Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions

Thomas F. Bloom; Olof Sisask

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arXiv · 2007.03528

Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions

Thomas F. Bloom·Olof Sisask

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Abstract

We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some absolute constant $c>0$. In particular, this proves the first non-trivial case of a conjecture of Erd\H{o}s on arithmetic progressions.

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Thomas F. Bloom, Olof Sisask. 2020-07-07. Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. https://arxiv.org/abs/2007.03528

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