On the distribution of the set \(\{\sum^ n_{i=1}\varepsilon _ i q^ i : \varepsilon_ i \in\{0,1\}, n \in \mathbb{N}\}\) (Q1187312)

Fix \(1<q<\sqrt{2}\) and let \(\{y_ n\}=\{\sum_{i=1}^ n \varepsilon_ i q^{2(n-i)}\): \(\varepsilon_ i\in\{0,1\}\}\), \(n=1,2,3,\dots\;\). The author shows that if \(y_{n+1}-y_ n\to 0\) as \(n\to\infty\) then there exists an expansion \(1=\sum_{i=1}^ \infty q^{-n_ i}\) such that \(\sup_ i(n_{i+1}-n_ i)=\infty\).