On the dimension of the graph of the classical Weierstrass function

Abstract

This paper examines dimension of the graph of the famous Weierstrass non-differentiable function \[ W_{λ, b} (x) = \sum_{n=0}^{\infty}λ^n\cos(2πb^n x) \] for an integer $b\ge 2$ and $1/b<\lambda <1$. We prove that for every $b$ there exists (explicitly given) ${\lambda }_b\in (1/b,1)$ such that the Hausdorff dimension of the graph of $W_{\lambda ,b}$ is equal to $D=2+\frac{\log \lambda }{\log b}$ for every $\lambda \in ({\lambda }_b,1)$. We also show that the dimension is equal to $D$ for almost every $\lambda$ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function \[ f (x) = \sum_{n=0}^{\infty}λ^nϕ(b^n x) \] for an integer $b\ge 2$ and $1/b<\lambda <1$ is equal to $D$ for a typical $\mathbb{Z}$-periodic $C^3$ function $\varphi$.

Final authors' version with a correction of an inexact statement in the introduction to the published version, concerning the box dimension of the graphs of functions of the form (1.1) and (1.2)

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