The high-dimensional Weierstrass functions

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