Product of difference sets of set of primes

Abstract

In a recent work \cite{key-11}, A. Fish proved that if $E_1$ and $E_2$ are two subsets of $\mathbb{Z}$ of positive upper Banach density, then there exists $k\in \mathbb{Z}$ such that $k\cdot \mathbb{Z}\subset \left(E_1-E_1\right)\cdot \left(E_2-E_2\right).$ In this article we will show that a similar result is true for the set of primes $\mathbb{P}$ (which has density $0$). We will prove that there exists $k\in \mathbb{N}$ such that $k\cdot \mathbb{N}\subset \left(\mathbb{P}-\mathbb{P}\right)\cdot \left(\mathbb{P}-\mathbb{P}\right),$ where $\mathbb{P}-\mathbb{P}=\left\{p-q:p>q\text{and}p,q\in \mathbb{P}\right\}.$

arXiv commentChanges as per suggestions of the referees of Proc. AMS