Möbius Disjointness for Nilsequences Along Short Intervals

Abstract

For a nilmanifold $G/\Gamma$, a $1$-Lipschitz continuous function $F$ and the Möbius sequence $\mu (n)$, we prove a bound on the decay of the averaged short interval correlation $$\frac{1}{HN}\sum _{n\le N}|\sum _{h\le H}\mu (n+h)F(g^{n+h}x)|$$ as $H,N\to \infty$. The bound is uniform in $g\in G$, $x\in G/\Gamma$ and $F$.

35 pages