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A certain finiteness property of Pisot number systems. (English) Zbl 1052.11055
Let \(\beta\) be a real number \(>1\), and let \(\text{Fin}(\beta)\) be the set of nonnegative real numbers having a finite \(\beta\)-expansion. The number \(\beta\) is said to have property (F) if \(\text{Fin}(\beta)=\mathbb Z[1/\beta]\cap \{x\geq 0\}\). This property was introduced by C. Frougny and B. Solomyak [Ergodic Theory Dyn. Syst. 12, No. 4, 713–723 (1992; Zbl 0814.68065)] who proved in particular that real numbers \(>1\) with property (F) must be Pisot numbers, (the reciprocal is not true). This property is also related to tilings of \(\mathbb R^k\).
A weaker property called (W) was introduced by Hollander in his PhD thesis (1996). It reads:
(W) \(\forall x\in\mathbb Z[1/\beta]\cap\{x\geq 0\}\) \(\forall\varepsilon > 0\), \(\exists y,z\in\text{Fin}(\beta)\) such that \(x=y-z\) and \(z<\varepsilon\).
This property was studied in relation with tilings (instead of property (F)) by the first-named author of this paper [J. Math. Soc. Japan 54, No. 2, 283–308 (2002; Zbl 1032.11033)] and in relation with ergodic results by N. A. Sidorov [J. Dyn. Control Syst. 7, No. 4, 447–472 (2002; Zbl 1134.37313), and also Acta Math. Hung. 101, 345–355 (2003; Zbl 1059.28014)]. Sidorov conjectured in particular that only Pisot numbers can have property (W). In the paper under review the authors study property (W) in more detail. We mention in particular their result that numbers with property (W) must be Salem or Pisot, and their algorithmic construction to show (W) or (F) for a given \(\beta\).

MSC:
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11A63 Radix representation; digital problems
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37B10 Symbolic dynamics
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References:
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