
Blog post by Terence Tao: Proposition 3 shows that if A^{2} < ^{3}⁄_{2}A then there exists a subgroup H with H ≤ A^{2} and an element x such that A ⊂ xH = Hx.


Paper by Ben Green and Imre Ruzsa: Theorem 1.1 is Freiman's theorem for an arbitrary abelian group; the first three paragraphs of Section 5 show that if A is a finite subset of an abelian group such that A + A ≤ KA then 2A  2A contains a coset progression of dimension at most K^{O(1)} and size at least exp(K^{O(1)}).


Proof of Minkowski's second theorem (Proposition 4.1 from the above Green―Ruzsa paper) 

Blog post by Tim Gowers: This post describes Petridis's proof of the Plünnecke―Ruzsa inequalities, which state that if A + A ≤ KA then for all natural numbers k, ℓ we have kA  ℓA ≤ K^{k+ℓ}A.


Petridis's paper 

Paper by Terence Tao: Lemma 3.4 shows that if A^{3} ≤ KA then A^{n} ≤ K^{f(n)}A for all n; Theorem 4.6 shows that if A^{2} ≤ KA then A is covered by K^{C} translates of a K^{C}approximate group of size at most K^{C}A. 

Paper by Emmanuel Breuillard and me: Corollary 3.16 shows that nilprogressions are approximate groups. 

Paper: Proposition C.1 shows that a nilprogression is contained in a bounded power of the corresponding ordered progression. 