  We give a self-contained exposition of the recent remarkable result of Kelley
and Meka: if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term
arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/11})N$ for
some constant $c>0$.
  Although our proof is identical to that of Kelley and Meka in all of the main
ideas, we also incorporate some minor simplifications relating to Bohr sets.
This eases some of the technical difficulties tackled by Kelley and Meka and
widens the scope of their method. As a consequence, we improve the lower bounds
for finding long arithmetic progressions in $A+A+A$, where $A\subseteq
\{1,\ldots,N\}$.
