Doubling constant ($|A+A|/|A|$)

The doubling constant of a nonempty finite set $A$ is $|A+A|/|A|$. A small doubling constant indicates that $A$ behaves approximately like a coset of a subgroup.

If finite sets satisfy $|A+B|\le K|A|$, then the Plünnecke-Ruzsa inequality bounds iterated sumsets and difference sets by

If $A\subseteq {\mathbb{F}}_p^n$ and $|A+A|\le K|A|$, then $A$ is contained in a vector subspace $H$ satisfying