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

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arxiv 2007.03528 v2 pith:WSFYAQ2B submitted 2020-07-07 math.NT math.CO

Thomas F. Bloom , Olof Sisask This is my paper · ORCID
classification math.NTmath.CO
keywords arithmeticprogressionsnon-trivialabsolutebarrierbreakingcaseconjecture
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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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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Roth theorem in $\mathbb R^2$ and a related ergodic theorem

    math.CA 2026-07 accept novelty 6.5 of 10

    A quantitative Roth theorem holds in R^{2} for the genuinely two-dimensional polynomial pattern (t1,t2) and (t1^{2}+t2^{2}, t1^{3}+t2^{3}), with a matching pointwise ergodic theorem.

  2. Large Sets of Integers with No Harmonic Triples

    math.NT 2026-07 accept novelty 6.0 of 10

    The author proves f(N) ≫ N exp(−(2√(log(24/7))+o(1))√(log log N)) for the largest harmonic-triple-free subset of [N], matching the form of the best 3-AP-free lower bound with log N replaced by log log N.

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