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Multiple recurrence and convergence along the primes
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 6, December 2012
- pp. 1705-1732
- 10.1353/ajm.2012.0048
- Article
- Additional Information
Let $E\subset \Bbb{Z}$ be a set of positive upper density. Suppose that
$P_1,P_2,\ldots,P_k\in {\Bbb Z}[X]$ are polynomials having zero constant
terms. We show that the set $E\cap (E-P_1(p-1))\cap\cdots\cap
(E-P_k(p-1))$ is non-empty for some prime number $p$. Furthermore, we
prove convergence in $L^2$ of polynomial multiple averages along the
primes.