New bounds for Szemeredi's theorem, II: A new bound for r_4(N)

Details New bounds for Szemeredi's theorem, II: A new bound for r_4(N) New bounds for Szemeredi's theorem, II: A new bound for r_4(N)

2006-10-19

arxiv.org/abs/math/0610604v1

Mathematics

Number Theory

26 pages

Scientific paper

Define r_4(N) to be the largest cardinality of a set A in {1,...,N} which does not contain four elements in arithmetic progression. In 1998 Gowers proved that r_4(N) << N(log log N)^{-c} for some absolute constant c> 0. In this paper (part II of a series) we improve this to r_4(N) << N e^{-c\sqrt{log log N}}. In part III of the series we will use a more elaborate argument to improve this to r_4(N) << N(log N)^{-c}.

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