Dynamical characterizations of combinatorially rich sets near zero

Abstract

Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of ((0,∞),+) and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone–Čech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central sets theorem near zero. Algebraically one can define quasi-central sets near zero for dense subsemigroups of ((0,∞),+), and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of ((0,∞),+), C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. We shall produce dynamical characterizations of these combinatorially rich sets near zero.