On some problems of I. Joó (Q1187309)

The authors provide negative answers to two problems raised bo I. Joó: Does there exist an expansion \(1=\sum_{i=1}^ \infty q^{-n_ i}\) satisfying \(\sup(n_{i+1}-n_ i)=\infty\) for every \(1<q<(1+\sqrt{5})/2\) and, for every \(1<q<2\) is the existence of such an expansion equivalent to the existence of \(2^{\aleph_ 0}\) such different expansions?