IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products

Abstract

If $(n_k{)}_{k\ge 1}$ is a strictly increasing sequence of integers, a continuous probability measure $\sigma$ on the unit circle $\mathbb{T}$ is said to be IP-Dirichlet with respect to $(n_k{)}_{k\ge 1}$ if $\hat{\sigma }({\sum }_{k\in F}{n}_k)\to 1$ as $F$ runs over all non-empty finite subsets $F$ of $\mathbb{N}$ and the minimum of $F$ tends to infinity. IP-Dirichlet measures and their connections with IP-rigid dynamical systems have been investigated recently by Aaronson, Hosseini and Lemańczyk. We simplify and generalize some of their results, using an approach involving generalized Riesz products.