The M\"obius function and distal flows

TL;DR

This paper proves that the Möbius function is linearly disjoint from certain irregular distal flows on the 2-torus, extending previous results from regular flows to more general cases.

Contribution

It establishes the linear disjointness of the Möbius function from analytic skew product flows on the 2-torus, including irregular distal flows.

Findings

Möbius function is linearly disjoint from analytic skew product flows.

Disjointness also holds for various distal homogeneous flows.

Extends previous regular flow results to irregular distal flows.

Abstract

We prove that the M\"{o}bius function is linearly disjoint from an analytic skew product on the $2$-torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of M\"{o}bius from various distal homogeneous flows.