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 $\varphi (x\Gamma )=uA(x)\Gamma$ 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 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: $\frac{1}{M}{\sum }_{M\le m<2M}\left|\frac{1}{H}{\sum }_{m\le n<m+H}f({\varphi }^n(x\Gamma ))\mu (n)\right|\to 0$ as $H\to \infty$ and $H/M\to 0$ 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 Möbius 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.

48 pages