On the topology of sums in powers of an algebraic number

Let $1<q<2$ and $$\mathrm{\Lambda }(q)=\sum _{k=0}^n{a}_k{q}^k\mid a_k\in \{-1,0,1\},n\ge 1\mathrm{.}$$ It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm 1$, then $\mathrm{\Lambda }(q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $\mathrm{\Lambda }(q)$ 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 $q\mathrm{\mid }\alpha \mathrm{\mid }<1$, then $\mathrm{\Lambda }(q)$ is dense in $\mathbb{R}$.