Multiple recurrence and convergence along the primes

Abstract

Let $E\subset \mathbb{Z}$ be a set of positive upper density. Suppose that $P_1,P_2,...,P_k\in \mathbb{Z}[X]$ are polynomials having zero constant terms. We show that the set $E\cap (E-P_1(p-1))\cap ...\cap (E-P_k(p-1))$ is non-empty for some prime number $p$. Furthermore, we prove convergence in $L^2$ of polynomial multiple averages along the primes.

Some changes made in light of comments from the referees

References in corpus (1)

• The primes contain arbitrarily long polynomial progressions