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Purely periodic $$\beta$$-expansions with Pisot unit base. (English) Zbl 1099.11062
In [Bull. Lond. Math. Soc. 12, 269–278 (1980; Zbl 0494.10040)] K. Schmidt proved that every $$x \in {\mathbb Q}(\beta) \cap [0,1)$$ has a periodic $$\beta$$-expansion $$x=\sum_{i=1}^{\infty} x_i \beta^{-i},$$ where $$x_i \in \{0,1,\dots,[\beta]\},$$ if $$\beta$$ is a Pisot number. Sometimes, but not always, the periodic $$\beta$$-expansion is purely periodic, that is, there is a $$t \in {\mathbb N}$$ such that $$x_{k+t}=x_k$$ for every $$k \in {\mathbb N}.$$ For example, if $$\beta$$ satisfies the equality $$\beta^2=n\beta+1$$ with $$n \in {\mathbb N},$$ then it is a Pisot unit and every $$x \in {\mathbb Q}(\beta) \cap [0,1)$$ has purely periodic $$\beta$$-expansion. For which Pisot units $$\beta$$ is the $$\beta$$-expansion of the number $$x \in {\mathbb Q}(\beta) \cap [0,1)$$ purely periodic?
Some special cases of this problem have been treated earlier by Y. Hara and S. Ito (quadratic Pisot units) [ Tokyo J. Math. 12, No. 2, 357–370 (1989; Zbl 0695.10030)] and by Y. Sano and S. Ito (Pisot units whose minimal polynomials have some special form) [Osaka J. Math. 38, No. 2, 349–368 (2001; Zbl 0991.11040)].
In this paper, the authors prove a theorem which characterizes all Pisot units $$\beta$$ and all real numbers $$x \in {\mathbb Q}(\beta) \cap [0,1)$$ such that $$x$$ has a purely periodic $$\beta$$-expansion. As an example, they show that the $$\beta$$-expansion of a rational number $$x \in [0,1),$$ where $$\beta^2=n\beta-1$$ and $$n \geq 3,$$ is not purely periodic.

##### MSC:
 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
##### Keywords:
PV-numbers; units; $$\beta$$-expansion; atomic surface; tiling
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##### References:
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