Weierstrass functions with random phases
Résumé
Consider the function $$f_\te(x)=\sum_{n=0}^{+\infty}b^{-n\al}g(b^nx+\te_n)$$ where $b>1$, $0<\al<1$ and $g$ is a non constant 1-periodic Lipschitz function. The phases $\te_n$ 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 $\abs{I}=\delta_k$, the oscillation of $f_\te$ satisfies $\osc(f_\te,I)\geq C\abs{I}^\al$. Moreover, the inequality $\osc(f_\te,I)\geq C\abs{I}^{\al+\varepsilon}$ is almost surely true at every scale. When $b$ is a transcendental number, these results can be improved : the minoration $\osc(f_\te,I)\geq C\abs{I}^\al$ is true for every choice of the phases $\te_n$ and at every scale.