Small difference set forces a three-term arithmetic progression

For every $C$ there is $N(C)$ such that a set $A\subseteq \mathbb{Z}$ with $|A|\ge N(C)$ and $|A-A|\le C|A|$ contains a nontrivial three-term arithmetic progression. The Freiman-Ruzsa theorem places $A$ densely in a bounded-rank coset progression, where the Szemerédi theorem in a bounded-rank coset progression applies.