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

By el Houcein el Abdalaoui, Mariusz Lemanczyk, Thierry de la Rue. 2015

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We show that Sarnak's conjecture on M\"obius disjointness holds in every uniquely ergodic modelof 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\[ \frac{1}{M} \sum\_{M\le m\textless{}2M} \frac{1}{H} \left| \sum\_{m\le n \textless{} m+H} e^{2\pi iP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5549fb8e0e8cbdc1bdea3bbf1c09f4298a13a71b