On the Hausdorff dimension of graph of random vector-valued Weierstrass function

TL;DR

This paper proves that the Hausdorff dimension of the graph of a random vector-valued Weierstrass function is almost surely 3 minus twice the parameter beta, extending Hunt's 1998 result.

Contribution

It establishes the Hausdorff dimension of the graph for a class of random vector-valued Weierstrass functions, generalizing previous deterministic results.

Findings

Hausdorff dimension of the graph is 3 - 2β with probability one.

Extends Hunt's 1998 deterministic result to a probabilistic setting.

Confirms the almost sure Hausdorff dimension for the random Weierstrass function.

Abstract

Let $\Theta=\{\theta_n\}, \Lambda=\{\lambda_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by \[ f_{\Theta,\Lambda}(t)= \left( \sum_{n=0}^{\infty} b^{-\beta n}\cos\bigl(2\pi (b^n t+\theta_n)\bigr),\ \sum_{n=0}^{\infty} b^{-\beta n}\sin\bigl(2\pi (b^n t+\lambda_n)\bigr) \right),\quad t\in[0,1], \] where $b>1, \beta\in (0,1/2)$. We prove that, with probability one, the Hausdorff dimension of the graph of this function is \[ \dim_H G(f_{\Theta,\Lambda})=3-2\beta, \] extending a result of Hunt in 1998.