The finite products of shifted primes and Moreira's Theorem

Abstract

Let $r\in \mathbb{N}$ and $\mathbb{N}={\bigcup }_{i=1}^r{C}_i$. Do there exist $x,y\in \mathbb{N}$ and $i\in \left\{1,2,\dots ,r\right\}$ such that $\left\{x,y,xy,x+y\right\}\subseteq C_i$? This is still an unanswered question asked by N. Hindman. Joel Moreira in [Annals of Mathematics 185 (2017) 1069-1090] established a partial answer to this question and proved that for infinitely many $x,y\in \mathbb{N}$, $\left\{x,xy,x+y\right\}\subseteq C_i$ for some $i\in \left\{1,2,\dots ,r\right\}$, which is called Moreira's Theorem. Recently, H. Hindman and D. Strauss established a refinement of Moreira's Theorem and proved that for infinitely many $y$, $\left\{x\in \mathbb{N}:\left\{x,xy,x+y\right\}\subseteq C_i\right\}$ is a piecewise syndetic set. In this article, we will prove infinitely many $y\in FP\left(\mathbb{P}-1\right)$ such that $\left\{x\in \mathbb{N}:\left\{xy,x+f(y):f\in F\right\}\subseteq C_i\right\}$ is piecewise syndetic, where $F$ is a finite subset of $x\mathbb{Z}\left[x\right]$. We denote $\mathbb{P}$ is the set of prime numbers in $\mathbb{N}$ and $FP\left(\mathbb{P}-1\right)$ is the set of all finite products of distinct elements of $\mathbb{P}-1$.

4 pages