On the Mahler measure of the spectrum of rank one maps

2021/08/29 by El Houcein El Abdalaoui, Abdalaoui, el Houcein el
Mathematics · #37A05 #37A30 #37A40 #42A05 #42A55 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2108.13416

openalex publication_date 2021/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^β), β≤ 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.

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