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

Abstract

Let $x = [0;a_1,a_2,...]$ be the decomposition of the irrational number $x \in [0,1]$ into regular continued fraction. Then for the derivative of the Minkowski function $?(x)$ we prove that $?'(x) = +\infty$ provided $\limsup_{t\to \infty}\frac{a_1+...+a_t}{t} <κ_1 =\frac{2\log λ_1}{\log 2} = 1.388^+$, and $?'(x) = 0$ provided $\liminf_{t\to \infty}\frac{a_1+...+a_t}{t} >κ_2 = \frac{4L_5-5L_4}{L_5-L_4}= 4.401^+$ (here $L_j = \log (\frac{j+\sqrt{j^2+4}}{2}) - j\cdot\frac{\log 2}{2}$). Constants $κ_1,κ_2$ are the best possible. Also we prove that $?'(x) = +\infty$ holds for all $x$ with partial quotients bounded by 4.