## Abstract

In a previous paper of the authors, we showed that for any polynomials P_{1}…P_{k} ε Z[m] with P_{1}(0) = … = P_{k}(0) and any subset A of the primes in [N] = {1, …, N} of relative density at least φ > 0, one can find a "polynomial progression" a+P_{1}(r), …, a+P_{k}(r) in A with 0 <|r|≤N^{o(1)}, if N is sufficiently large depending on k, P_{1}, …, P_{k} and φ. In this paper we shorten the size of this progression to 0 <|r|≤log^{L}N, where L depends on k, P_{1}, …, P_{k} and φ. In the linear case P_{i} =(i-1)m, we can take L independent of φ. The main new ingredient is the use of the densification method of Conlon, Fox, and Zhao to avoid having to directly correlate the enveloping sieve with dual functions of unbounded functions.

Original language | American English |
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Title of host publication | Analytic Number Theory |

Subtitle of host publication | In Honor of Helmut Maier's 60th Birthday |

Publisher | Springer International Publishing |

Pages | 357-379 |

Number of pages | 23 |

ISBN (Electronic) | 9783319222400 |

ISBN (Print) | 9783319222394 |

DOIs | |

State | Published - 1 Jan 2015 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing Switzerland 2015.