Additively stable sets, critical sets for the $3k-4$ theorem in $\mathbb Z$ and $\mathbb R$

We describe additively left-stable sets, i.e. sets satisfying $((A+A)-\inf (A))\cap [\inf (A),\sup (A)]=A$ (meaning that $A-\inf (A)$ is stable by addition with itself on its convex hull), when $A$ is a finite set of integers and when $A$ is a bounded set of real numbers. More precisely we give a sharp upper bound for the density of $A$ in $[\inf (A),x]$ for $x\le \sup (A)$, and construct sets reaching this density for any given $x$ in this range. This gives some information on sets involved in the structural description of some critical sets in Freiman’s $3k-4$ theorem in both cases.