Möbius disjointness for homogeneous dynamics

Abstract

We prove Sarnak's Möbius 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 $f(x{u}^n)$ cannot correlate with the Möbius function on average.

35 pages, proof of proposition 3.10 contained an error that has been corrected

References in corpus (2)

• Uniformity in the Wiener-Wintner theorem for nilsequences
• Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

Cited by in corpus (11)

• Automatic sequences fulfill the Sarnak conjecture
• Möbius disjointness for topological models of ergodic systems with discrete spectrum
• Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals
• Measure complexity and Möbius disjointness
• Disjointness of Möbius from asymptotically periodic functions
• Möbius disjointness for analytic skew products
• Möbius disjointness for $C^{1+\epsilon }$ skew products
• Möbius disjointness for a class of exponential functions
• Möbius disjointness for skew products on a circle and a nilmanifold
• Bracket words along Hardy field sequences
• Möbius Disjointness for Skew Products on the Heisenberg Nilmanifold