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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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