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

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We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F₂ⁿ, improving the previously known bounds in such theorems. For instance, if A is a subset of F₂ⁿ such that |A+A| <= K|A| (thus A has small additive doubling), we show that there exists an affine subspace V of F₂ⁿ of cardinality |V| >> K-O(√K) |A| such that |A ∩ V| >> |V|/2K. Under the assumption that A contains at least |A|³/K quadruples with a₁ + a₂ + a₃ + a₄ = 0 we obtain a similar result, albeit with the slightly weaker condition |V| >> K⁻ᵒ⁽ᵏ⁾|A|.