Product of difference sets of set of primes

TL;DR

This paper extends a result about difference sets of positive density subsets of integers to the set of primes, showing that a scaled natural number set is contained within the product of prime difference sets despite primes having zero density.

Contribution

It proves that a scaled natural number set is contained in the product of prime difference sets, generalizing a known result to the prime numbers despite their zero density.

Findings

Existence of k such that k·N is within the product of prime difference sets

Extension of difference set product results to zero-density sets

Prime difference sets contain scaled natural numbers

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 $\mathbb{P}-\mathbb{P}=\left\{ p-q:p>q\,\text{and}\,p,q\in\mathbb{P}\right\} .$