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 $H,B\in [1,2]$, $H\le B$ we provide examples of such functions with $\dim_H(\graph f) = \underline{\dim}_B(\graph f) = H$, $\bar{\dim}_B(\graph f) = B$.

arXiv comment18 pages

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• On the box dimension of graph of harmonic functions on the Sierpiński gasket
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