Type: Article
Publication Date: 2009-02-01
Citations: 17
DOI: https://doi.org/10.1017/s1446788708000359
Abstract We prove quantitative versions of the Balog–Szemerédi–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 $A \subseteq \mathbb {F}_2^n$ is such that ∣ A + A ∣≤ K ∣ A ∣ (thus A has small additive doubling), we show that there exists an affine subspace H of 𝔽 2 n of cardinality $|H| \gg K^{-O(\sqrt {K})} |A|$ such that $|A \cap H| \geq (2K)^{-1} |H|$ . 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 ∣ H ∣≫ K − O ( K ) ∣ A ∣.