On the Mahler measure of the spectrum of rank one maps

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.