Erdős Problem #310

Let \(\alpha >0\) and \(N\geq 1\). Is it true that for any \(A\subseteq \{1,\ldots,N\}\) with \(\lvert A\rvert \geq \alpha N\) there exists some \(S\subseteq A\) such that \[\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\] with \(a\leq b =O_\alpha(1)\)? Liu and Sawhney [LiSa24] observed that the main result of Bloom [Bl21] implies a positive solution to this conjecture. They prove a more precise version, that if \((\log N)^{-1/7+o(1)}\leq \alpha \leq 1/2\) then there is some \(S\subseteq A\) such that \(\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\) with \(a\leq b \leq \exp(O(1/\alpha))\). They also observe that the dependence \(b\leq \exp(O(1/\alpha))\) is sharp.