Ensembles de petite somme, structure de sous-criticit\'e

If $A$ and $B$ are two bounded sets of reals, Ruzsa proved a precise lower bound of the measure of the sumset $A+B$ involving the ratio $\lambda(A)/\lambda(B)$. De Roton established a structural result about the critical sets of this lower bound. Here, we prove a generalization of de Roton's work by establishing a result in a neighborhood of the case of equality.