Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{ -\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ orthogonal to the invariant factor, the modulated, random averages \[ \sup_{ b} \Big| \frac{ 1}{ N} \sum_{ n = 1}^N b(n) T^{ a_{ n}} f \Big| \] converge to $0$ pointwise almost everywhere, where the supremum is taken over a set of bounded functions with certain uniform approximation properties.
Categories: math.CA math.DS