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Fundamentalnaya i Prikladnaya Matematika, 2010, Volume 16, Issue 6, Pages 33–44
(Mi fpm1349)
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This article is cited in 19 scientific papers (total in 19 papers)
On the derivative of the Minkowski question mark function $?(x)$
A. A. Dushistova, N. G. Moshchevitin M. V. Lomonosov Moscow State University
Abstract:
Let $x=[0;a_1,a_2,\dots]$ be the regular continued fraction expansion an irrational number $x\in[0,1]$. For the derivative of the Minkowski function $?(x)$ we prove that $?'(x)=+\infty$, provided that $\limsup_{t\to\infty}\frac{a_1+\dots+a_t}t<\kappa_1=\frac{2\log\lambda_1}{\log 2} = 1.388^+$, and $?'(x) = 0$, provided that $\liminf\limits_{t\to \infty}\frac{a_1+\dots+a_t}t>\kappa_2=\frac{4L_5-5L_4}{L_5-L_4}= 4.401^+$, where $L_j=\log\bigl(\frac{j+\sqrt{j^2+4}}2\bigr)-j\cdot\frac{\log2}2$. Constants $\kappa_1$, $\kappa_2$ are the best possible. It is also shown that $?'(x)=+\infty$ for all $x$ with partial quotients bounded by $4$.
Citation:
A. A. Dushistova, N. G. Moshchevitin, “On the derivative of the Minkowski question mark function $?(x)$”, Fundam. Prikl. Mat., 16:6 (2010), 33–44; J. Math. Sci., 182:4 (2012), 463–471
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https://www.mathnet.ru/eng/fpm1349 https://www.mathnet.ru/eng/fpm/v16/i6/p33
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Abstract page: | 788 | Full-text PDF : | 319 | References: | 50 | First page: | 1 |
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