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.
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机译：我们改进了三项算术级数对罗斯定理的定量估计，表明如果{1，...，N}的子集不包含非平凡的三项算术级数，则竖线A竖线 N （log log N）（4）/ logN。通过相同的方法，我们还改善了Fq [t]上的类似问题的边界，也改善了在求和中找到长算术级数的问题。
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