Narrow Arithmetic Progressions in the Primes

Abstract

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any k≥3 and N large, there exist nontrivial k-term arithmetic progressions in (any positive density subset of) the primes up to N with common difference O((logN)Lk)⁠, for an unspecified constant Lk. In this work, we obtain this statement with the precise value Lk=(k−1)2k−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.

Locations

• International Mathematics Research Notices - View
• arXiv (Cornell University) - View - PDF