Central set Theorem near zero

TL;DR

This paper introduces new concepts of $J$-sets and $C$-sets near zero within dense subsemigroups of positive real numbers, and extends the Central Sets Theorem to this near-zero context.

Contribution

It defines $J$-sets and $C$-sets near zero and proves a version of the Central Sets Theorem applicable near zero in dense subsemigroups.

Findings

Defined $J$-sets and $C$-sets near zero.

Proved the Central Sets Theorem near zero.

Abstract

In this paper, we introduce notions of $J$-set near zero and $C$-set near zero for a dense subsemigroup of $((0,+\infty),+)$ and obtain some results for them. Also we derive the Central Sets Theorem near zero.