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 $i\in\left\{1,2,\ldots,r\right\}$ 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 $i\in\left\{1,2,\ldots,r\right\}$, 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 $\mathbb{P}-1$.