On dynamical $C^{\star}$-set and its combinatorial consequences

Using the methods from topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. Later D. De, Neil Hindman, and D. Strauss [Fund. Math.199 (2008), 155-175.] established a stronger version of the Central Sets Theorem and then introduced the notion of $C$-sets satisfying the Central Sets Theorem and studied the properties of these sets. For any weak mixing system $\left(X, \mathcal{B},\mu, T\right),$ and $A_{0},A_{1}\in\mathcal{B}$, with $\mu\left(A_{0}\right)\mu\left(A_{1}\right)>0$, R. Kung and X.Ye [Disc. Cont. Dyn. sys., 18 (2007) 817-827.] proved that the set $N\left(A,B\right)= \left\{n:\mu\left(A_{0}\cap T^{-n}A_{1}\right)>0\right\}$ intersects all sets of positive upper Banach density. However, later N. Hindman and D. Strauss [New York J. Math. 26 (2020) 230-260.] proved that there exist $C$-sets having zero upper Banach density. Inspired by this result, in this article, we prove that $N\left(A, B \right)$ intersects with all $C$-sets. Then we introduce the notion of a dynamical $C^{\star}$-set and then we study their combinatorial properties.