Box dimension of the graphs of the generalized Weierstrass-type functions

Abstract

For a Lipschitz $\mathbb{Z}-$periodic function $\varphi :\mathbb{R}\to {\mathbb{R}}^2$ satisfied that ${\mathbb{R}}^2\setminus \{\varphi (x):x\in \mathbb{R}\}$ is not connected, an integer $b\ge 2$ and $\lambda \in (c/b^{\frac{1}{2}},1)$, we prove the following for the generalized Weierstrass-type function $W(x)=\sum _{n=0}^{\infty }{\lambda }^n\varphi (b^{n}x)$: the box dimension of its graph is equal to $3+2{\log }_b\lambda$, where $c$ is a constant depending on $\varphi$.