A NOTE ON THE FREIMAN AND BALOG-SZEMEREDI-GOWERS THEOREMS IN FINITE FIELDS

We prove 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 subset of F(2)(n) is such that vertical bar A + A vertical bar <= K vertical bar A vertical bar (thus A has small additive doubling), we show that there exists an affine subspace H of F(2)(n) of cardinality vertical bar H vertical bar >> K(-O(root K)) vertical bar A vertical bar such that vertical bar A boolean AND H vertical bar >= (2K)(-1) vertical bar H vertical bar. Under the assumption that A contains at least vertical bar A vertical bar(3)/K quadruples with a(1) + a(2) + a(3) + a(4) = 0, we obtain a similar result, albeit with the slightly weaker condition vertical bar H vertical bar >> K(-O(K))vertical bar A vertical bar.