M\"obius disjointness for homogeneous dynamics

TL;DR

This paper proves Sarnak's M"obius disjointness conjecture for unipotent translations on homogeneous spaces of real connected Lie groups, establishing a significant link between number theory and homogeneous dynamics.

Contribution

It establishes the M"obius disjointness conjecture for all unipotent flows on homogeneous spaces of real connected Lie groups, a major advancement in the field.

Findings

M"obius function does not correlate with unipotent orbits on homogeneous spaces.

The result applies to all such spaces with any lattice and unipotent element.

This confirms the conjecture in a broad class of dynamical systems.

Abstract

We prove Sarnak's M\"obius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma\subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then for every $x\in\Gamma\backslash G$ and every continuous, bounded function $f$ on $\Gamma\backslash G$, the sequence $f(xu^{n})$ cannot correlate with the M\"obius function on average.