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• W2084705636 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." @default.
• W2084705636 created "2016-06-24" @default.
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• W2084705636 date "2008-01-01" @default.
• W2084705636 modified "2023-09-29" @default.
• W2084705636 title "The primes contain arbitrarily long polynomial progressions" @default.
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• W2084705636 doi "https://doi.org/10.1007/s11511-008-0032-5" @default.
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