On the spectral type of rank one flows and Banach problem with calculus of generalized Riesz products on the real line

TL;DR

This paper demonstrates that certain Riesz product measures on the real line can serve as spectral types for rank one flows, establishing singular spectra and extending classical theorems to new contexts.

Contribution

It introduces new methods extending Bourgain-Klemes-Reinhold-Peyri ière and Salem-Zygmund CLT to analyze spectral types of rank one flows on the real line.

Findings

Certain Riesz product measures are realized as spectral types of rank one flows.

Some rank one flows have singular spectra, even in the  Z-action case.

Extended formulas for Radon-Nikodym derivatives and Mahler measures in this context.

Abstract

It is shown that a certain class of Riesz product type measures on $\mathbb{R}$ is realized a spectral type of rank one flows. As a consequence, we will establish that some class of rank one flows has a singular spectrum. Some of the results presented here are even new for the $\mathbb{Z}$-action. Our method is based, on one hand, on the extension of Bourgain-Klemes-Reinhold-Peyri\`ere method, and on the other hand, on the extension of the Central Limit Theorem approach to the real line which gives a new extension of Salem-Zygmund Central Limit Theorem. We extended also a formula for Radon-Nikodym derivative between two generalized Riesz products obtained by el Abdalaoui-Nadkarni and a formula of Mahler measure of the spectral type of rank one flow but in the weak form. We further present an affirmative answer to the flow version of the Banach problem, and we discuss some issues related to flat trigonometric polynomials on the real line in connection with the famous Banach-Rhoklin problem in the spectral theory of dynamical systems.