Weierstrass functions with random phases

Abstract

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