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⊂{1,2,…,N} is free of three-term progressions, then |A|≤N/(logN)1−o(1). Unlike previous proofs, this is almost entirely done in physical space using almost-periodicity.