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 Host
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Submitted on : Wednesday, February 3, 2016 - 7:37:26 PM
Last modification on : Thursday, July 18, 2019 - 3:00:04 PM
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⟩
