Advertisement

Journal of Mathematical Sciences

, Volume 182, Issue 4, pp 463–471 | Cite as

On the derivative of the Minkowski question mark function ?(x)

  • Anna A. DushistovaEmail author
  • Nikolai G. Moshchevitin
Article

Abstract

Let x = [0; a 1 , a 2 , …] be the regular continued fraction expansion of an irrational number x ∈ [0, 1]. For the derivative of the Minkowski function ?(x) we prove that ?′(x) = +, provided that \( \mathop {{\lim \sup }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} < {\kappa_1} = \frac{{2\log {\lambda_1}}}{{\log 2}} = {1.388^{+} } \), and ?′(x) = 0, provided that \( \mathop {{\lim \inf }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} > {\kappa_2} = \frac{{4{L_5} - 5{L_4}}}{{{L_5} - {L_4}}} = {4.401^{+} } \), where \( {L_j} = \log \left( {\frac{{j + \sqrt {{{j^2} + 4}} }}{2}} \right) - j \cdot \frac{{\log 2}}{2} \). Constants κ1, κ2 are the best possible. It is also shown that ?′(x) = + for all x with partial quotients bounded by 4.

Keywords

Irrational Number Multifractal Analysis Paradis Continue Fraction Expansion Successive Point 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. Jenkinson, “On the density of Hausdorff dimension of bounded type continued fraction sets: the Texan conjecture,” Stoch. Dyn., 4, 63–76 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    I. D. Kan, “Refining of the comparison rule for continuants,” Discrete Math. Appl., 10, No. 5, 477–480 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    M. Kesseböhmer and B. O. Stratmann, Fractal Analysis for Sets of Nondifferentiability of Minkowski Question Mark Function, arXiv:math.NT/0706.0453v1 (2007).Google Scholar
  4. 4.
    J. R. Kinney, “Note on a singular function of Minkowski,” Proc. Am. Math. Soc., 11, 788–789 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    H. Minkowski, Gesammelte Abhandlungen, Vol. 2 (1911).Google Scholar
  6. 6.
    T. S. Motzkin and E. G. Straus, “Some combinatorial extremum problems,” Proc. Am. Math. Soc., 7, 1014–1021 (1956).MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Paradis, P. Viader, and L. Bibiloni, “A new light on Minkowski’s ?(x) function,” J. Number Theory, 73, 212–227 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    J. Paradis, P. Viader, and L. Bibiloni, “The derivative of Minkowski’s ?(x) function,” J. Math. Anal. Appl., 253, 107–125 (2001).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    R. Salem, “On some singular monotone functions which are strictly increasing,” Trans. Am. Math. Soc., 53, 427–439 (1943).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations