Narrow arithmetic progressions in the primes

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Abstract

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any κ ≥ 3 and N large, there exist nontrivial κ-term arithmetic progressions in (any positive density subset of) the primes up to N with common difference O((logN)Lκ ), for an unspecified constant Lκ. In this work, we obtain this statement with the precise value Lκ = (κ - 1)2κ-2. This is achieved by proving a relative version of Szemerédi's theorem for narrow progressions requiring simpler pseudorandomness hypotheses in the spirit of recent work of Conlon, Fox, and Zhao.

Original languageEnglish
Pages (from-to)391-428
Number of pages38
JournalInternational Mathematics Research Notices
Volume2017
Issue number2
DOIs
StatePublished - Jan 2017

Bibliographical note

Publisher Copyright:
© The Author(s) 2016. Published by Oxford University Press. All rights reserved.

ASJC Scopus subject areas

  • Mathematics (all)

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