On the dimension of graphs of Weierstrass-type functions with rapidly growing frequencies

Abstract

We determine the Hausdorff and box dimension of the fractal graphs for a general class of Weierstrass-type functions of the form $f(x) = \sum_{n=1}^\infty a_n \, g(b_n x + \theta_n)$, where $g$ is a periodic Lipschitz real function and $a_{n+1}/a_n \to 0$, $b_{n+1}/b_n \to \infty$ as $n \to \infty$. Moreover, for any $H, B \in [1, 2]$, $H \leq B$ we provide examples of such functions with $\dim_H(\graph f) = \underline{\dim}_B(\graph f) = H$, $\bar{\dim}_B(\graph f) = B$.