The order type of the set of Pisot numbers
An algebraic integer \(\theta\) is a Pisot (or Pisot-Vijayaraghavan or PV-) number if \(\theta\) is real, \(\theta>1\) and its remaining conjugates lie inside the unit circle. The main result of this paper (answering the question raised by the second author) is the following theorem: The order type of the set \(S\) of all PV-numbers is the ordered sum, \(\sum_{n = 1}^\infty a_n\), where \(a_1 = \omega+1 + \omega^*\) and \(a_{n+1} = a_n \omega+1 + (a_n \omega)^*\) for \(n\geq 1\). The authors are also occupied with some properties of the \(n\)th derived sets \(S^{(n)}\); for example, it is proved that each \(\theta \in S^{(n)}\) \((n \geq 1)\) is a two-sided limit of elements of \(S^{(n-1)}\).
scientific article; zbMATH DE number 896586