An improvement to the Kelley-Meka bounds on three-term arithmetic progressions

In a recent breakthrough Kelley and Meka proved a quasipolynomial upper bound for the density of sets of integers without non-trivial three-term arithmetic progressions. We present a simple modification to their method that strengthens their conclusion, in particular proving that if $A\subset \{1,\dots ,N\}$ has no non-trivial three-term arithmetic progressions then $$\mid A\mid \le \exp (-c(\log N{)}^{1\mathrm{/}9})N$$ for some $c>0$.