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We describe in this paper 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 byaddition with itself on its own support), when $A$ is a finite subset ofintegers and when $A$ is a bounded subset of real numbers. More precisely wegive a sharp upper bound for the size of $A\cap[\inf(A),x]$ for $x\le\sup(A)$,and construct sets reaching this bound for any given $x$ in this range. Thisimplies a precise characterization of the structure of some critical sets inFreiman's $3k-4$ theorem in both cases.

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