Multiple recurrence and convergence for sequences related to the prime numbers
Résumé
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 $p$.