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

Abstract

Let $\hat{I}˜=\{\hat{I}{¸}_n\},\hat{I}›=\{\hat{I}{»}_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by $$f_{\hat{I}˜,\hat{I}›}(x)=\left(\sum _{n=0}^{\infty }{a}^n\cos (2\ddot{I}€(b^{n}x+\hat{I}{¸}_n)),\sum _{n=0}^{\infty }{a}^n\sin (2\ddot{I}€(b^{n}x+\hat{I}{»}_n))\right),x\in [0,1],$$ where $0<a<1<b,ab>1$. The Hausdorff dimension of the graph of this function is proved to be $${\dim }_{H}G(f_{\hat{I}˜,\hat{I}›})=\min \left\{-\frac{\log b}{\log a},3+2\frac{\log a}{\log b}\right\}\text{a.s.}$$

14 pages, 1 figure