The primes contain arbitrarily long polynomial progressions

Abstract

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0)=...=P_k(0)=0$ and any $\eps > 0$, we show that there are infinitely many integers $x,m$ with $1 \leq m \leq x^\eps$ such that $x+P_1(m),...,x+P_k(m)$ are simultaneously prime. The arguments are based on those in Green and Tao, which treated the linear case $P_i = (i-1)\m$ and $\eps=1$; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.

82 pages, 1 figure. A minor error in the paper (concerning the definition of the polynomial forms condition, which had too weak of a requirement on the degree of the polynomials involved) has been fixed

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