Freiman's theorem

Remarks

Here $A + A$ is the sumset of $A$ with itself. Ruzsa's proof has four steps. First, the Plünnecke–Ruzsa inequality bounds $\abs{2A - 2A}$. Second, a large part of $A$ is modelled by a set in a cyclic group $\Z/N\Z$ with $N$ not much larger than $\abs{A}$. Third, Bogolyubov's lemma finds a large Bohr set inside the set $2A' - 2A'$ of the model $A'$. Fourth, geometry of numbers finds a progression inside the Bohr set, and a covering argument finishes. A converse holds: if $A$ is a large subset of a proper progression $P$ of dimension $d$, then $\abs{A + A} \le \abs{P + P} \le 2^d \abs{P}$, so $A$ has small doubling. Green and Ruzsa extended the theorem to all abelian groups, with coset progressions in place of progressions.

Going deeper

For a student

Freiman's theorem is the central inverse theorem of additive combinatorics. Good bounds on $d(K)$ and $f(K)$ matter in applications, and finding the right ones is related to the polynomial Freiman–Ruzsa conjecture. It is used in Gowers' proof of Szemerédi's theorem, together with the Balog–Szemerédi–Gowers theorem.