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;a1,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 ?'(x) = +¥, provided that lim supt → ¥(a1+...+at)/(t) < k1 = (2log l1)/(log2) = 1.388+, and ?'(x) = 0, provided that lim inft → ¥(a1+...+at)/(t) > k2 = (4L5 - 5L4)/(L5 - L4) = 4.401+, where $ L_j = \log (\frac {j+\sqrt {j^2+4}}{2}) - j\cdot \frac {\log 2}{2} $. Constants k1k2 are the best possible. It is also shown that ?'(x) = +¥ for all x with partial quotients bounded by 4.

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