Theory of Computing Library Graduate Surveys --------------------------------------------------- Title : Selected Results in Additive Combinatorics: An Exposition Authors : Emanuele Viola Number : 3 Pages : 1-15 URL : https://theoryofcomputing.org/articles/gs003 Abstract -------- We give a stripped-down, self-contained exposition of selected results in additive combinatorics over the vector space F_2^n, leading to the result by Samorodnitsky (STOC 2007) stating that linear transformations are efficiently testable. In particular, we prove the theorems known as the Balog-Szemeredi-Gowers theorem (Combinatorica 1994 and GAFA 1998) and the Freiman-Ruzsa theorem (AMS 1973 and Astérisque 1999).