Monocromaticity of additive and multiplicative central sets and Goswami's theorem in large Integral Domains

Abstract

In \cite{Fi} A. Fish proved that if $E_1$ and $E_2$ are two subsets of $\mathbb{Z}$ of positive upper Banach density, then there exists $k\in \mathbb{Z}\setminus 0$ such that $k\cdot \mathbb{Z}\subset \left(E_1-E_1\right)\cdot \left(E_2-E_2\right)$. In \cite{G}, S. Goswami proved the same but a fundamental result on the set of prime numbers $\mathbb{P}$ in $\mathbb{N}$ and proved that for some $k\in \mathbb{N}$, $k\cdot \mathbb{N}\subset \left(\mathbb{P}-\mathbb{P}\right)\cdot \left(\mathbb{P}-\mathbb{P}\right)$. To do so, Goswami mainly proved that the product of an $I{P}^{\star }$-set with an $I{P}_r^{\star }$-set contains $k\mathbb{N}$. This result is very important and surprising to mathematicians who are aware of combinatorially rich sets. In this article, we extend Goswami's result to large Lntegral Domain, that behave like $\mathbb{N}$ in the sense of some combinatorics. We prove that for a combinatorially rich ($CR$-set) set, $A$, for some $k\in \mathbb{N}$, $k\cdot \mathbb{N}\subset \left(A-A\right)\cdot \left(A-A\right)$. We provide a new proof that if we partition a large Integral Domain, $R$, into finitely many cells, then at least one cell is both additive and multiplicative central, and we prove the converse part, which is a new and unknown result.