# Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals

Abstract : We show that Sarnak's conjecture on M\"obius disjointness holds in every uniquely ergodic model of a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function $\bnu:\N\to\C$, $|\bnu|\leq1$, we have $\frac{1}{M} \sum_{M\le m<2M} \frac{1}{H} \left| \sum_{m\le n < m+H} e^{2\pi iP(n)}\bnu(n) \right|\longrightarrow 0$ as $M\to\infty$, $H\to\infty$, $H/M\to 0$.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-01176039
Contributor : Thierry de La Rue <>
Submitted on : Tuesday, July 14, 2015 - 8:55:00 AM
Last modification on : Thursday, February 7, 2019 - 5:24:12 PM
Document(s) archivé(s) le : Thursday, October 15, 2015 - 10:12:31 AM

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AOP-QDS2.pdf
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### Identifiers

• HAL Id : hal-01176039, version 1
• ARXIV : 1507.04132

### Citation

El Houcein El Abdalaoui, Mariusz Lemanczyk, Thierry de La Rue. Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals. International Mathematics Research Notices, Oxford University Press (OUP), 2017, 2017 (14), pp.4350-4368. ⟨hal-01176039⟩

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