An approximation property of Pisot numbers

Let $q>1$. Initiated by P. Erd\H os et al. in \cite{ErdJooKom1}, several authors studied the numbers $l^m(q)=\inf \{y\ :\ y\in\Lambda_m,\ y\ne 0\}$, $m=1,2,\dots$, where $\Lambda_m$ denotes the set of all finite sums of the form $y=\eps_0 + \eps_1 q + \eps_2 q^2 + \dots + \eps_n q^n$ with integer coefficients $-m\le \eps_i \le m$. It is known (\cite{Bug}, \cite{ErdJooKom1}, \cite{ErdKom}) that $q$ is a Pisot number if and only if $l^m(q)>0$ for all $m$. The value of $l^1(q)$ was determined for many particular Pisot numbers, but the general case remains widely open. In this paper we determine the value of $l^m(q)$ in other cases.