# Multiple recurrence 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 $\mu\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$. Furthermore, we show the existence of the limit in $L^2(\mu)$ of the associated double 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 $p-1$ for some prime $p$.
Document type :
Journal articles

https://hal-upec-upem.archives-ouvertes.fr/hal-01267074
Contributor : Bernard Host <>
Submitted on : Wednesday, February 3, 2016 - 7:37:26 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

### Identifiers

• HAL Id : hal-01267074, version 1

### Citation

Nikos Frantzikinakis, Bernard Host, Bryna Kra. Multiple recurrence and convergence for sequences related to the prime numbers. journal für die reine und angewandte Mathematik (Crelles Journal), de Gruyter, 2007, 611, pp.131-144. ⟨hal-01267074⟩

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