The high-dimensional Weierstrass functions

Abstract

For a real analytic periodic function $\varphi :\mathbb{R}\to {\mathbb{R}}^d$, an integer $b\ge 2$ and $\lambda \in (1/b,1)$, we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function $W(x)={\sum }_{n=0}^{\infty }{\lambda }^n\varphi (b^{n}x)$ are both equal to $$\min \left\{{\log }_{{\lambda }^{-1}}b,1+\left(d-q\right)\left(1+{\log }_b\lambda \right)\right\},$$ where $q=q(\varphi ,b,\lambda )$ denotes the maximum dimension of all linear spaces $V<{\mathbb{R}}^d$ such that the projection ${\pi }_{V}W$ is Lipschitz.