论文标题
差异集的产品集
Product of difference sets of set of primes
论文作者
论文摘要
在最近的一项工作\ cite {key-11}，A。Fish证明，如果$ e_ {1} $和$ e_ {2} $是$ \ mathbb {z} $的两个子集，则是阳性上层Banach密度的$ \ Mathbb {z} $ $ k \ cdot \ mathbb {z} \ subset \ left（e_ {1} -e_ {1} \ right）\ cdot \ left（e_ {2} -e_ {2} - {2} \ right）。$ $在本文中，我们将表明类似的结果对普莱斯$ $ \ mathbb} $ 0是正确的。我们将证明存在$ k \ in \ mathbb {n} $中的$ k \ cdot \ cdot \ mathbb {n} $ \ mathbb {p} - \ Mathbb {p} = \ left \ {p-q：p> q \，\ text {and} \，p，q \ in \ mathbb {p} \ right \}。$
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 $\mathbb{P}-\mathbb{P}=\left\{ p-q:p>q\,\text{and}\,p,q\in\mathbb{P}\right\} .$
