The typical structure of sets with large sumset

Given $m \in \mathbb{N}$ and a $p$-random subset $A \subseteq \mathbb{N}$, we asymptotically determine $\log \Pr(|\mathbb{N} \setminus (A + A)| \ge m)$ for $p$ above the threshold for this property. As an application, we show that a uniformly at random chosen subset $A \subseteq \{1, \ldots, n\}$ of