Polynomial extension of the Stronger Central Sets Theorem

TL;DR

This paper extends the Stronger Central Sets Theorem to polynomial cases, broadening the understanding of central sets in topological dynamics and combinatorics.

Contribution

It introduces a polynomial extension of the stronger version of the Central Sets Theorem, expanding the theoretical framework of central sets.

Findings

Established a polynomial extension of the stronger Central Sets Theorem

Discussed properties of the resulting families of sets

Enhanced the theoretical understanding of central sets in dynamical systems

Abstract

In [F81] Furstenberg introduced the notion of central set and established his famous Central Sets Theorem. Since then, several improved versions of Furstenberg's result have been found. The strongest generalization has been published by De, Hindman and Strauss in [DHS08], whilst a polynomial extension by Bergelson, Johnson and Moreira appeared in [BJM17]. In this article, we will establish a polynomial extension of the stronger version of the central sets theorem, and we will discuss properties of the families of sets that this result leads to consider