The finite products of shifted primes and Moreira's Theorem

TL;DR

This paper advances the understanding of partition regularity involving shifted primes and products, extending Moreira's Theorem by demonstrating the existence of infinitely many structured sets within certain partitions.

Contribution

It proves the existence of infinitely many shifted prime products y such that specific sets involving y are piecewise syndetic, generalizing previous results related to Moreira's Theorem.

Findings

Existence of infinitely many y in finite products of shifted primes with structured properties.

Sets involving y are shown to be piecewise syndetic under certain conditions.

Generalization of Moreira's Theorem to broader classes of sets.

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 $\mathbb{P}-1$.