Preservation of notion of large sets near zero over reals

TL;DR

This paper extends Hindman–Strauss type results from natural numbers to subsets of the positive reals that lie near zero. It develops a common framework using the Stone–Čech compactification of dense subsemigroups of $(0,\infty)$, defining $0^+(S^v)$ and a broad collection of near-zero largeness notions (IP*, central*, Q*, P*, J0*, etc.). The main contribution is a preservation theorem showing that for a finite matrix $M$ that is image partition regular over $\mathbb{R}^+$, the set of inputs $\vec{x}$ mapping into a near-zero large set $C^u$ remains large across a wide range of near-zero notions when viewed in $({\mathbb{R}}^+)^v$. The results unify and extend previous work on central and C-sets to the near-zero context, with implications for Ramsey theory in real semigroups and related dynamical systems.

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 $Ψ$ is a notion of large sets in $\mathbb{N}$ then $\{\vec{x} \in \mathbb{N}^v: M\vec{x} \in Ψ^u\}$ is large set in $\mathbb{N}^v$. In this article, we investigate the above result for various notions of largeness near zero in $\mathbb{R}^+$.

Theorems & Definitions (45)

• Definition 1.1
• Theorem 1.2
• proof
• Theorem 1.3
• proof
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Lemma 2.2