Kenyon Theorem Revisited

We study sets $E(Σ,q)=\left\{\sum_{i=1}^\infty σ_iq^i\colon(σ_i)\inΣ^{\mathbb N}\right\}$ for a finite set $Σ\subset \mathbb R$ and $q\in(0,1)$. Under the assumption $q|Σ|=1$ we prove several new equivalent conditions for $E(Σ,q)$ to contain an interval. We give a full characterization, if additionally $|Σ|$ is prime.