Derivative of the Minkowski function: optimal estimates

Abstract

It is well known that the derivative of the Minkowski function $?(x)$, if it exists, can take only two values, $0$ and $+\infty$. It is also known that the value of $?'(x)$ at a point $x=[0;a_1,a_2,…,a_t,…]$ is related to the limiting behaviour of the arithmetic mean $(a_1+a_2+…+a_t)/t$. In particular, as shown by Moshchevitin and Dushistova, if $a_1+a_2+…+a_t>(\kappa_2+\varepsilon)t$, where $\varepsilon>0$ and $\kappa_2\approx 4.4010487$ is some explicitly given constant, then $?'(x)=0$. They also showed that $\kappa_2$ cannot be replaced by a smaller constant. We consider the dual problem: how small can the quantity $\kappa_2t-a_1+a_2+…+a_t$ be if it is known that $?'(x)=0$? We obtain optimal estimates in this problem. Bibliography: 9 titles.