  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
$\tfrac{3+\sqrt{5}}{2}$ 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.
