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 $p$.
https://hal-upec-upem.archives-ouvertes.fr/hal-01267074 Contributor : Bernard HostConnect in order to contact the contributor Submitted on : Wednesday, February 3, 2016 - 7:37:26 PM Last modification on : Saturday, January 15, 2022 - 4:03:56 AM
Nikos Frantzikinakis, Bernard Host, Bryna Kra. Multiple recurrence and convergence for sequences related to the prime numbers. Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2007, 611, pp.131-144. ⟨hal-01267074⟩