Multiple recurrence and convergence along the primes

Trevor Wooley and Tamar Ziegler. Multiple recurrence and convergence along the primes

Abstract

Let E ⊂ Z be a set of positive upper density. Suppose that P₁,P₂,... ,P_k∈ Z[X] are polynomials having zero constant terms. We show that the set E ∩ (E-P₁(p-1))∩ ... ∩ (E-P_k(p-1)) is non-empty for some prime number p. Furthermore, we prove convergence in L² of polynomial multiple averages along the primes.

Online Video: Multiple recurrence along the shifted primes. [MSRI Discrete Rigidity Phenomena in Additive Combinatorics workshop].