A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite
fields

We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman
theorems in the model case of a finite field geometry F_2^n, improving the
previously known bounds in such theorems. For instance, if A is a subset of
F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show
that there exists an affine subspace V of F_2^n of cardinality |V| >>
K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that
A contains at least |A|^3/K quadruples with a_1 + a_2 + a_3 + a_4 = 0 we obtain
a similar result, albeit with the slightly weaker condition |V| >>
K^{-O(K)}|A|.