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arXiv · 1405.5800

A quantitative improvement for Roth's theorem on arithmetic progressions

Thomas F. Bloom
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Abstract

We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if $A\subset\{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert\ll N(\log\log N)^4/\log N$. By the same method we also improve the bounds in the analogous problem over $\mathbb{F}_q[t]$ and for the problem of finding long arithmetic progressions in a sumset.

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Thomas F. Bloom. 2014-05-22. A quantitative improvement for Roth's theorem on arithmetic progressions. https://doi.org/10.1112/jlms%2Fjdw010

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