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For any measure preserving system (X,
, μ,T) and A ∈ with μ(A) > 0, we show that there exist infinitely many primes p such that 
(the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L2(μ) 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.
Received: 2006-01-31
Revised: 2006-07-18
Published Online: 2007-11-14
Published in Print: 2007-10-26
© Walter de Gruyter
