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arXiv · 1210.4659

Polynomial configurations in the primes

Thai Hoang Le·Julia Wolf
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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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Thai Hoang Le, Julia Wolf. 2012-10-17. Polynomial configurations in the primes. https://doi.org/10.1093/imrn%2Frnt169

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