A NOTE ON THE FREIMAN AND BALOG–SZEMERÉDI–GOWERS THEOREMS IN FINITE FIELDS

Abstract We prove quantitative versions of the Balog–Szemerédi–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 $A \subseteq \mathbb {F}_2^n$ is such that ∣A+A∣≤K∣A∣ (thus A has small additive doubling), we show that there exists an affine subspace H of 𝔽2n 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 a1+a2+a3+a4=0, we obtain a similar result, albeit with the slightly weaker condition ∣H∣≫K−O(K)∣A∣.