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Cubic Pisot units with finite beta expansions. (English) Zbl 1001.11038
Halter-Koch, Franz (ed.) et al., 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).
Let \(\beta>1\) be a fixed real number. Any positive real number \(x\) can by expanded in base \(\beta;\) this expansion is called a beta expansion. The author investigates the case when the set of all finite beta expansions coincides with the set \({\mathbb Z}[\beta^{-1}]_{\geq 0}.\) It is known that then \(\beta\) must be a Pisot number, but the converse is false. In his main result, the author gives a necessary and sufficient condition for the above equality to occur for cubic Pisot numbers in terms of their minimal polynomials. More precisely, the equality for cubic \(\beta\) occurs if and only if \(\beta\) is a root of the polynomial \(x^3-ax^2-bx-1\) with integral \(a \geq 0\) and \(b\) satisfying \(-1 \leq b \leq a+1.\) Such results are connected with fractal tilings of Euclidean space.
For the entire collection see [Zbl 0940.00025].

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11A63 Radix representation; digital problems
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A80 Fractals