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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 ofBourgain-Klemes-Reinhold-Peyri\`ere method, and on the other hand, on theextension of the Central Limit Theorem approach to the real line which gives anew extension of Salem-Zygmund Central Limit Theorem. We extended also aformula for Radon-Nikodym derivative between two generalized Riesz productsobtained by el Abdalaoui-Nadkarni and a formula of Mahler measure of thespectral type of rank one flow but in the weak form. We further present anaffirmative answer to the flow version of the Banach problem, and we discusssome issues related to flat trigonometric polynomials on the real line inconnection with the famous Banach-Rhoklin problem in the spectral theory ofdynamical systems.

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