Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$), ordered by size as $0=x_1<x_2<\cdots$.

Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$), ordered by size as $0=x_1<x_2<\cdots$.

Is it true that, provided $\epsilon>0$ is sufficiently small, $x_{k+1}-x_k \to 0$?

This was solved affirmatively by Erdős and Komornik [ErKo98], who proved the conclusion whenever $1<q<\sqrt{q_1}$, where $q_1$ is the second Pisot-Vijayaraghavan number.