Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

Abstract

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