Hausdorff dimension of the graphs of the classical Weierstrass functions

Abstract

We show that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log\lambda/\log b$, for every integer $b\ge 2$ and every $\lambda\in (1/b,1)$. Replacing $\cos(2\pi x)$ by a general non-constant $C^2$ periodic function, we obtain the same result under a further assumption that $\lambda b$ is close to $1$.