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

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.