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Sarnak’s conjecture for sequences of almost quadratic word growth

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

Published online by Cambridge University Press:  04 December 2020

REDMOND MCNAMARA
REDMOND MCNAMARA*
Affiliation:
UCLALos Angeles, CA, USA (e-mail: [email protected])
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Abstract

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We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

Keywords

arithmetic and algebraic dynamicsnumber theoryclassical ergodic theory

MSC classification

Primary: 37A45: Relations with number theory and harmonic analysis 11N37: Asymptotic results on arithmetic functions
Secondary: 11K65: Arithmetic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 10 , October 2021 , pp. 3060 - 3115
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Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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