Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds
Volume 244 / 2019
Abstract
Let be a connected, simply connected nilpotent Lie group and \Gamma \lt 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\colon 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 each ergodic affine unipotent diffeomorphism;
(ii) for ergodic affine unipotent diffeomorphisms themselves, Möbius orthogonality holds on so-called typical short intervals: \frac1M\sum_{M\leq m \lt 2M}\bigg|\frac1H\sum_{m\leq n \lt m+H} f(\phi^n(x\Gamma))\boldsymbol{\mu} (n)\bigg|\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, (i) and (ii) hold for ergodic niltranslations. Moreover, we prove that each nilsequence is orthogonal to the Möbius function \boldsymbol{\mu} on a typical short interval.
We also study the problem of lifting the AOP property to induced actions, and derive some applications to uniform distribution.