A structure theorem for sets of small popular doubling, revisited

We prove that every set $Asubsetmathbb{Z}/pmathbb{Z}$ with $mathbb{E}_xmin(1_A*1_A(x),t)le(2+delta)tmathbb{E}_x 1_A(a)$ is very close to an arithmetic progression. Here $p$ stands for a large prime and $delta,t$ are small real numbers. This shows that the Vosper theorem is stable in the case of a single set.