M\"obius Disjointness for Nilsequences Along Short Intervals

TL;DR

This paper establishes bounds on the decay of short interval correlations between the Möbius function and nilsequences, demonstrating a form of Möbius disjointness along short intervals on nilmanifolds.

Contribution

It proves a uniform decay bound for short interval averages of Möbius and nilsequence correlations, extending Möbius disjointness results to short intervals on nilmanifolds.

Findings

Bound on short interval correlation decay as H,N grow

Uniformity of bounds across nilmanifold elements and functions

Extension of Möbius disjointness to short intervals

Abstract

For a nilmanifold $G/\Gamma$, a $1$-Lipschitz continuous function $F$ and the M\"obius sequence $\mu(n)$, we prove a bound on the decay of the averaged short interval correlation $$\frac1{HN}\sum_{n\leq N}\Big|\sum_{h\leq H} \mu(n+h)F(g^{n+h}x)\Big|$$ as $H,N\to\infty$. The bound is uniform in $g\in G$, $x\in G/\Gamma$ and $F$.