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 $A{+}_{\mathrm{\Gamma }}B$, which applies when the doubling constant is at most $\frac{3+\sqrt{5}}{2}$ 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.