Narrow arithmetic progressions in the primes

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 L_κ. In this work, we obtain this statement with the precise value L_κ = (κ - 1)2^κ-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.