Some class of cubic Pisot numbers with finiteness property
A real base \(\beta > 1\) has the finiteness property (F) if each number in \(\mathbb{Z}[\beta] \cap [0,1)\) has a finite \(\beta\)-expansion. \textit{S. Akiyama} [in: Algebraic number theory and diophantine analysis. Proceedings of the international conference, Graz, Austria, August 30--September 5, 1998. Berlin: Walter de Gruyter. 11--26 (2000; Zbl 1001.11038)] showed that there are exactly 4 classes of cubic Pisot units with this property. The authors extend these classes to 5 classes of cubic Pisot numbers, namely the algebraic numbers with minimal polynomial \(x^3-ax^2-bx-c\) satisfying \(a\geq b\geq c\geq 1\) or \(a>b+c>b\geq 0\) or \(b+c\geq a\geq b=c-1\geq 0\) or \(a\geq c-b>b+c\geq 0\) or \(0\leq a<b\leq a+(c+1)/2\leq a+(b+3)/4\); the proofs are very technical. Some classes of cubic Pisot numbers not satisfying the finiteness property are also given. From the theory of shift radix systems, we know that a complete classification of the cubic Pisot numbers with finiteness property is complicated; see e.g. [\textit{P. Kirschenhofer} and \textit{J. M. Thuswaldner}, RIMS Kôkyûroku Bessatsu B46, 1--59 (2014; Zbl 1352.11014)].
scientific article; zbMATH DE number 7598529