PROVED
This has been solved in the affirmative.
Is there a lacunary sequence $A\subseteq \mathbb{N}$ (so that $A=\{a_1<a_2<\cdots\}$ and there exists some $\lambda>1$ such that $a_{n+1}/a_n\geq \lambda$ for all $n\geq 1$) such that\[\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}\]contains all rationals in some open interval?
Bleicher and Erdős conjectured the answer is no.
In fact the answer is yes, with any lacunarity constant $\lambda\in (1,2)$ (though not $\lambda=2$), as proved by van Doorn and Kovač
[DoKo25].
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This page was last edited 18 November 2025.
Additional thanks to: Will Sawin and Stefan Steinerberger
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #355, https://www.erdosproblems.com/355, accessed 2026-01-16
