Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

TL;DR

This paper proves that ergodic affine unipotent diffeomorphisms on nilmanifolds exhibit asymptotic orthogonality of powers, leading to M"obius orthogonality results and applications to nilsequences and distribution properties.

Contribution

It establishes the AOP property for ergodic affine unipotent diffeomorphisms on nilmanifolds, with implications for M"obius orthogonality and distribution theory.

Findings

Ergodic affine unipotent diffeomorphisms have AOP property.

M"obius orthogonality holds in uniquely ergodic models.

Nilsequences are orthogonal to M"obius function on typical short intervals.

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.