Kenyon Theorem Revisited

We study sets $E(\mathrm{\Sigma },q)=\{{\sum }_{i=1}^{\mathrm{\infty }}{\sigma }_i{q}^i\text{\hspace{0.17em}⁣}:({\sigma }_i)\in {\mathrm{\Sigma }}^{\mathbb{N}}\}$ for a finite set $\mathrm{\Sigma }\subset \mathbb{R}$ and $q\in (0,1)$. Under the assumption $q\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }=1$ we prove several new equivalent conditions for $E(\mathrm{\Sigma },q)$ to contain an interval. We give a full characterization, if additionally $\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }$ is prime.