What are some interesting applications of Freiman's theorem or, better-yet, its recent generalizations (eg Green-Ruzsa) that could be included in a graduate course in additive combinatorics?

I'm looking for an application or two which would take no more than a total of two weeks to present in class. Nathanson includes a proof which relies on Freiman's theorem that if a sufficiently large subset of the integers contains many three term arithmetic progressions, then it contains a long arithmetic progression. This application meets my criteria, but is so old at this point -- surely newer and more interesting applications are out there.

  • $\begingroup$ This is not exactly additive combinatorics, but a surprising application nevertheless. I learned from a talk by Figalli that he and Jerison proved a stability result, of Brunn-Minkowski type, for sumsets in $\mathbb R^n$; the proof is done by induction on the dimension, and the basic step $n=1$ is done by discretisation, splitting a subset of $\mathbb R$ in $p$ parts for a large prime $p$, and then applying Freiman or some qualitatively equivalent theorem. See ma.utexas.edu/users/figalli/papers/… for the details. $\endgroup$ – Emanuele Tron Mar 12 '15 at 17:56

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