Kenyon Theorem Revisited

Abstract

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

20 pages