Preservation of notion of C sets near zero over reals
Kilangbenla Imsong, Ram Krishna Paul
TL;DR
This work extends the Hindman–Strauss program to the setting of $C$-sets near zero in $\mathbb{R}^+$, providing a complete equivalence: a matrix $M\in\mathbb{R}^{u\times v}$ is image partition regular near zero iff it preserves $C$-sets near zero, with the preimage of a $C$-set near zero under $M$ also forming a $C$-set near zero. The authors develop a new characterization of $C$-sets near zero via idempotents in $0^+(\mathbb{R}^+)$ and deploy Stone–Čech ultrafilter techniques, first-entry matrix reductions, and IP-set methods to prove preservation results. The results deepen the combinatorial understanding of near-zero largeness, linking topological dynamics, ultrafilter algebra, and linear configurations in $\mathbb{R}^+$. This provides a robust framework for analyzing the impact of linear maps on near-zero largeness and opens avenues for further applications in Ramsey theory on dense subsemigroups of the positive reals.
Abstract
There are several notions of largeness in a semigroup. N. Hindman and D. Strauss established that if $u,v \in \mathbb{N}$, $A$ is a $u \times v$ matrix with entries from $\mathbb{Q}$ and $ψ$ is a notion of a large set in $\mathbb{N}$, then $\{\vec{x} \in \mathbb{N}^v: A\vec{x} \in ψ^u \}$ is large in $\mathbb{N}^v$. Among the several notions of largeness, C sets occupies an important place of study because they exhibit strong combinatorial properties. The analogous notion of C set appears for a dense subsemigroup $S$ of $((0, \infty),+)$ called a C-set near zero. These sets also have very rich combinatorial structure. In this article, we investigate the above result for C sets near zero in $\mathbb{R}^+$ when the matrix has real entries. We also develop a new characterisation of C-sets near zero in $\mathbb{R}^+$.