We generalize Bourgain's discretized sum-product theorem to matrix algebras. 0 k 1 Freiman problem asks, in a given abelian group, which subsets grow slowly under addition. Freiman's Theorem asserts that they are "close" to generalized arithmetic progressions. Obtaining a polynomial bound for this theorem is one of the fundamental open problems in additive combinatorics. See [31, Chapter 5].
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