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Algebra i Analiz, 2022, Volume 34, Issue 5, Pages 23–52 (Mi aa1830)  

Research Papers

On the derivative of the Minkowski question-mark function for numbers with bounded partial quotients

D. R. Gayfulin

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
References:
Abstract: 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.
Keywords: fraction, continuant, Minkowski function.
Received: 14.12.2021
English version:
St. Petersburg Mathematical Journal, 2023, Volume 34, Issue 5, Pages 737–758
DOI: https://doi.org/10.1090/spmj/1777
Document Type: Article
Language: Russian
Citation: D. R. Gayfulin, “On the derivative of the Minkowski question-mark function for numbers with bounded partial quotients”, Algebra i Analiz, 34:5 (2022), 23–52; St. Petersburg Math. J., 34:5 (2023), 737–758
Citation in format AMSBIB
\Bibitem{Gay22}
\by D.~R.~Gayfulin
\paper On the derivative of the Minkowski question-mark function for numbers with bounded partial quotients
\jour Algebra i Analiz
\yr 2022
\vol 34
\issue 5
\pages 23--52
\mathnet{http://mi.mathnet.ru/aa1830}
\transl
\jour St. Petersburg Math. J.
\yr 2023
\vol 34
\issue 5
\pages 737--758
\crossref{https://doi.org/10.1090/spmj/1777}
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    Алгебра и анализ St. Petersburg Mathematical Journal
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