Problem Statement
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?
Categories:
Number Theory Unit Fractions
Progress
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].
Source: erdosproblems.com/355 | Last verified: January 14, 2026