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

Journal article

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}.

Authors: Green, B; Tao, T

• Publication date: 2006-10-19
• Keywords: math.NT; math.CO