A counter-example for polynomial version of Sarnak's conjecture

TL;DR

This paper constructs a minimal Toeplitz system counter-example demonstrating that the polynomial version of Sarnak's conjecture does not hold, showing the M"obius function can be linearly disjoint from polynomial subsequences in such systems.

Contribution

It provides the first known counter-example within minimal Toeplitz systems for the polynomial Sarnak's conjecture, challenging previous assumptions.

Findings

Counter-example in minimal Toeplitz systems

Disproof of polynomial Sarnak's conjecture in this setting

Shows M"obius function can be linearly disjoint from polynomial subsequences

Abstract

We construct the counter-example for polynomial version of Sarnak's conjecture for minimal systems, which assets that the M\"obius function is linearly disjoint from subsequences along polynomials of deterministic sequences realized in minimal systems. Our example is in the class of Toeplitz systems, which are minimal.