Erdős Problem #1096

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.