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 $IP^{\star}$-set with an $IP_{r}^{\star}$-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.