Product of difference sets of set of primes

Sayan Goswami

Published 2022-10-28Version 1

A. Fish proved that for any two positive density sets $A$ and $B$ in two measure preserving systems $\left(X,\mu,T\right)$ and $\left(Y,\nu,S\right)$, there exists $k\in\mathbb{Z}$ such that $k\cdot \mathbb{Z}\subseteq R\left(A\right)\cdot R\left(B\right)$, where $R\left(A\right)$ and $R\left(B\right)$ are the return times sets. As a consequence it follows that for any two sets of positive density $E_{1}$ and $E_{2}$ in $\mathbb{Z}$, 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 the result is still true under sufficiently weak assumption. As a consequence we show that there exists $k\in\mathbb{N}$ such that $k\cdot \mathbb{N}\subseteq\left(\mathbb{P}-\mathbb{P}\right)\cdot\left(\mathbb{P}-\mathbb{P}\right)$, where $\mathbb{P}$ is the set of primes.