Multiple recurrence and convergence along the primes

TL;DR

This paper proves that sets of positive density in integers contain polynomial configurations along primes and establishes convergence of polynomial multiple averages in the primes, advancing understanding of prime number patterns and ergodic averages.

Contribution

It introduces new results on polynomial recurrence and convergence along primes, extending classical combinatorial and ergodic theorems to prime number settings.

Findings

Existence of polynomial configurations in dense sets along primes

Convergence of polynomial multiple averages in the primes

Extension of polynomial recurrence results to prime numbers

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.