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

Abstract

Let $Θ=\{θ_n\}, Λ=\{λ_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by $$ f_{Θ,Λ}(x)= \left( \sum_{n=0}^{\infty} a^n\cos\bigl(2π(b^n x+θ_n)\bigr),\ \sum_{n=0}^{\infty} a^n\sin\bigl(2π(b^n x+λ_n)\bigr) \right), \; x\in[0,1], $$ where $0<a<1<b,\ ab> 1$. The Hausdorff dimension of the graph of this function is proved to be $$\dim_H G(f_{Θ,Λ}) = \min\left\{-\frac{\log b}{\log a}, \, 3 +2\frac{\log a}{\log b}\right\} \quad \text{a.s.}$$