Journal of the Australian Mathematical Society

Research Article

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

BEN GREENa1 c1 and TERENCE TAOa2

a1 Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK (email: b.j.green@dpmms.cam.ac.uk)

a2 Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA (email: tao@math.ucla.edu)

Abstract

We prove quantitative versions of the Balog–Szemerédi–Gowers and Freiman theorems in the model case of a finite field geometry xs1D53D2n, improving the previously known bounds in such theorems. For instance, if $A \subseteq \mathbb {F}_2^n$ is such that xs2223A+Axs2223Kxs2223Axs2223 (thus A has small additive doubling), we show that there exists an affine subspace H of xs1D53D2n 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 xs2223Axs22233/K quadruples with a1+a2+a3+a4=0, we obtain a similar result, albeit with the slightly weaker condition xs2223Hxs2223xs226BKO(K)xs2223Axs2223.

(Received November 02 2006)

(Accepted November 12 2007)

2000 Mathematics subject classification

  • primary 11B99

Keywords and phrases

  • Freiman’s theorem;
  • finite field models;
  • Balog–Szemerédi–Gowers theorem

Correspondence:

c1 For correspondence; e-mail: b.j.green@dpmms.cam.ac.uk

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.