Erdős Problem #355 — Lacunary Reciprocal Sums Representing an Interval

Does there exist a lacunary sequence $A=(A_n)_{n\in\mathbb N}$ of natural numbers and real numbers $u<v$ such that every rational number in the open interval $(u,v)$ is a finite subsum of reciprocal terms of $A$? More precisely, $A$ is lacunary if there is some constant $\lambda>1$ such that

$$\frac{A_{n+1}}{A_n}\ge\lambda$$

for all $n\ge1$, and the required representation property is

$$\forall q\in\mathbb Q,\quad u<q<v\ \Longrightarrow\ \exists A'\subseteq A\text{ finite such that }q=\sum_{a\in A'}\frac1a.$$