M\"{o}bius random law and infinite rank-one maps

TL;DR

This paper proves M"{o}bius disjointness for infinite measure symbolic rank-one maps, extending key results in ergodic theory and number theory, and providing new proofs and classes of maps satisfying these properties.

Contribution

It extends Bourgain-Sarnak and Bourgain's theorems to infinite measure and rank-one maps, establishing M"{o}bius disjointness in broader contexts.

Findings

M"{o}bius disjointness holds for infinite measure symbolic rank-one maps.

Extension of Bourgain-Sarnak's result to $\sigma$-finite measure spaces.

Identification of a class of maps with M"{o}bius disjointness for bounded continuous functions.

Abstract

We prove that Sarnak's conjecture holds for any infinite measure symbolic rank-one map. We further extended Bourgain-Sarnak's result, which says that the M\"{o}bius function is a good weight for the ergodic theorem, to maps acting on $\sigma$-finite measure spaces. We also discuss and extend Bourgain's theorem by establishing that there is a class of maps for which the M\"{o}bius disjointness property holds for any continuous bounded function. Our proof allows us to obtain an extension of Bourgain's theorem on M\"{o}bius disjointness for bounded rank one maps and a simple and self-contained proof of this fact.