Disjointness of Möbius from asymptotically periodic functions

We investigate Sarnak’s Möbius Disjointness Conjecture through asymptotically periodic functions. It is shown that Sarnak’s conjecture for rigid dynamical systems is equivalent to the disjointness of Möbius 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–Lemańczyk–Radziwiłł.

Keywords

asymptotically periodic function, mean state, Möbius function, Sarnak’s Möbius Disjointness Conjecture

2010 Mathematics Subject Classification

Primary 37A55. Secondary 11N37.

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This research was supported in part by the University of New Hampshire, by the AMSS of the Chinese Academy of Sciences, and by Grant TRT 0159 from the Templeton Religion Trust, and by the fellowship of China Postdoctoral Science Foundation 2020M670273.

Received 18 August 2021

Received revised 16 January 2022

Accepted 31 January 2022

Published 24 July 2022