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 $1$.

minor corrections

References in corpus (2)

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

Cited by in corpus (3)

• On the abelian complexity of the Rudin-Shapiro sequence
• Invariant Graphs for Chaotically Driven Maps
• Resistance metric, and spectral asymptotics, on the graph of the Weierstrass function