An approximation property of Pisot numbers (Q1971702)

Given a positive real number \(q\) and an integer \(m\geq 1\), denote by \(\Lambda= \Lambda_m\) the set of all real numbers \(y\) having at least one representation of the form \[ y= \varepsilon_0+ \varepsilon_1q+ \varepsilon_2 q^2+\cdots+ \varepsilon_n q^n \] with some integer \(n> 0\) and \(\varepsilon_i\in \{-m, -m+1,\dots, -1,0,1,\dots, m-1,m\}\), and let \[ l^m(q)= \inf\{|y|: y\in \Lambda_m,\;y\neq 0\}. \] It is known that \(q\) is a Pisot number if and only if \(l^m(q)> 0\) for all \(m\). In this paper the authors determine formulae for \(l^m(A)\) for all \(m\), where \(A\) is the golden ratio \((\sqrt{5}+1)/2\).