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 $k\ge 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset of) the primes up to $N$ with common difference $O((\log N{)}^{L_k})$, for an unspecified constant $L_k$. In this work we obtain this statement with the precise value $L_k=(k-1)2^{k-2}$. 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.

30 pages

References in corpus (2)

• An averaged form of Chowla's conjecture
• New Proofs of the Green-Tao-Ziegler Dense Model Theorem: An Exposition