An arithmetic transference proof of a relative Szemeredi theorem

Recently, Conlon, Fox and the author gave a new proof of a relative Szemeredi theorem, which was the main novel ingredient in the proof of the celebrated Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions. Roughly speaking, a relative Szemeredi theorem says that if S is a set of integers satisfying certain conditions, and A is a subset of S with positive relative density, then A contains long arithmetic progressions, and our recent results show that S only needs to satisfy a so-called linear forms condition. This paper contains an alternative proof of the new relative Szemeredi theorem, where we directly transfer Szemeredi's theorem, instead of going through the hypergraph removal lemma. This approach provides a somewhat more direct route to establishing the result, and it gives better quantitative bounds. The proof has three main ingredients: (1) a transference principle/dense model theorem of Green-Tao and Tao-Ziegler (with simplified proofs given later by Gowers, and independently, Reingold-Trevisan-Tulsiani-Vadhan) applied with a discrepancy/cut-type norm (instead of a Gowers uniformity norm as it was applied in earlier works); (2) a counting lemma established by Conlon, Fox and the author; and (3) Szemeredi's theorem as a black box.