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.
Similar content being viewed by others
References
O. Jenkinson, “On the density of Hausdorff dimension of bounded type continued fraction sets: the Texan conjecture,” Stoch. Dyn., 4, 63–76 (2004).
I. D. Kan, “Refining of the comparison rule for continuants,” Discrete Math. Appl., 10, No. 5, 477–480 (2000).
M. Kesseböhmer and B. O. Stratmann, Fractal Analysis for Sets of Nondifferentiability of Minkowski Question Mark Function, arXiv:math.NT/0706.0453v1 (2007).
J. R. Kinney, “Note on a singular function of Minkowski,” Proc. Am. Math. Soc., 11, 788–789 (1960).
H. Minkowski, Gesammelte Abhandlungen, Vol. 2 (1911).
T. S. Motzkin and E. G. Straus, “Some combinatorial extremum problems,” Proc. Am. Math. Soc., 7, 1014–1021 (1956).
J. Paradis, P. Viader, and L. Bibiloni, “A new light on Minkowski’s ?(x) function,” J. Number Theory, 73, 212–227 (1998).
J. Paradis, P. Viader, and L. Bibiloni, “The derivative of Minkowski’s ?(x) function,” J. Math. Anal. Appl., 253, 107–125 (2001).
R. Salem, “On some singular monotone functions which are strictly increasing,” Trans. Am. Math. Soc., 53, 427–439 (1943).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 6, pp. 33–44, 2010.
Rights and permissions
About this article
Cite this article
Dushistova, A.A., Moshchevitin, N.G. On the derivative of the Minkowski question mark function ?(x). J Math Sci 182, 463–471 (2012). https://doi.org/10.1007/s10958-012-0750-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-0750-2