Let K be a real algebraic number field. A well-known result, due to Meyer, states that the set \(\wp _{K}\) of Pisot numbers generating K is uniformly discrete and relatively dense in the interval \([1,\infty ).\) A recent theorem of Dubickas yields that an algebraic integer, generating K, belongs to the set \(\wp _{K}-\wp _{K}\) if and only if its other conjugates are of modulus less than 2. In the present paper, we show that \(1\in \wp _{K}-\wp _{K},\) and if K is totally real, then the elements of \(\wp _{K}-\wp _{K}\) are all algebraic integers of K whose images under the action of all embeddings of K into \(\mathbb {R},\) other than the identity of K, belong to the interval \((-2,2).\) Also, we prove that set \(\{\theta ^{\prime }-\theta \mid \theta ^{\prime }\in \wp _{K},\) \(\theta \in \wp _{K},\) \(\theta ^{\prime }>\theta \) and \((\theta ,\theta ^{\prime })\cap \wp _{K}=\varnothing \}\) is finite and contains at least two elements (resp. at least \(2^{\deg (K)-1})\) elements when \(K\ne \mathbb {Q}\) (resp. when K is totally real).
