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

Part of: Sequences and sets

Published online by Cambridge University Press:  01 February 2009

BEN GREEN  and
TERENCE TAO
BEN GREEN*
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK (email: b.j.green@dpmms.cam.ac.uk)
TERENCE TAO
Affiliation:
Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA (email: tao@math.ucla.edu)
*
✉For correspondence; e-mail: b.j.green@dpmms.cam.ac.uk
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Abstract

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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∣.

Keywords

Freiman’s theoremfinite field modelsBalog–Szemerédi–Gowers theorem

MSC classification

Secondary: 11B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 1 , February 2009 , pp. 61 - 74
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

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.

References

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