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

TL;DR

This paper calculates the Hausdorff and box dimensions of fractal graphs of a broad class of Weierstrass-type functions with rapidly increasing frequencies, revealing how these dimensions can be precisely controlled.

Contribution

It provides explicit formulas and examples for the Hausdorff and box dimensions of graphs of generalized Weierstrass functions with rapidly growing frequencies.

Findings

Hausdorff and box dimensions of the graphs are determined.

Examples show dimensions can be prescribed within [1, 2].

Dimensions depend on the decay of coefficients and growth of 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 $\dim_H(\graph f) = \underline{\dim}_B(\graph f) = H$, $\bar{\dim}_B(\graph f) = B$.