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Transactions of the American Mathematical Society

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Weierstrass functions with random phases


Author: Yanick Heurteaux
Journal: Trans. Amer. Math. Soc. 355 (2003), 3065-3077
MSC (2000): Primary 26A27, 28A80, 37A05; Secondary 60F20
DOI: https://doi.org/10.1090/S0002-9947-03-03221-5
Published electronically: March 19, 2003
MathSciNet review: 1974675
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Abstract: Consider the function

\begin{displaymath}f_\theta(x)=\sum_{n=0}^{+\infty}b^{-n\alpha}g(b^nx+\theta_n),\end{displaymath}

where $b>1$, $0<\alpha<1$, and $g$ is a non-constant 1-periodic Lipschitz function. The phases $\theta_n$ are chosen independently with respect to the uniform probability measure on $[0,1]$. We prove that with probability one, we can choose a sequence of scales $\delta_k\searrow 0$ such that for every interval $I$ of length $\vert I\vert=\delta_k$, the oscillation of $f_\theta$ satisfies $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^\alpha$. Moreover, the inequality $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^{\alpha+\varepsilon}$ is almost surely true at every scale. When $b$ is a transcendental number, these results can be improved: the minoration $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^\alpha$ is true for every choice of the phases $\theta_n$ and at every scale.


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Additional Information

Yanick Heurteaux
Affiliation: Laboratoire de Mathématiques pures, Université Blaise Pascal, F-63177 Aubière cedex, France
Email: Yanick.Heurteaux@math.univ-bpclermont.fr

DOI: https://doi.org/10.1090/S0002-9947-03-03221-5
Keywords: Weierstrass functions, almost periodic functions, oscillations, fractal dimension
Received by editor(s): July 8, 2002
Published electronically: March 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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