Geodesic
On the derivative of the Minkowski question-mark function for numbers with bounded partial quotients
Algebra i analiz, Tome 34 (2022) no. 5, pp. 23-52 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is well known that the derivative of the Minkowski function $?(x)$ may take only the values $0$ and $+\infty$ (provided it exists). Let $\mathbf{E}_n$ be the set of irrational numbers on the interval $[0; 1]$ whose contitued fraction expansion has all convergents of at most $n$. It is known also that the quantity $?'(x)$ at the point $x=[0;a_1,a_2,\ldots,a_t,\ldots]$ is related to the limit behavior of the arithmetic mean $(a_1+a_2+\ldots+a_t)/t$. In particular, as was shown by A. Dushistova, I. Kan, and N. Moshchevitin, if for $x\in \mathbf{E}_n$ we have $a_1+a_2+\ldots+a_t>(\kappa^{(n)}_1-\varepsilon) t$, where $\varepsilon>0$, and $\kappa^{(n)}_1$ is a certain explicit constant, then $?'(x)=+\infty$. They also showed that the quantity $\kappa^{(n)}_1$ cannot be replaced by a greater constant. In the present paper, a dual problem is treated, specifically, how small the quantity $a_1+a_2+\ldots+a_t-\kappa^{(n)}_1 t$ may be if it is known that $?'(x)=0$/ Optimal estimates in this problem are deduced.
Mots-clés : fraction, continuant
Keywords: Minkowski function. 
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D. R. Gayfulin. On the derivative of the Minkowski question-mark function for numbers with bounded partial quotients. Algebra i analiz, Tome 34 (2022) no. 5, pp. 23-52. http://geodesic.mathdoc.fr/item/AA_2022_34_5_a1/

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