Polynomial configurations in the primes

TL;DR

This paper extends the Bergelson-Leibman theorem to show that polynomial configurations in the primes can be found with the step variable m restricted to primes minus 1, combining previous results on primes and polynomial patterns.

Contribution

It proves that the step m in polynomial configurations within primes can be specifically taken from the set of primes minus 1, unifying prior generalizations.

Findings

Polynomial configurations exist in primes with step m in primes minus 1

The result generalizes previous theorems by Tao-Ziegler and Wooley-Ziegler

Confirms the existence of polynomial patterns in primes with restricted step sets

Abstract

The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are integers. Various generalizations of this theorem are known. Wooley and Ziegler showed that the variable m can in fact be taken to be a prime minus 1, and Tao and Ziegler showed that the Bergelson-Leibman theorem holds for subsets of the primes of positive relative upper density. Here we prove a hybrid of the latter two results, namely that the step m in the Tao-Ziegler theorem can be restricted to the set of primes minus 1.