Dynamical models for Liouville and obstructions to further progress on sign patterns

TL;DR

This paper introduces a class of dynamical systems related to the Liouville function, identifies obstructions to progress on sign pattern conjectures, and constructs explicit examples demonstrating these obstructions.

Contribution

It defines a new class of dynamical systems, shows how they relate to existing problems, and constructs explicit polynomial-based examples illustrating obstructions.

Findings

Dynamical systems with anomalous local behavior can obstruct progress on sign pattern conjectures.

Explicit bounds on sign patterns from polynomial phases are established.

Constructed examples demonstrate the practical realization of these obstructions.

Abstract

We define a class of dynamical systems by modifying a construction due to Tao, which includes certain Furstenburg limits arising from the Liouville function. Most recent progress on the Chowla conjectures and sign patterns of the Mobius and Liouville functions uses methods that apply to any dynamical system in this class. Hence dynamical systems in this class with anomalous local behavior present obstructions to further progress on these problems by the same techniques. We construct straightforward examples of dynamical systems in this class based on polynomial phases and calculate the resulting obstruction. This requires explicit bounds for the number of sign patterns arising in a certain way from polynomials, which is elementary but not completely trivial.