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A probabilistic property of Katsuura’s continuous nowhere differentiable function. (English) Zbl 1167.26002

Let \(f\) be H. Katsuura’s continuous nowhere differentiable function [Am. Math. Mon. 98, No. 5, 411–416 (1991; Zbl 0752.26005)]. Let \(\gamma=((2/3)(1/3)(2/3))^{1/3}\), \(\mu=-\log_2\gamma\), \(\sigma=\sqrt{2}/3\), and
\[ \Delta_k(x,h)=\log_2|f(x+3^{-k}h)-f(x)|+k\mu, \]
where \((x,h)\in [0,1)^2\) and \(k\geq 0\). Suppose that \(m_1\) and \(m_2\) denote the linear and planar Lebesgue measures, respectively.
The main result of the paper states that:
(a) For almost every \((x,h)\in [0,1)^2\) with respect to \(m_2\), we have
\[ \lim_{k\rightarrow\infty}\frac{\Delta_k(x,h)}{k}=0. \]
(b) For all numbers \(a<b\), as \(k\rightarrow\infty\) we have
\[ m_2\left\{(x,h)\in [0,1)^2:\frac{\Delta_k(x,h)}{\sigma\sqrt{k}}\in (a,b]\right\}\rightarrow\frac1{\sqrt{2\pi}}\int_a^b e^{-x^2/2}\,dx. \]
(c) For almost every \((x,h)\in [0,1)^2\), with respect to \(m_2\) we have
\[ \limsup_{k\rightarrow\infty}\frac{\Delta_k(x,h)}{\sigma\sqrt{2k\log\log k}}=1. \]
In the second step of the work, the author lets \(\nu(A)=m_1\{x\in[0,1]:f(x)\in A\}\) for each Borel set \(A\subseteq \mathbb{R}^1\), and proves that \(\nu([a,b])\leq 6(b-a)^q\), where \(q=\ln(2/3)\ln(1/3)\approx 0.369\), and \([a,b]\subset [0,1]\).

MSC:

26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
26A46 Absolutely continuous real functions in one variable

Citations:

Zbl 0752.26005
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References:

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