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 ${?}^{\prime }(x)=+\infty$ provided ${\mathrm{limsup}}_{t\to \infty }\frac{a_1+...+a_t}{t}<{\kappa }_1=\frac{2\log {\lambda }_1}{\log 2}=1.388^{+}$, and ${?}^{\prime }(x)=0$ provided ${\mathrm{liminf}}_{t\to \infty }\frac{a_1+...+a_t}{t}>{\kappa }_2=\frac{4L_5-5L_4}{L_5-L_4}=4.401^{+}$ (here $L_j=\log (\frac{j+\sqrt{j^2+4}}{2})-j\cdot \frac{\log 2}{2}$). Constants ${\kappa }_1,{\kappa }_2$ are the best possible. Also we prove that ${?}^{\prime }(x)=+\infty$ holds for all $x$ with partial quotients bounded by 4.

10 pages, submitted to Discrete Mathematics and Applications, minor correction of misprints

Cited by in corpus (1)

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