A Freiman-type Theorem for restricted sumsets

Let A and B be nonempty finite subsets of Z. Freiman's 3k - 4 Theorem states that if vertical bar A + A vertical bar <= 3 vertical bar A vertical bar - 4, then A is contained in a short arithmetic progression. Freiman generalized his theorem establishing that if vertical bar A+ B vertical bar <= vertical bar A vertical bar +vertical bar B vertical bar + min{vertical bar A vertical bar, vertical bar B vertical bar} - 4, then A and B are contained in short arithmetic progressions with common difference. Take S subset of A x B and write A (S)(+) B = {a + b : (a, b) is an element of S}. There have been several attempts to generalize Freiman's statements for restricted sumsets A (S)(+) B. In the last few years, there have been some results establishing that (under reasonable technical conditions) if vertical bar A (S)(+) B vertical bar < vertical bar A vertical bar + vertical bar B vertical bar + (c - d) min{vertical bar A vertical bar, vertical bar B vertical bar} for an absolute constant c and d such that d -> 0 whenever vertical bar(AxB)\S vertical bar vertical bar AxB vertical bar -> 0, then there are arithmetic progressions C and D with common difference such that vertical bar A\C vertical bar/vertical bar A vertical bar and vertical bar B\D vertical bar/vertical bar B vertical bar are small (in terms of vertical bar(AxB)\S vertical bar/vertical bar AxB vertical bar), vertical bar C vertical bar <= vertical bar A (S)(+) B vertical bar - vertical bar B vertical bar(1 - e) and vertical bar D vertical bar = vertical bar A (S)(+) B vertical bar - vertical bar A vertical bar( 1 - e) where e -> 0 whenever vertical bar(AxB)\S vertical bar vertical bar AxB vertical bar -> 0. Furthermore, in some of the papers where these results appear, it was conjectured that the best possible value of c such that the same conclusion is reached is c = 1. In this paper we confirm this conjecture.