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[...