Measure and Hausdorff dimension of randomized Weierstrass-type functions

Abstract

In this paper we consider functions of the type $$f(x)=\sum _{n=0}^{\infty }{a}_{n}g(b_{n}x+{\theta }_n),$$ where $(a_n)$ are independent random variables uniformly distributed on $(-a^n,a^n)$ for some $0<a<1$, $b_{n+1}/b_n\ge b>1$, $a^{2}b>1$ and $g$ is a $C^1$ periodic real function with finite number of critical points in every bounded interval. We prove that the occupation measure for $f$ has $L^2$ density almost surely. Furthermore, the Hausdorff dimension of the graph of $f$ is almost surely equal to $D=2+\log a/\log b$ provided $b={\lim }_{n\to \infty }{b}_{n+1}/b_n>1$ and $ab>1$.

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• A Dichotomy for the Weierstrass-type functions