Weierstrass functions with random phases

Abstract:

Consider the function \[ f_\theta (x)=\sum _{n=0}^{+\infty }b^{-n\alpha }g(b^nx+\theta _n),\] where $b>1$, $0<\alpha <1$, and $g$ is a non-constant 1-periodic Lipschitz function. The phases $\theta _n$ are chosen independently with respect to the uniform probability measure on $[0,1]$. We prove that with probability one, we can choose a sequence of scales $\delta _k\searrow 0$ such that for every interval $I$ of length $|I|=\delta _k$, the oscillation of $f_\theta$ satisfies $\operatorname {osc}(f_\theta ,I)\geq C|I|^\alpha$. Moreover, the inequality $\operatorname {osc}(f_\theta ,I)\geq C|I|^{\alpha +\varepsilon }$ is almost surely true at every scale. When $b$ is a transcendental number, these results can be improved: the minoration $\operatorname {osc}(f_\theta ,I)\geq C|I|^\alpha$ is true for every choice of the phases $\theta _n$ and at every scale.