On the Mahler measure of the spectrum of rank one maps

TL;DR

This paper investigates the spectral properties of rank one maps, establishing a link between Mahler measure and singular spectrum, and providing new examples and bounds related to these measures.

Contribution

It extends the Kakutani-Zygmund dichotomy to generalized Riesz-products and links Mahler measure zero to singularity, also analyzing spectral measures of rank one maps with specific parameters.

Findings

Generalized Riesz-product measures are singular iff their Mahler measure is zero.

Constructs new subclasses of rank one maps with singular spectrum.

Shows Mahler measure of certain rank one map spectra is zero for p_n=O(n^β), β ≤ 1.

Abstract

We extend partially the Kakutani-Zygmund dichotomy theorem to a class of generalized Riesz-product type measures by proving that the generalized Riesz-product is singular if and only if its Mahler measure is zero. As a consequence, we exhibit a new subclass of rank one maps acting on a finite measure space with singular spectrum. In our proof the $H^p$ theory coming to play. Furthermore, by appealing to a deep result of Bourgain, we prove that the Mahler measure of the spectrum of rank one map with cutting parameter $p_n=O(n^\beta)$, $\beta \leq 1$ is zero, and we establish that the integral of the absolute part of any generalized Riesz-product is strictly less than 1. This answer partially a question asked by M. Nadkarni.