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arXiv · 1506.07778

M\"obius disjointness for homogeneous dynamics

Ryan Peckner
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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.

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Ryan Peckner. 2015-06-25. M\"obius disjointness for homogeneous dynamics. https://doi.org/10.1215/00127094-2018-0026

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