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Claim. Every $A\subseteq\mathbb N$ with $\sum_{n\in A}1/n=\infty$ contains infinitely many non-trivial three-term arithmetic progressions. This is Corollary 1.2 of T. F. Bloom and O. Sisask, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions, arXiv:2007.03528 (v1 2020-07-07, v2 2021-09-01), deduced by partial summation from their Theorem 1.1: a subset of $\{1,\ldots,N\}$ with no non-trivial three-term progression has size $\ll N/(\log N)^{1+c}$ for an absolute constant $c>0$. A set with no three-term progression meets each dyadic block $[2^m,2^{m+1})$ in $\ll2^m/m^{1+c}$ elements, so its reciprocal sum converges; a set with divergent reciprocal sum therefore contains a three-term progression, and removing finitely many progressions leaves the sum divergent, so it contains infinitely many. The proof of Theorem 1.1 is a density increment over Bohr sets with additive frameworks, a structure theorem for non-smoothing sets and a spectral boosting step; the constant $c$ is in principle effective but not computed. The paper is digested on its library card.

Covers. The case $k=3$ of Problem 3: progressions of length three. Longer progressions are the full question, answered yes by the OpenAI release on its claim page, whose bounds for $k\ge4$ are of a different form. The later bound $r_3(N)\ll N\exp(-c(\log N)^{1/12})$ of Kelley and Meka, cited on the problem page, reproves this case with room to spare and has no page of its own: it appeared in the FOCS 2023 proceedings, not a journal, and the site credits the case $k=3$ to Bloom and Sisask.

Depends on. Nothing in this wiki: the deduction is the paper's own.

Standing. Claimed. Not refereed: the arXiv record lists no journal reference. Not reviewed: the site's commentary (page last edited 4 April 2026) credits the case $k=3$ to [BlSi20], but the site's curator is a co-author of the paper, so that credit is not an independent review, and the site's label for the problem is OPEN. Not formalized: the formal-conjectures statement file lists the three-term case as a solved variant (erdos_3.variants.three) without a proof, and this corpus has built no Lean for it.