Abstract

We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if $A\subseteq \{1,\dots <!--FUNCTION\; APPLICATION-->,N\}$ has no nontrivial three-term arithmetic progressions then $\left|A\right|\le \exp <!--FUNCTION\; APPLICATION-->(-c{(\log <!--FUNCTION\; APPLICATION-->N)}^{1∕12})N$, where $c>0$ is a constant.

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 the problem of finding long arithmetic progressions in $A+A+A$, where $A\subseteq \{1,\dots <!--FUNCTION\; APPLICATION-->,N\}$.

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