Disjointness of M\"{o}bius from asymptotically periodic functions

TL;DR

This paper explores the relationship between Sarnak's M"obius Disjointness Conjecture and asymptotically periodic functions, establishing equivalences and providing new conditions under which the conjecture holds for certain dynamical systems.

Contribution

It demonstrates the equivalence of Sarnak's conjecture for rigid systems with M"obius disjointness from asymptotically periodic functions and offers new sufficient conditions for this disjointness.

Findings

Sarnak's conjecture for rigid systems is equivalent to M"obius disjointness from asymptotically periodic functions.

Provided sufficient conditions for M"obius disjointness from asymptotically periodic functions.

Confirmed Sarnak's conjecture for a new class of rigid dynamical systems, extending previous results.

Abstract

We investigate Sarnak's M\"obius Disjointness Conjecture through asymptotically periodic functions. It is shown that Sarnak's conjecture for rigid dynamical systems is equivalent to the disjointness of M\"obius from asymptotically periodic functions. We give sufficient conditions and a partial answer to the later one. As an application, we show that Sarnak's conjecture holds for a class of rigid dynamical systems, which improves an earlier result of Kanigowski-Lema{\'{n}}czyk-Radziwi{\l}{\l}.