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

TL;DR

This paper introduces a generalized Riesz product approach to study IP-Dirichlet measures and IP-rigid dynamical systems, simplifying and extending recent results in the field.

Contribution

It provides a simplified and generalized framework for analyzing IP-Dirichlet measures and their relation to IP-rigid systems using Riesz products.

Findings

Generalized Riesz products effectively characterize IP-Dirichlet measures.

The approach simplifies previous proofs and extends the class of measures studied.

Connections between IP-Dirichlet measures and dynamical systems are clarified.

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 Lema\'nczyk. We simplify and generalize some of their results, using an approach involving generalized Riesz products.