# Multiple recurrence and convergence for sequences related to the prime numbers

@inproceedings{Frantzikinakis2006MultipleRA,
title={Multiple recurrence and convergence for sequences related to the prime numbers},
author={Nikos Frantzikinakis and Bernard Host and Bryna Kra},
year={2006}
}
• Published 25 July 2006
• Mathematics
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 L 2(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…
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