A polynomial version of Sarnak's conjecture

TL;DR

This paper introduces polynomial variants of Sarnak's conjecture for minimal systems, motivated by recent variations and the behavior of the Mobius function as a good weight in polynomial ergodic theorems.

Contribution

It proposes new polynomial forms of Sarnak's conjecture tailored for minimal systems, expanding the scope of the original conjecture.

Findings

Mobius function acts as a good weight for polynomial ergodic theorems in L^q

Introduction of polynomial Sarnak conjecture variants for minimal systems

Connections to recent variations of Sarnak's conjecture

Abstract

Motivated by the variations of Sarnak's conjecture due to El Abdalaoui, Kulaga-Przymus, Lemanczyk, De La Rue and by the observation that the Mobius function is a good weight (with limit zero) for the polynomial pointwise ergodic theorem in $L^q$, q>1, we introduce polynomial versions of the Sarnak conjecture for minimal systems.