Freiman-Ruzsa theorem

For every $K$, a finite set $A$ in an abelian group with $|A+A|\le K|A|$ is contained in a coset progression of rank and relative size bounded in terms of $K$ alone.

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

For fixed density $\delta >0$, rank $r$, and progression length, every sufficiently large proper coset progression of rank at most $r$ has the property that each subset of relative density at least $\delta$ contains a nontrivial arithmetic progression of that length. This follows from the multidimensional Szemerédi theorem.