Dynamical characterization of combinatorially rich sets near zero

TL;DR

This paper explores the dynamical properties of combinatorially rich sets near zero within dense subsemigroups of positive real numbers, extending classical combinatorial theorems using algebraic and topological methods.

Contribution

It provides dynamical characterizations of C-sets and related sets near zero, extending the algebraic combinatorial framework to a dynamical setting.

Findings

Dynamical characterizations of C-sets near zero are established.

Extension of central sets theorem to the near zero context.

Generalization of algebraic combinatorial results using Stone-Cech compactification.

Abstract

Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,\infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central theorem near zero. Algebraically one can define quasi-central set near zero for dense subsemigroup of $((0,\infty),+)$, and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of $((0,\infty),+)$, C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. Like discrete case, we shall produce dynamical characterizations of these combinatorically rich sets near zero.