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Blog post by Terence Tao: Proposition 3 shows that if |A2| < 3⁄2|A| then there exists a subgroup H with |H| ≤ |A2| and an element x such that A ⊂ xH = Hx.
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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| ≤ K|A| then 2A - 2A contains a coset progression of dimension at most KO(1) and size at least exp(-KO(1)).
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Proof of Minkowski's second theorem (Proposition 4.1 from the above Green―Ruzsa paper) |
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Blog post by Tim Gowers: This post describes Petridis's proof of the Plünnecke―Ruzsa inequalities, which state that if |A + A| ≤ K|A| then for all natural numbers k, ℓ we have |kA - ℓA| ≤ Kk+ℓ|A|.
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Petridis's paper |
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Paper by Terence Tao: Lemma 3.4 shows that if |A3| ≤ K|A| then |An| ≤ Kf(n)|A| for all n; Theorem 4.6 shows that if |A2| ≤ K|A| then A is covered by KC translates of a KC-approximate group of size at most KC|A|. |
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Paper by Emmanuel Breuillard and me: Corollary 3.16 shows that nilprogressions are approximate groups. |
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Paper: Proposition C.1 shows that a nilprogression is contained in a bounded power of the corresponding ordered progression. |