Measure and Hausdorff dimension of randomized Weierstrass-type functions

Abstract

In this paper we consider functions of the type $$f(x) = \sum_{n=0}^\infty a_n g(b_nx+\theta_n),$$ where $(a_n)$ are independent random variables uniformly distributed on $(-a^n, a^n)$ for some $0<a<1$, $b_{n+1}/b_n \geq b >1$, $a^2b> 1$ 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\rightarrow \infty}b_{n+1}/b_n>1$ and $ab>1$.