On an approximation property of Pisot numbers (Q1872906)

For a fixed real number \(\alpha\in(1,2)\) denote for \(m=1,2,\dots\) by \(l_m(\alpha)\) the infimum of non-zero absolute values of \(P(\alpha)\), where \(P\) runs over all polynomials from \(Z[X]\) having height \(H(P)\leq m\). The author shows that for every fixed \(m\) one has \(\sup_\alpha l_m(\alpha)=l_m((1+\sqrt 5)/2)\), \(\limsup_\alpha l_1(\alpha)=1/2\), and if one restricts \(\alpha\) to Pisot numbers in \((1,2)\) then \(\inf_\alpha l_m(\alpha)=0\).