Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds
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 Mobius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the Mobius 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 Mobius 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.