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

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 $P\in \mathbb{R}[x]$ with irrational leading coefficient and for each multiplicative function $\text{bnu}:\mathbb{N}\to \text{C}$, $\mathrm{\mid }\text{bnu}\mathrm{\mid }\le 1$, we have$$\frac{1}{M}\sum \mathrm{\_}M\le m\text{textless}2M\frac{1}{H}\mid \sum \mathrm{\_}m\le n\text{textless}m+H{e}^{2\pi iP(n)}\text{bnu}(n)\mid ⟶0$$ as $M\to \mathrm{\infty }$, $H\to \mathrm{\infty }$, $H\mathrm{/}M\to 0$.