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Claim. Theorem 1.1 of Leng, Sah and Sawhney, Improved Bounds for Szemerédi's Theorem, states that for each fixed $k\ge5$ there is $c_k\in(0,1)$ with

$$r_k(N)\ll N\exp\bigl(-(\log\log N)^{c_k}\bigr),$$

where $r_k(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ with no non-trivial $k$-term arithmetic progression. Since the exponential factor tends to zero, the bound gives $r_k(N)=o(N)$ for every $k\ge5$, those instances of Problem 139, with a rate the problem does not ask for. The paper improves Gowers's bound $N(\log\log N)^{-2^{-2^{k+9}}}$, the only earlier bound for $k\ge5$, by feeding the authors' quasipolynomial inverse theorem for the Gowers $U^{k+1}$ norm into the density-increment strategy of Heath-Brown and Szemerédi in the form Green and Tao gave it. The library card is Leng, Sah and Sawhney 2024.

Covers. Every instance $k\ge5$ of the statement, $r_k(N)=o(N)$, which Szemerédi's accepted full claim already settles; the page records the bound's rate, which no claim of this problem requires. Nothing about $k=3$ or $k=4$.

Depends on. Nothing in this wiki; the theorem is the paper's own.

Standing. Claimed. The paper is an arXiv preprint with no journal record on its arXiv listing, so refereed is not listed. The site's curator labels the problem proved on Szemerédi's theorem and cites this paper in the commentary only as the best known bound for $k\ge5$, which credits the bound and not a settlement of the problem, so no reviewed evidence is listed. The proof is not checked here.