On the distances between Pisot numbers generating the same number field

A well-known result, due to Meyer, states that the set P of Pisot numbers, generating a real algebraic number field K, is uniformly discrete and relatively dense in the set of positive real number. In the present paper, we show that P is contained is the set, say D, of the differences of elements of P, and the complement of P in D is not finite. Also, we prove that if K is totally real, then the elements of D are the algebraic integers of K whose images under the action of all embeddings of K into R, other than the identity of K, belong to the interval (-2,2).