New bounds for Szemeredi's theorem, II: A new bound for r_4(N) - Green, B; Tao, T (2006)
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}.