OPEN

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 \[\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}\] contain all rationals in some open interval?

#355: [ErGr80,p.58]

number theory, unit fractions