(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 6, PAGES 33-44

## On the derivative of the Minkowski question mark function $?\left(x\right)$

A. A. Dushistova
N. G. Moshchevitin

Abstract

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Let $x = \left[0;a$1,a2, ¼] be the regular continued fraction expansion an irrational number $x$Î [0,1]. For the derivative of the Minkowski function $?\left(x\right)$ we prove that $?\text{'}\left(x\right) = +$¥, provided that $lim sup$t → ¥(a1+...+at)/(t) < k1 = (2log l1)/(log2) = 1.388+, and $?\text{'}\left(x\right) = 0$, provided that $lim inf$t → ¥(a1+...+at)/(t) > k2 = (4L5 - 5L4)/(L5 - L4) = 4.401+, where $L_j = \log \left(\frac \left\{j+\sqrt \left\{j^2+4\right\}\right\}\left\{2\right\}\right) - j\cdot \frac \left\{\log 2\right\}\left\{2\right\}$. Constants k1k2 are the best possible. It is also shown that $?\text{'}\left(x\right) = +$¥ for all $x$ with partial quotients bounded by $4$.

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