流形上遍历仿射幂单同态的幂的近似正交性
Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds
摘要：
设$ G $是连通单连通幂零李群，$ Γ< G $是格.本文证明了零流形$ G/Γ$上的每个遍历自同构$ ϕ(xΓ)=uA(x)Γ$，其中$ u\in G $和$ A:G\to G $是满足$ A(Γ)=Γ$的单幂自同构，具有渐近正交幂（AOP）性质.由此产生两个后果：（i）关于莫比乌斯正交性的Sarnak猜想在遍历仿射幂单同态的每个唯一遍历模型中成立;（ii）对于遍历仿射幂单同态本身，莫比乌斯正交性在所谓的典型短区间上成立：对于每个$ xΓ\in G/Γ$和每个$ f\in C(G/Γ) $，$ \frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H} f(ϕ^n(xΓ))μ(n)\right|\to 0$为$ H\to\infty $和$ H/M\to0 $。特别地，（i）和（ii）中的结果对遍历零平移成立。此外，我们还证明了在一个典型的短区间上，每一个零序列都与Möbius函数$μ$正交.我们还研究了AOP性质对诱导作用的提升问题，并给出了在均匀分布上的一些应用。
英文摘要：
Let $ G $ be a connected, simply connected nilpotent Lie group and $ Γ< G $ a lattice. We prove that each ergodic diffeomorphism $ ϕ(xΓ)=uA(x)Γ$ on the nilmanifold $ G/Γ$, where $ u\in G $ and $ A:G\to G $ is a unipotent automorphism satisfying $ A(Γ)=Γ$, 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: $ \frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H} f(ϕ^n(xΓ))μ(n)\right|\to 0$ as $ H\to\infty $ and $ H/M\to0 $ for each $ xΓ\in G/Γ$ and each $ f\in C(G/Γ) $. 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 $μ$ 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.
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