Hausdorff dimension of images and graphs of some random complex series

Abstract

Let $\{X_n=e^{2\pi i{\theta }_n}\}$ be a sequence of Steinhaus random variables, where ${\theta }_n$ are independent and uniformly distributed on $[0,1]$. We compute the almost sure Hausdorff dimension of the images and graphs of the random complex series $S(x)={\sum }_{n=1}^{\infty }{a}_n{X}_n{\varphi }_n({\lambda }_{n}x)$, where ${\lambda }_n$ is an increasing sequence with ${\sup }_n{\lambda }_{n+1}/{\lambda }_n<\infty$ and ${\varphi }_n$ satisfies some uniform Lipschitz and boundedness conditions. This class of series includes the famous Weierstrass and Riemann functions as well as others appeared in literature. These results help predict the exact values of the deterministic cases.