Logarithmic bounds for Roth's theorem via almost-periodicity

We give a new proof of logarithmic bounds for Roth’s theorem on arithmetic progressions, namely that if $A\subset \{1,2,\dots ,N\}$ is free of three-term progressions, then $\mid A\mid \le N\mathrm{/}(\log N{)}^{1-o(1)}$. Unlike previous proofs, this is almost entirely done in physical space using almost-periodicity.