Polynomial extension of the stronger central sets theorem (Q6117255)

Summary: In [Recurrence in ergodic theory and combinatorial number theory. Princeton, NJ: Princeton University Press (1981; Zbl 0459.28023)] \textit{H. Furstenberg} introduced the notion of central set and established his famous Central Sets Theorem. Since then, several improved versions of Furstenberg's result have been found. The strongest generalization has been published by \textit{D. De} et al. in [Fundam. Math. 199, No. 2, 155--175 (2008; Zbl 1148.05052)], whilst a polynomial extension by \textit{V. Bergelson} et al. in [J. Comb. Theory, Ser. A 147, 119--154 (2017; Zbl 1352.05023)]. In this article, we will establish a polynomial extension of the stronger version of the central sets theorem, and we will discuss properties of the families of sets that this result leads to consider.