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

Abstract

Let $W_{\lambda ,b}(x)={\sum }_{n=0}^{\infty }{\lambda }^{n}g(b^{n}x)$ where $b⩾2$ is an integer and $g(u)=\cos (2\pi u)$ (classical Weierstrass function). Building on work by Ledrappier (1992), Baránsky, Bárány 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 Baránsky, Bárány 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.

The proof of Proposition 3.3 is clarified and a mistake is corrected

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