Multiple recurence and convergence for sequences related to the prime numbers

Abstract

For any measure preserving system $(X,\mathcal{X},\mu ,T)$ and $A\in \mathcal{X}$ with $\mu (A)>0$, we show that there exist infinitely many primes $p$ such that $\mu (A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A)>0$ (the same holds with $p-1$ replaced by $p+1$). Furthermore, we show the existence of the limit in $L^2(\mu )$ of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form $p-1$ (or $p+1$) for some prime $p$.

14 pages. To appear in Crelle's Journal

Cited by in corpus (1)

• The inverse conjecture for the Gowers norm over finite fields in low characteristic