A refined saturation theorem for polynomials and applications

TL;DR

This paper introduces a refined saturation theorem for polynomials in dynamical systems, revealing new structural properties and answering a conjecture about pro-nilsystems with applications to polynomial configurations.

Contribution

It establishes a refined saturation theorem for polynomials and characterizes the structure of systems with trivial regionally proximal relations along polynomial sets.

Findings

For minimal systems, $RP_C^{[d]}(X,T)= riangle$ implies the system is an almost one-to-one extension of a pro-nilfactor.

When $C$ consists of linear polynomials, $RP_C^{[d]}(X,T)= riangle$ characterizes $d$-step pro-nilsystems.

The paper provides a negative answer to a conjecture in the literature regarding polynomial configurations.

Abstract

For a dynamical system $(X,T)$, $d\in\mathbb{N}$ and distinct non-constant integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, the notion of regionally proximal relation along $C=\{p_1,\ldots,p_d\}$ (denoted by $RP_C^{[d]}(X,T)$) is introduced. It turns out that for a minimal system, $RP_C^{[d]}(X,T)=\Delta$ implies that $X$ is an almost one-to-one extension of $X_k$ for some $k\in\mathbb{N}$ only depending on a set of finite polynomials associated with $C$ and has zero entropy, where $X_k$ is the maximal $k$-step pro-nilfactor of $X$. Particularly, when $C$ is a collection of linear polynomials, it is proved that $RP_C^{[d]}(X,T)=\Delta$ implies $(X,T)$ is a $d$-step pro-nilsystem, which answers negatively a conjecture in \cite{5p}. The results are obtained by proving a refined saturation theorem for polynomials.