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

A. A. Dushistova ; N. G. Moshchevitin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 6, pp. 33-44

Résumé

Let $x=[0;a_1,a_2,\dots]$ be the regular continued fraction expansion an irrational number $x\in[0,1]$. For the derivative of the Minkowski function $?(x)$ we prove that $?'(x)=+\infty$, provided that $\limsup_{t\to\infty}\frac{a_1+\dots+a_t}t\kappa_1=\frac{2\log\lambda_1}{\log 2} = 1.388^+$, and $?'(x) = 0$, provided that $\liminf\limits_{t\to \infty}\frac{a_1+\dots+a_t}t>\kappa_2=\frac{4L_5-5L_4}{L_5-L_4}= 4.401^+$, where $L_j=\log\bigl(\frac{j+\sqrt{j^2+4}}2\bigr)-j\cdot\frac{\log2}2$. Constants $\kappa_1$, $\kappa_2$ are the best possible. It is also shown that $?'(x)=+\infty$ for all $x$ with partial quotients bounded by $4$.