---

Claim. The answer to Problem 310 is yes: for every fixed density $\alpha>0$ there is a bound $B(\alpha)$ such that every $A\subseteq\{1,\ldots,N\}$ with $|A|\ge\alpha N$ contains a subset $S$ whose reciprocal sum is a rational $a/b$ with $a\le b\le B(\alpha)$. Liu and Sawhney prove the quantitative form recorded on the result page Proposition 1.4 of their paper On further questions regarding unit fractions: there is an absolute constant $C\ge1$ such that for every $\varepsilon>0$, every $N$ large in terms of $\varepsilon$, and every $A\subseteq[1,N]$ with $|A|\ge\alpha N$ and $(\log N)^{-1/7+\varepsilon}\le\alpha\le1/2$, some $B\subseteq A$ has $\sum_{n\in B}1/n=s/t$ with $1\le s\le t\le\exp(C/\alpha)$. For fixed $\alpha\le1/2$ this is the statement with $a=s$, $b=t$ and $b\le\exp(C/\alpha)=O_\alpha(1)$; for fixed $\alpha>1/2$ the case $\alpha=1/2$ applies to any subset of $A$ of size at least $N/2$, a one-line reduction recorded on the problem page. The paper remarks (p. 3) that a direct application of Bloom's Proposition 1 with standard estimates already gives $t=O_\alpha(1)$, which is the qualitative answer the site attributes to Bloom's density theorem through this observation, while the range of $\alpha$ and the bound $\exp(C/\alpha)$ need the paper's own methods; the dependence on $\alpha$ is sharp up to the constant, since the integers in $[N/2,N]$ with all prime factors above $\exp(1/\alpha)$ have density $\gg\alpha$, reciprocal sum below $1$, and no nontrivial subsum with denominator below $\exp(1/\alpha)$.

Acceptance. Refereed: the paper is published in International Mathematics Research Notices 2026, no. 2, rnaf382 (published online 14 January 2026; DOI 10.1093/imrn/rnaf382). Reviewed: the site's curator, T. F. Bloom, marks the problem proved and credits the answer to Liu and Sawhney's observation and their precise version; Bloom is not an author of their paper (Bloom's density theorem is the input their remark applies), so the credit is independent of the claimants. arXiv v1 (10 April 2024) is the only arXiv version, and its result page records the proof (p. 21) as a pointer and sketch with four parameter questions to be compared against the published text; the corpus has not verified the proof, and the acceptance rests on the refereed publication and the curator's credit.