Preservation of notion of large sets near zero over reals

TL;DR

This paper explores how large set properties near zero in the positive reals relate to combinatorial and algebraic structures, extending known results from natural numbers to real numbers close to zero.

Contribution

It generalizes the concept of largeness from natural numbers to the positive reals near zero, analyzing the preservation of large set properties under linear transformations.

Findings

Large set properties near zero are preserved under certain linear transformations.

The results extend combinatorial largeness concepts from ℕ to ℝ^+ near zero.

New conditions for largeness in the real numbers are established.

Abstract

The study of the size of subsets in a semigroup have shown that many of these subsets have strong combinatorial properties and contribute richly to the algebraic structure of the Stone-Cech compactification of a discrete semigroup. N. Hindman and D. Strauss have proved that if u, v $\in \mathbb{N}$, M is a u \times v matrix satisfying restrictions that vary with the notion of largeness and if $\Psi$ is a notion of large sets in $\mathbb{N}$ then $\{\vec{x} \in \mathbb{N}^v: M\vec{x} \in \Psi^u\}$ is large set in $\mathbb{N}^v$. In this article, we investigate the above result for various notions of largeness near zero in $\mathbb{R}^+$.