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Furstenberg systems of pretentious and MRT multiplicative functions

Part of: Multiplicative number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  03 February 2025

NIKOS FRANTZIKINAKIS ,
MARIUSZ LEMAŃCZYK  and
THIERRY DE LA RUE Open the ORCID record for THIERRY DE LA RUE [Opens in a new window]
NIKOS FRANTZIKINAKIS*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Crete, Voutes University Campus, Heraklion 71003, Greece
MARIUSZ LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland (e-mail: mlem@mat.umk.pl)
THIERRY DE LA RUE
Affiliation:
Université de Rouen Normandie, CNRS, Laboratoire de Mathématiques Raphaël Salem, Avenue de l’Université - 76801 Saint Étienne du Rouvray, France (e-mail: Thierry.de-la-Rue@univ-rouen.fr)
*
e-mail: frantzikinakis@gmail.com
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Abstract

We prove structural results for measure-preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions, these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function, all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwiłł, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that a trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro averages.

Keywords

multiplicative functionspretentiousSarnak conjectureChowla conjectureFurstenberg systems

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11K65: Arithmetic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 80
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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