A quantitative improvement for Roth's theorem on arithmetic progressions

We improve the quantitative estimate for Roth's theorem on three‐term arithmetic progressions, showing that if A⊂{1,...,N} contains no non‐trivial three‐term arithmetic progressions, then |A|≪N(loglogN)4/logN . By the same method, we also improve the bounds in the analogous problem over Fq[t] and for the problem of finding long arithmetic progressions in a sumset.

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