Dynamical characterizations of combinatorially rich sets near zero

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.