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

Ben Green·Terence Tao
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Abstract

We obtain 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 is a subset of F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show that there exists an affine subspace V of F_2^n of cardinality |V| >> K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. 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 |V| >> K^{-O(K)}|A|.

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Ben Green, Terence Tao. 2007-11-12. A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields. https://arxiv.org/abs/math/0701585

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