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