Multiple recurrence and convergence for sequences related to the prime numbers

Nikos Frantzikinakis*, Bernard Host, Bryna Kra

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


For any measure preserving system (X, ,T) and A with (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() 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.

Original languageEnglish (US)
Pages (from-to)131-144
Number of pages14
JournalJournal fur die Reine und Angewandte Mathematik
Issue number611
StatePublished - Oct 26 2007

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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