The Interplay between Additive and Multiplicative Central Sets Theorems

The concept of Central sets, introduced by Furstenberg through the framework of topological dynamics, has played a pivotal role in combinatorial number theory. Furstenberg’s Central Sets Theorem highlighted their rich combinatorial structure. Later, De, Hindman, and Strauss strengthen this theorem using the algebraic framework of the Stone–Čech compactification. In this article, we establish a unified version of the Central Sets Theorem that simultaneously captures both additive and multiplicative structures.