Sarkozy's Theorem for P-Intersective Polynomials

TL;DR

This paper characterizes when polynomials guarantee the presence of prime-difference patterns in dense sets and provides bounds on the size of sets avoiding such patterns, extending to primes and quadratic cases.

Contribution

It establishes a necessary and sufficient condition for polynomials to ensure prime-difference patterns in positive density sets and offers quantitative bounds on sets avoiding these patterns.

Findings

Sets of positive density contain nonzero differences of the form h(p) for some prime p.

The density of sets avoiding such differences is at most a constant times (log N)^{-c}.

Results extend to quadratic polynomials and relative prime settings with additional methods.

Abstract

We define a necessary and sufficient condition on a polynomial $h\in \mathbb{Z}[x]$ to guarantee that every set of natural numbers of positive upper density contains a nonzero difference of the form $h(p)$ for some prime $p$. Moreover, we establish a quantitative estimate on the size of the largest subset of ${1,2,\dots,N}$ which lacks the desired arithmetic structure, showing that if deg$(h)=k$, then the density of such a set is at most a constant times $(\log N)^{-c}$ for any $c<1/(2k-2)$. We also discuss how an improved version of this result for $k=2$ and a relative version in the primes can be obtained with some additional known methods.