Narrow arithmetic progressions in the primes

TL;DR

This paper determines the exact minimal size of common differences in prime arithmetic progressions, improving previous bounds by providing explicit constants and simplifying pseudorandomness conditions.

Contribution

It establishes the precise value of the minimal common difference exponent for prime progressions and introduces a simplified pseudorandomness framework for narrow progressions.

Findings

Exact value of L_k = (k-1) 2^{k-2} for prime progressions

Simplified pseudorandomness hypotheses for narrow progressions

Extension of Szemerédi's theorem to narrower arithmetic progressions

Abstract

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ 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 Szemer\'{e}di's theorem for narrow progressions requiring simpler pseudorandomness hypotheses in the spirit of recent work of Conlon, Fox, and Zhao.