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 \[ W_{\lambda, b} (x) = \sum_{n=0}^{\infty}\lambda^n\cos(2\pi b^n x) \] 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 \[ f (x) = \sum_{n=0}^{\infty}\lambda^n\phi(b^n x) \] for an integer $b \ge 2$ and $1/b < \lambda < 1$ is equal to $D$ for a typical $\mathbb Z$-periodic $C^3$ function $\phi$.