On Sums of Generating Sets in (Z₂)^n

Let A and B be two affinely generating sets of (Z_2)^n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of (Z_2)^n. These cosets are arranged as Hamming balls, the smaller of which has radius 1. By similar methods, we re-prove the Freiman-Ruzsa theorem in (Z_2)^n, with an optimal upper bound. Denote by F(K) the maximal spanning constant |<A>|/|A|, over all subsets A of (Z_2)^n with doubling constant |A+A|/|A| < K. We explicitly calculate F(K), and in particular show that 4^K / 4K < F(K) (1+o(1)) < 4^K / 2K. This improves the estimate F(K) = poly(K) 4^K, found recently by Green and Tao and by Konyagin.