On a family of singular continuous measures related to the doubling map

Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asym...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Baake, Michael, Coons, Michael, Evans, James, Gohlke, Philipp
Format: Article
Language:English
Subjects:
Distribution functions
Mathematical analysis
Periodic functions
Polynomials
Online Access:Get full text
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Summary:Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asymptotics near the origin, thus providing a family of extremal examples of singular measures, including the Thue--Morse measure.
ISSN:2331-8422
DOI:10.48550/arxiv.2006.09755