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.