Numbers expressible as a difference of two Pisot numbers

We characterize algebraic integers which are differences of twoPisot numbers. Each such number \(\alpha\) $\alpha$ must be real and its conjugates over \(\mathbb{Q}\) $\mathbb{Q}$ mustall lie in the union of the disc \(|z|<2\) $|z|<2$ and the strip \(|\Im(z)|<1\) $|\Im (z)|<1$ . In particular, weprove that every real algebraic integer \(\alpha\) $\alpha$ whose conjugates over \(\mathbb{Q}\) $\mathbb{Q}$ , except possiblyfor \(\alpha\) $\alpha$ itself, all lie in the disc \(|z|<2\) $|z|<2$ can always be written as a difference oftwo Pisot numbers. We also show that a real quadratic algebraic integer \(\alpha\) $\alpha$ withconjugate \(\alpha'\) ${\alpha }^{\prime }$ over \(\mathbb{Q}\) $\mathbb{Q}$ is always expressible as a difference of two Pisot numbers exceptfor the cases \(\alpha<\alpha'<-2\) $\alpha <{\alpha }^{\prime }<-2$ or \(2<\alpha'<\alpha\) $2<{\alpha }^{\prime }<\alpha$ when \(\alpha\) $\alpha$ cannot be expressed in thatform. A similar complete characterization of all algebraic integers \(\alpha\) $\alpha$ expressibleas a difference of two Pisot numbers in terms of the location of their conjugatesis given in the case when the degree \(d\) $d$ of \(\alpha\) $\alpha$ is a prime number.