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The polynomial multidimensional Szemerédi Theorem along shifted primes

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Abstract

If \(\vec q_1 ,...,\vec q_m \): ℤ → ℤℓ are polynomials with zero constant terms and E ⊂ ℤℓ has positive upper Banach density, then we show that the set E ∩ (E − \(\vec q_1 \)(p − 1)) ∩ … ∩ (E − \(\vec q_m \)(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.

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Authors and Affiliations

  1. Department of Mathematics, University of Crete, Knossos Avenue, Heraklion, 71409, Greece

    Nikos Frantzikinakis

  2. Laboratoire d’analyse et de mathématiques appliquées, Université de Marne la Vallée & CNRS UMR 8050, 5 Bd. Descartes, Champs sur Marne, 77454, Marne la Vallée Cedex 2, France

    Bernard Host

  3. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, USA

    Bryna Kra

Authors
  1. Nikos Frantzikinakis
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  2. Bernard Host
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  3. Bryna Kra
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Corresponding author

Correspondence to Nikos Frantzikinakis.

Additional information

Partially supported by Marie Curie IRG 248008.

Partially supported by the Institut Universitaire de France.

Partially supported by NSF grant 0900873.

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Frantzikinakis, N., Host, B. & Kra, B. The polynomial multidimensional Szemerédi Theorem along shifted primes. Isr. J. Math. 194, 331–348 (2013). https://doi.org/10.1007/s11856-012-0132-y

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  • DOI: https://doi.org/10.1007/s11856-012-0132-y

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