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May 2018 The logarithmic Sarnak conjecture for ergodic weights
Nikos Frantzikinakis, Bernard Host
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Ann. of Math. (2) 187(3): 869-931 (May 2018). DOI: 10.4007/annals.2018.187.3.6

Abstract

The Möbius disjointness conjecture of Sarnak states that the Möbius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument, and the key ingredient is a structural result for measure preserving systems naturally associated with the Möbius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity that was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.

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Nikos Frantzikinakis. Bernard Host. "The logarithmic Sarnak conjecture for ergodic weights." Ann. of Math. (2) 187 (3) 869 - 931, May 2018. https://doi.org/10.4007/annals.2018.187.3.6

Information

Published: May 2018
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2018.187.3.6

Subjects:
Primary: 11N37
Secondary: 37A45

Keywords: Chowla conjecture , Liouville function , M\"obius function , Sarnak conjecture

Rights: Copyright © 2018 Department of Mathematics, Princeton University

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Vol.187 • No. 3 • May 2018
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