Improved Bounds for 3-Progressions

We prove that if $A\subset \{1,\dots ,N\}$ has no nontrivial three-term arithmetic progressions, then $\mathrm{\mid }A\mathrm{\mid }\le \exp (-c\log (N{)}^{1\mathrm{/}6}\log \log (N{)}^{-1})N$ for some absolute constant $c>0$. To obtain this bound, we use an iterated variant of the sifting argument of Kelley and Meka, as well as an improved bootstrapping argument for Croot-Sisask almost-periodicity due to Bloom and Sisask.