On the derivative of the Minkowski question-mark function

TL;DR

This paper investigates the derivative of the Minkowski question-mark function, establishing bounds on the partial sums of continued fraction coefficients that determine whether the derivative is zero or infinite.

Contribution

It provides non-improvable estimates for the partial sums of continued fraction coefficients related to the derivative's behavior, complementing previous results on the derivative being infinite.

Findings

Established bounds for the sum of continued fraction coefficients when the derivative is zero.

Extended understanding of the relationship between continued fractions and the derivative of the question-mark function.

Provided non-improvable estimates for the dual problem to previous work.

Abstract

The Minkowski question-mark function $?(x)$ is a continuous strictly increasing function defined on $[0,1]$ interval. It is well known fact that the derivative of this function, if exists, can take only two values: $0$ and $+\infty$. It is also known that the value of the derivative $?'(x)$ at the point $x=[0;a_1,a_2,\ldots,a_t,\ldots]$ is connected with the limit behavior of the arithmetic mean $(a_1+a_2+\ldots+a_t)/t$. Particularly, N. Moshchevitin and A. Dushistova showed that if $a_1+a_2+\ldots+a_t<\kappa_1 t$, where $\kappa_1 = 2\log\bigl({\frac{1+\sqrt{5}}{2}}\bigr)/\log{2}= 1.3884\ldots$, then $?'(x)=+\infty$. They also proved that the constant $\kappa_1$ is non-improvable. We consider a dual problem: how small can be the quantity $a_1+a_2+\ldots+a_t-\kappa_1 t$ if $?'(x)=0$? We obtain the non-improvable estimates of this quantity.