Multiple recurrence and convergence for sequences related to the prime numbers

For any measure preserving system (X, X, mu, T) and A is an element of X with mu(A)>0, we show that there exist infinitely many primes p such that mu(A boolean AND T(-(P-1))A boolean AND 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.