New bounds for Szemerédi's theorem, III: A polylogarithmic bound for $r_4(N)$

Abstract: Define $r_4(N)$ to be the largest cardinality of a set $A \subset {1,\dots,N}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that [ r_4(N) \ll N(\log \log N)^{-c}] for some absolute constant $c>0$. In 2005, the authors improved this to [ r_4(N) \ll N e^{-c\sqrt{\log\log N}}.] In this paper we further improve this to [ r_4(N) \ll N(\log N)^{-c},] which appears to be the limit of our methods.