• DOI:10.1007/S10958-012-0750-2
• Corpus ID: 115156022

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

@article{Dushistova2007OnTD,
  title={On the derivative of the Minkowski question mark function ?(x)},
  author={Anna Aleksandrovna Dushistova and Nikolai Germanovich Moshchevitin},
  journal={Journal of Mathematical Sciences},
  year={2007},
  volume={182},
  pages={463-471}
}

• A. Dushistova, N. Moshchevitin
• Published 15 June 2007
• Mathematics
• Journal of Mathematical Sciences

Let x = [0; a1, a2, …] 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.