A structure theorem for sets of small popular doubling

In this paper we prove that every set $Asubsetmathbb{Z}$ satisfying the inequality $sum_{x}min(1_A*1_A(x),t)le(2+delta)t|A|$ for $t$ and $delta$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset $Asubsetmathbb{N}$ satisfies $|mathbb{N}setminus(A+A)|ge k$; specifically we show that $mathbb{P}(|mathbb{N}setminus(A+A)|ge k)=Theta(2^{-k/2})$.