The polynomial multidimensional Szemerédi Theorem along shifted primes

Nikos Frantzikinakis*, Bernard Host, Bryna Kra

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


If q1,..., qm: Z → Z are polynomials with zero constant terms and E ⊂ Z has positive upper Banach density, then we show that the set E ∩ (E - q1 (p - 1)) ∩... ∩ (E - qm (p - 1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.

Original languageEnglish (US)
Pages (from-to)331-348
Number of pages18
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - Mar 2013

ASJC Scopus subject areas

  • Mathematics(all)


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