On the topology of sums in powers of an algebraic number

Abstract

Let $1<q<2$ and \[ \Lambda(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $\Lambda(q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $\Lambda(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $\alpha$ such that $q|\alpha|<1$, then $\Lambda(q)$ is dense in $\mathbb{R}$.