Combinatorial properties of some big sets

In this paper, we review certain combinatorial properties of some big sets, namely, sets with positive upper Banach density and piecewise syndeticity. In the process to do so, we give a proof of the upper density polynomial version of Furstenberg-Glasner's 定理, and using ergodic tools, we show that N_A(C) is piecewise syndetic in Z² if C=A-B, where A,B⊆ Z are two subsets of Z with positive upper Banach density. Moreover, we provide a simpler dynamical proof of Jin's 定理, i.e., the above mentioned C=A-B is piecewise syndetic, and give an affirmative answer to a question by Kra et al. when the set involved is piecewise syndetic.