Hausdorff dimension of images and graphs of some random complex series

TL;DR

This paper determines the almost sure Hausdorff dimension of images and graphs of certain random complex series, including classical functions like Weierstrass and Riemann functions, extending understanding of their fractal properties.

Contribution

It provides explicit formulas for the Hausdorff dimension of images and graphs of a broad class of random complex series, including well-known deterministic functions.

Findings

Computed Hausdorff dimensions for a class of random complex series.

Included classical functions like Weierstrass and Riemann functions as special cases.

Extended results to predict dimensions in deterministic scenarios.

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 $S(x)=\sum_{n=1}^{\infty}a_n X_n\phi_n(\lambda_nx)$, where $\lambda_n$ is an increasing sequence with $\sup_n\lambda_{n+1}/\lambda_n<\infty$ and $\phi_n$ 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.