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Hausdorff dimension of level sets of generalized Takagi functions

Published online by Cambridge University Press:  19 June 2014

PIETER C. ALLAART
PIETER C. ALLAART*
Affiliation:
Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017, U.S.A. e-mail: [email protected]
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Abstract

This paper examines the Hausdorff dimension of the level sets f−1(y) of continuous functions of the form

\begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*}
where φ(x) is the distance from x to the nearest integer, and for each n, ωn is a {−1,1}-valued function which is constant on each interval [j/2n,(j+1)/2n), j=0,1,. . .,2n − 1. This class of functions includes Takagi's continuous but nowhere differentiable function. It is shown that the largest possible Hausdorff dimension of f−1(y) is $\log ((9+\sqrt{105})/2)/\log 16\approx .8166$, but in case each ωn is constant, the largest possible dimension is 1/2. These results are extended to the intersection of the graph of f with lines of arbitrary integer slope. Furthermore, two natural models of choosing the signs ωn(x) at random are considered, and almost-sure results are obtained for the Hausdorff dimension of the zero set and the set of maximum points of f. The paper ends with a list of open problems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 2 , September 2014 , pp. 253 - 278
Copyright
Copyright © Cambridge Philosophical Society 2014 

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