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 is an element of G and A : G -> 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 each ergodic affine unipotent diffeomorphism; (ii) for ergodic affine unipotent diffeomorphisms themselves, Mobius orthogonality holds on so-called typical short intervals: 1/M Sigma(M <= m<2M)vertical bar 1/H Sigma(m <= n<m+H) f(phi(n)(x Gamma))mu(n)vertical bar -> 0 as H -> infinity and H/M -> 0 for each x Gamma is an element of G/Gamma and each f is an element of C(G/Gamma). In particular, (i) and (ii) hold for ergodic niltranslations. 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 the AOP property to induced actions, and derive some applications to uniform distribution.