Essential CR set near zero

Abstract

Combinatorially Rich sets were introduced by Bergelson and Glasscock for commutative semigroup. Latter Hindman, Hosseini, Strauss and Tootkaboni extended the definition of Combinatorially Rich sets for arbitrary semigroup. Recently Goswami proved that product of two Combinatorially Rich sets is also a Combinatorially Rich set. On the other hand Hindman and Leader were the first to introduce the concept of central sets near zero for dense semigroups of $\left(\left(0,\infty\right),+\right)$ and demonstrated an important combinatorial consequence regarding these sets. In this article we provided dynamical and combinatorial characterization of essential CR-sets near zero and explore the cartesian product of these sets.