Polynomial Configurations in the Primes

The Bergelson–Leibman theorem states that if P1,…,Pk∈Z[x], then any subset of the integers of positive upper density contains a polynomial configuration x+P1(m),…,x+Pk(m), where x,m∈Z. 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.