A quantitative improvement for Roth's theorem on arithmetic progressions

摘要

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.