Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

Let $G$ be a connected, simply connected nilpotent Lie group and $Γ< G$ a lattice. We prove that each ergodic diffeomorphism $ϕ(xΓ)=uA(x)Γ$ on the nilmanifold $G/Γ$, where $u\in G$ and $A:G\to G$ is a unipotent automorphism satisfying $A(Γ)=Γ$, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow: (i) Sarnak's conjecture on Möbius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the Möbius orthogonality holds on so called typical short interval: $\frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H} f(ϕ^n(xΓ))μ(n)\right|\to 0$ as $H\to\infty$ and $H/M\to0$ for each $xΓ\in G/Γ$ and each $f\in C(G/Γ)$. In particular, the results in (i) and (ii) hold for ergodic nil-translations. Moreover, we prove that each nilsequence is orthogonal to the Möbius function $μ$ on a typical short interval. We also study the problem of lifting of the AOP property to induced actions and derive some applications on uniform distribution.

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