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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.
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