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On the distribution of certain Pisot numbers. (English) Zbl 1306.11059

Summary: A Pisot number \(\theta\) is said to be simple if the beta-expansion of its fractional part, in base \(\theta\), is finite. Let \(\mathbb P\) be the set of such numbers, and \(\mathbb S \setminus \mathbb P\) be the complement of \(\mathbb P\) in the set \(\mathbb S\) of Pisot numbers. We show several results about the derived sets of \(\mathbb P\) and of \(\mathbb S \setminus \mathbb P\). A Pisot number \(\theta \), with degree greater than 1, is said to be strong, if it has a proper real positive conjugate which is greater than the modulus of the remaining conjugates of \(\theta \). The set, say \(X\), of such numbers has been defined by D. W. Boyd [Math. Comput. 65, No. 214, 841–860 (1996; Zbl 0855.11039)], and is contained in \(\mathbb S \setminus \mathbb P\). We also prove that the infimum of the \(j\)-th derived set of \(X\), where \(j\) runs through the set of positive rational integers, is at most \(j+2\).

MSC:

11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11J71 Distribution modulo one
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Citations:

Zbl 0855.11039
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Full Text: DOI

References:

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