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Claim. Theorem 1 of Bloom and Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions, states that if $A\subseteq\{1,\ldots,N\}$ contains only trivial three-term arithmetic progressions, then

$$\lvert A\rvert\le\exp\bigl(-c(\log N)^{1/9}\bigr)N$$

for some constant $c>0$; the note remarks that a more elaborate version of its idea reaches the exponent $5/41$. The result page Theorem 1 records the statement. The bound gives $r_3(N)=o(N)$, the instance $k=3$ of Problem 139, with a sharper rate than Kelley and Meka's exponent $1/12$; the problem asks for no rate. The note modifies the almost-periodicity step of Kelley and Meka's argument and otherwise follows it as presented in the authors' exposition.

Covers. The instance $k=3$ of the statement, $r_3(N)=o(N)$, which Szemerédi's accepted full claim and Kelley and Meka's accepted partial claim already settle; the page records the sharper rate. Nothing about any $k\ge4$.

Depends on. Kelley and Meka's claim, whose argument the note modifies.

Standing. Claimed. The note is an arXiv preprint, posted 2023-09-05 and not revised with no journal record (Crossref, 2026-09-18; the authors' exposition in Essential Number Theory is a separate paper), so refereed is not listed. The site's curator labels the problem proved on Szemerédi's theorem and cites this note in the commentary only as the improvement of the best known bound for $k=3$, which credits the bound and not a settlement of the problem, so no reviewed evidence is listed. The proof is not checked here.