Multidimensional central sets theorem near zero

TL;DR

This paper generalizes the Multidimensional Central Sets Theorem to the context of near zero, extending previous results for polynomials and near zero concepts in additive combinatorics.

Contribution

It introduces a generalized version of the Multidimensional Central Sets Theorem applicable near zero, unifying previous polynomial and near zero results.

Findings

Extended the theorem to the near zero setting

Unified previous polynomial and near zero results

Provided new combinatorial structures near zero

Abstract

In [B] Beiglb\"ock gave a Multidimension Central sets theorem. Recently, [GP] extended this result for polynomials. They proved the Multidimensional Polynomial Central sets theorem. Earlier, Hindman and Leader introduced the near zero concept and proved the Central sets theorem near 0 in [HL]. In this article, we generalize the Multidimensional Central sets theorem for near 0.