Proceedings of the London Mathematical Society Advance Access originally published online on July 31, 2008
Proceedings of the London Mathematical Society 2009 98(2):365-392; doi:10.1112/plms/pdn030
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© 2008 London Mathematical Society
New bounds for Szemerédi's theorem, I: progressions of length 4 in finite field geometries
Centre for Mathematical Sciences
Wilberforce Road
Cambridge
CB3 0WA
United Kingdom
Department of Mathematics
UCLA
Los Angeles, CA 90095-1555
USA
tao@math.ucla.edu
Received 26 September 2005. Revision received 4 February 2008.
Let k 
3 be an integer, and let G be a finite abelian group with |G| = N, where (N, (k – 1)!) = 1. We write rk(G) for the largest cardinality |A| of a set A 
G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that rk(
/N) = ok(N). It is known, in fact, that the estimate rk(G) = ok(N) holds for all G. There have been many papers concerning the issue of finding quantitative bounds for rk(G). A result of Bourgain states thatr3(G) << N(log log N/log N)1/2 for all G. In this paper we obtain a similar bound for r4(G) in the particular case G = Fn, where F is a fixed finite field with char (F) 
2, 3 (for example, F = 
5). We prove that r4(G) << F N(log N)–c for some absolute constant c > 0. In future papers we will treat the general abelian groups G, eventually obtaining a comparable result for an arbitrary G.
2000 Mathematics Subject Classification 11B99.
The first author is a Clay Research Fellow, and is pleased to acknowledge the support of the Clay Mathematics Institute. Some of this work was carried out while he was on a long-term visit to MIT. The second author is supported by a grant from the Packard Foundation.