Weierstrass functions with random phases

Consider the function

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 $\vert I\vert=\delta_k$, the oscillation of $f_\theta$ satisfies $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^\alpha$. Moreover, the inequality $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^{\alpha+\varepsilon}$ is almost surely true at every scale. When $b$ is a transcendental number, these results can be improved: the minoration $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^\alpha$ is true for every choice of the phases $\theta_n$ and at every scale.