Narrow progressions in the primes

In a previous paper of the authors, we showed that for any polynomials P₁…P_k ε Z[m] with P₁(0) = … = P_k(0) and any subset A of the primes in [N] = {1, …, N} of relative density at least φ > 0, one can find a "polynomial progression" a+P₁(r), …, a+P_k(r) in A with 0 <|r|≤N^o(1), if N is sufficiently large depending on k, P₁, …, P_k and φ. In this paper we shorten the size of this progression to 0 <|r|≤log^LN, where L depends on k, P₁, …, P_k and φ. In the linear case P_i =(i-1)m, we can take L independent of φ. The main new ingredient is the use of the densification method of Conlon, Fox, and Zhao to avoid having to directly correlate the enveloping sieve with dual functions of unbounded functions.