The polynomial multidimensional Szemerédi Theorem along shifted primes

Details The polynomial multidimensional Szemerédi Theorem along shifted primes The polynomial multidimensional Szemerédi Theorem along shifted primes

2010-09-08

arxiv.org/abs/1009.1484v2

Mathematics

Dynamical Systems

13 pages. Small changes suggested by the referee incorporated. To appear in Israel Journal of Mathematics

Scientific paper

If $\vf_1, ... \vf_m\colon\Z\to\Z^\ell$ are polynomials with zero constant terms and $E\subset\Z^\ell$ has positive upper Banach density, then we show that the set $E\cap (E-\vf_1(p-1))\cap\...\cap (E-\vf_m(p-1))$ 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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