Bloom 2023 improvement kelley meka bounds three term

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theorem_1: The sharpened Kelley–Meka bound, exponent 1/9 in place of 1/12, from a modification of the almost-periodicity step; the source of the exponent in the site's upper bound exp(O((log k)^9)) for W(3,k).

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Thomas F. Bloom, Olof Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).

This short note modifies the almost-periodicity step in Kelley and Meka's argument to sharpen their quasipolynomial bound. Theorem 1 shows that if A ⊆ {1,...,N} has only trivial three-term arithmetic progressions then |A| <= exp(-c(log N)^{1/9})N, improving the exponent 1/12 of Kelley-Meka; the authors remark a more elaborate version reaches 5/41. Theorem 2 gives the model-setting analog |A| << q^{n - c n^{1/7}} for progression-free A ⊆ F_q^n with q an odd prime, improving Kelley-Meka's 1/9. Theorem 3 shows any A ⊆ F_q^n of density alpha has, for each gamma in (0,1], an affine subspace V of codimension O(L(alpha)^5 L(gamma)^2) with |(A+A) ∩ V| >= (1-gamma)|V|, where L(x) = log(2/x), against Kelley-Meka's codimension O(L(alpha)^5 L(gamma)^4) and Sanders's O(L(alpha)^4 gamma^{-2}); Hunter and Pohoata use essentially this theorem for monochromatic subspaces in 2-colourings of the 1-dimensional subspaces of F_2^n. Theorem 4 gives long progressions in A+A+A of length exp(-O(L(alpha)^2)) N^{Omega(1/L(alpha)^7)}. The improved bootstrapping of almost-periodicity is the paper's technical contribution, and these are the quantitative bounds cited for problems 139, 160, 657 and 721.

Source: https://arxiv.org/abs/2309.02353.

The retained folder-name PDF is the arXiv v1 of 5 September 2023 (nine pages), the only version (listing read); no journal version was found (a Crossref bibliographic query on 2026-09-18 returned the authors' separate exposition in Essential Number Theory 2 (2023), 15--44, not this note), so the paper is held as a preprint. For problem 721 the bearing is indirect: the note never mentions van der Waerden numbers (no occurrence of "Waerden" in its text layer); the site's upper bound W(3,k) << exp(O((log k)^9)) takes the exponent 9 as the reciprocal of Theorem 1's 1/9 through the density argument stated in Hunter's footnote 1 and Schoen's remark, a derivation written in no held source. Read status for problem 721: claims checked for Theorem 1 and its surrounding paragraph, read clause by clause on the page image of p. 1; the proof was not read. Result page: theorem_1. The arXiv record (https://arxiv.org/abs/2309.02353, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #139, #160, #657, #721

Results to transcribe.

• Theorem 1: A ⊆ {1,...,N} without non-trivial three-term progressions satisfies |A| <= exp(-c(log N)^{1/9})N; exponent 5/41 attainable with more work.
• Theorem 2: For odd prime q and A ⊆ F_q^n progression-free, |A| << q^{n - c n^{1/7}}.
• Theorem 3: For A ⊆ F_q^n of density alpha and gamma in (0,1], some affine subspace V of codimension O(L(alpha)^5 L(gamma)^2) has |(A+A) ∩ V| >= (1-gamma)|V|.
• Theorem 4: If A ⊆ {1,...,N} has size alpha N then A+A+A contains an arithmetic progression of length at least exp(-O(L(alpha)^2)) N^{Omega(1/L(alpha)^7)}.
• Technical contribution: A quantitatively improved bootstrapping of almost-periodicity, inserted into the Kelley-Meka argument as presented in Bloom-Sisask.