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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].

##### MSC:
 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
##### Keywords:
cubic Pisot units; minimal polynomial; beta expansions