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 $A\subset \{1,2,\dots ,N\}$ is free of three-term progressions, then $|A|\le N/(\log N{)}^{1-o(1)}$. Unlike previous proofs, this is almost entirely done in physical space using almost-periodicity.

20 pages

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