Calculus of Generalized Riesz Products

TL;DR

This paper explores generalized Riesz products, integrating $H^p$ theory, Mahler measure, and polynomial zeros, establishing formulas for Radon-Nikodym derivatives, and presenting a Dichotomy theorem with applications to flat polynomials.

Contribution

It introduces new formulas for Radon-Nikodym derivatives of generalized Riesz products and proves a Dichotomy theorem, advancing understanding of their structure and properties.

Findings

Derived formulas for Radon-Nikodym derivatives

Established a Dichotomy theorem for generalized Riesz products

Discussed implications for flat polynomials

Abstract

In this paper we discuss generalized Riesz products bringing into consideration $H^p$ theory, the notion of Mahler measure, and the zeros of polynomials appearing in the generalized Riesz product. Formula for Radon-Nikodym derivative between two generalized Riesz product is established under suitable conditions. This is then used to formulate a Dichotomy theorem and prove a conditional version of it. A discussion involving flat polynomials is given.