a1 Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK (email: firstname.lastname@example.org)
a2 Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA (email: email@example.com)
We prove quantitative versions of the Balog–Szemerédi–Gowers and Freiman theorems in the model case of a finite field geometry 2n, improving the previously known bounds in such theorems. For instance, if is such that A+A≤KA (thus A has small additive doubling), we show that there exists an affine subspace H of 2n of cardinality such that . Under the assumption that A contains at least A3/K quadruples with a1+a2+a3+a4=0, we obtain a similar result, albeit with the slightly weaker condition HK−O(K)A.
(Received November 02 2006)
(Accepted November 12 2007)
2000 Mathematics subject classification
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The first author is a Clay Research Fellow, and is pleased to acknowledge the support of the Clay Mathematics Institute. The second author is supported by a grant from the MacArthur Foundation.