A Note on the Freiman and Balog–szemerédi–gowers Theorems in Finite Fields

We prove quantitative versions of the Balog–Szemerédi–Gowers and Freiman theorems in the model case of a finite field geometry F2 , improving the previously known bounds in such theorems. For instance, if A ⊆ F2 is such that |A + A|6 K |A| (thus A has small additive doubling), we show that there exists an affine subspace H of F2 of cardinality |H | K−O( √ K ) |A| such that |A ∩ H |> (2K )−1|H |. 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 |H | K−O(K |A|. 2000 Mathematics subject classification: primary 11B99.