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 $1/b < λ< 1$. We prove that for every $b$ there exists (explicitly given) $λ_b \in (1/b, 1)$ such that the Hausdorff dimension of the graph of $W_{λ, b}$ is equal to $D = 2+\frac{\logλ}{\log b}$ for every $λ\in(λ_b,1)$. We also show that the dimension is equal to $D$ for almost every $λ$ 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 $1/b < λ< 1$ is equal to $D$ for a typical $\mathbb Z$-periodic $C^3$ function $ϕ$.