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arXiv · 1406.3571

An elementary proof for the dimension of the graph of the classical Weierstrass function

Gerhard Keller
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Abstract

Let $W_{\lambda,b}(x)=\sum_{n=0}^\infty\lambda^n g(b^n x)$ where $b\geqslant2$ is an integer and $g(u)=\cos(2\pi u)$ (classical Weierstrass function). Building on work by Ledrappier (1992), Bar\'ansky, B\'ar\'any and Romanowska (2013) and Tsujii (2001), we provide an elementary proof that the Hausdorff dimension of $W_{\lambda,b}$ equals $2+\frac{\log\lambda}{\log b}$ for all $\lambda\in(\lambda_b,1)$ with a suitable $\lambda_b<1$. This reproduces results by Bar\'ansky, B\'ar\'any and Romanowska without using the dimension theory for hyperbolic measures of Ledrappier and Young (1985,1988), which is replaced by a simple telescoping argument together with a recursive multi-scale estimate.

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Gerhard Keller. 2014-06-13. An elementary proof for the dimension of the graph of the classical Weierstrass function. https://arxiv.org/abs/1406.3571

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