Large sets near idempotent and its product

Tootkaboni and Vahed introduced the notion of some large sets near idempotent along with some combinatorial properties. We characterize when the finite Cartesian product of central sets near idempotent is central near idempotent. Moreover, we provide a partial characterization for the infinite Cartesian product of the same. We then study the abundance of some large sets near idempotent. We also investigate the effect of tensor product near zero. Finally, as an application we provide a characterization of members of polynomials (with constant term 0) evaluated at idempotents in a near zero semigroup.