys., 207 (1999), 145–171.

    Article  MATH  MathSciNet  Google Scholar 

  29. B. Volkmann, Über Hausdorffsche Dimensionen von Mengen, die durch Zifferneigenschaften charakterisiert sind. VI, Math. Z., 68 (1958), 439–449.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, China

    B. Wang & J. Wu

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  1. B. Wang
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  2. J. Wu
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Correspondence to B. Wang.

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Wang, B., Wu, J. Divergence points with fast growth orders of the partial quotients in continued fractions. Acta Math Hung 125, 261–274 (2009). https://doi.org/10.1007/s10474-009-9016-y

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