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A NOTE ON THE FREIMAN AND BALOG–SZEMERÉDI–GOWERS THEOREMS IN FINITE FIELDS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 86 / Issue 1 / February 2009
- Published online by Cambridge University Press:
- 01 February 2009, pp. 61-74
- Print publication:
- February 2009
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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
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 
such that 
. 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∣.
is such that ∣
such that 
. Under the assumption that 