A dynamical characterization of C sets

Abstract

Furstenberg, using tools from topological dynamics, defined the notion of a central subset of positive integers, and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-\v{C}ech compactification, this combinatorial theorem has been generalized and extended to the Central Sets Theorem. The algebraic techniques also discovered many sets, which are not central, that satisfy the conclusion of the Central Sets Theorem. We call such sets C sets. Since C sets are defined combinatorially, it is natural to ask if this notion admits a dynamical characterization similar to Furstenberg's original definition of a central set? In this paper we give a positive answer to this question by proving a dynamical characterization of C sets.