A robust version of Freiman's 3k-4 Theorem and applications

We prove a robust version of Freiman's 3k - 4 theorem on the restricted sumset A + (Gamma) B, which applies when the doubling constant is at most (3 + root 5)/2 in general and at most 3 in the special case when A = -B. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz-Sobolev inequality.