A structure theorem for sets of small popular doubling, revisited

Abstract

We prove that every set $A\subset \mathbb{Z}/p\mathbb{Z}$ with ${\mathbb{E}}_x\min (1_A\ast 1_A(x),t)\le (2+\delta )t{\mathbb{E}}_x{1}_A(a)$ 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.