FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

On the derivative of the Minkowski question mark function $?(x)$

A. A. Dushistova
N. G. Moshchevitin

Abstract

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

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