Numbers expressible as a difference of two Pisot numbers

Abstract

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