Multidimensional Stronger Central Sets Theorem and its Polynomial Extension

TL;DR

This paper extends the Stronger Central Sets Theorem to multiple dimensions and introduces a polynomial generalization, using algebraic tools from the Stone-ch compactification, with various applications discussed.

Contribution

It provides the first multidimensional and polynomial extensions of the Stronger Central Sets Theorem, broadening its scope and applicability.

Findings

Complete characterization of the multidimensional extension.

Development of a polynomial generalization.

Discussion of multiple applications.

Abstract

We establish and fully characterize the multidimensional extension of the Stronger Central Sets Theorem. Additionally, we develop a polynomial generalization of this result. Our approach utilizes tools from the Algebra of the Stone-\v{C}ech compactification of discrete semigroups. Several applications of these results are also discussed.