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Claim. Let $A\subset(1,\infty)$ be a countably infinite set such that $\lvert kx-y\rvert\ge1$ for all distinct $x,y\in A$ and all integers $k\ge1$. Then

$$\sum_{\substack{x<n\\ x\in A}}\frac{1}{x}=o(\log n),$$

the second displayed assertion of Problem 143. This is the contrapositive of Theorem 1 of the paper on the library card koukoulopoulos_2025_erdos_s_integer_dilation_approximation_problem: if a discrete set $A$ of positive reals satisfies $\limsup_{x\to\infty}(\log x)^{-1}\sum_{\alpha\in A\cap[1,x]}1/\alpha>0$, then for every $\epsilon>0$ there are distinct $\alpha,\beta\in A$ and a positive integer $n$ with $\lvert n\alpha-\beta\rvert<\epsilon$. The problem's hypothesis with $k=1$ keeps distinct elements at least $1$ apart, so $A$ is discrete, and the conclusion with $\epsilon=1$ is exactly what the hypothesis forbids; the upper logarithmic density of $A$ is therefore $0$. By partial summation the same bound gives $\liminf_{x\to\infty}\lvert A\cap[1,x]\rvert/x=0$. The paper resolves the dilation approximation problem that Erdős posed in 1948 under his second hypothesis, where before it only Haight's theorem for sets with all ratios irrational was known; its proof uses the GCD graphs that Koukoulopoulos and Maynard built for the Duffin--Schaeffer conjecture inside a structure-versus-randomness dichotomy.

Covers. The $o(\log n)$ assertion for every set $A$ satisfying the hypothesis, and with it the vanishing of the lower asymptotic density. It says nothing about the first displayed assertion, the convergence of $\sum_{x\in A}1/(x\log x)$, which Apicella's pending claim asserts is false, and nothing about the full density statement $\lim\lvert A\cap[1,x]\rvert/x=0$, which Besicovitch's primitive sets of positive upper density refute.

Standing. The site's curator credits the paper on the problem page (last edited 24 April 2026) with a partial resolution, the $o(\log n)$ assertion, but the site labels the problem OPEN, and commentary on an open problem is not acceptance, so no reviewed evidence is listed. The paper is an arXiv preprint of 13 February 2025 (47 pages) with no journal reference found on 2026-10-07, and no Lean proof of it exists, so the claim stays claimed.