Logarithmic bounds for Roth's theorem via almost-periodicity

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