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 subset of {1,..., N} contains no non-trivial three-term arithmetic progressions, then vertical bar A vertical bar << N(log log N)(4)/log N. By the same method, we also improve the bounds in the analogous problem over F-q[t] and for the problem of finding long arithmetic progressions in a sumset.