Difference sets and Polynomials of prime variables

TL;DR

This paper proves that for sets of positive integers or primes with positive density, differences of elements can be expressed as polynomial functions of primes, extending understanding of additive structures in dense sets.

Contribution

It establishes the existence of polynomial-difference representations within dense sets of integers and primes, generalizing previous results to a broader class of polynomials.

Findings

Differences in dense integer sets can be expressed as polynomial functions of primes.

Similar results hold for dense sets of primes, showing a wide applicability.

The work extends classical additive number theory results to polynomial difference structures.

Abstract

Let \psi(x) be a polynomial with rational coefficients. Suppose that \psi has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x,y\in A and a prime p such that x-y=\psi(p-1). Furthermore, if P be a set of primes with the positive relative upper density, then there exist x,y\in P and a prime p such that x-y=\psi(p-1).