|
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)
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
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
Linking options:
https://www.mathnet.ru/eng/aa1830 https://www.mathnet.ru/eng/aa/v34/i5/p23
|
Statistics & downloads: |
Abstract page: | 143 | Full-text PDF : | 1 | References: | 33 | First page: | 24 |
|