TY - JOUR
T1 - The polynomial multidimensional Szemerédi Theorem along shifted primes
AU - Frantzikinakis, Nikos
AU - Host, Bernard
AU - Kra, Bryna
N1 - Funding Information:
∗ Partially supported by Marie Curie IRG 248008. ∗∗ Partially supported by the Institut Universitaire de France. † Partially supported by NSF grant 0900873. Received May 4, 2011 and in revised form July 26, 2011
PY - 2013/3
Y1 - 2013/3
N2 - If q1,..., qm: Z → Zℓ are polynomials with zero constant terms and E ⊂ Zℓ 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.
AB - If q1,..., qm: Z → Zℓ are polynomials with zero constant terms and E ⊂ Zℓ 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.
UR - http://www.scopus.com/inward/record.url?scp=84876935948&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84876935948&partnerID=8YFLogxK
U2 - 10.1007/s11856-012-0132-y
DO - 10.1007/s11856-012-0132-y
M3 - Article
AN - SCOPUS:84876935948
SN - 0021-2172
VL - 194
SP - 331
EP - 348
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -