Polynomial configurations in the primes

Abstract : The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are integers. Various generalizations of this theorem are known. Wooley and Ziegler showed that the variable m can in fact be taken to be a prime minus 1, and Tao and Ziegler showed that the Bergelson-Leibman theorem holds for subsets of the primes of positive relative upper density. Here we prove a hybrid of the latter two results, namely that the step m in the Tao-Ziegler theorem can be restricted to the set of primes minus 1.
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Submitted on : Monday, September 2, 2013 - 4:33:31 PM
Last modification on : Wednesday, March 27, 2019 - 4:10:22 PM

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  • HAL Id : hal-00856932, version 1
  • ARXIV : 1210.4659

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Thai Hoang Le, Julia Wolf. Polynomial configurations in the primes. International Mathematics Research Notices, Oxford University Press (OUP), 2013, 2013, in press. ⟨hal-00856932⟩

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