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On the successive derived sets of the Pisot numbers


Author: David W. Boyd
Journal: Proc. Amer. Math. Soc. 73 (1979), 154-156
MSC: Primary 12A15
DOI: https://doi.org/10.1090/S0002-9939-1979-0516454-5
MathSciNet review: 516454
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Abstract: Let S denote the set of Pisot (or Pisot-Vijayaraghavan) numbers and let $ {S^{(k)}}$ be the kth derived set of S. It is shown that $ {k^{1/2}} \leqslant \min {S^{(k)}}$ and that $ \lim \sup (\min {S^{(k)}}/k) < 1$. The lower bound improves the estimate $ {k^{1/4}} \leqslant \min {S^{(k)}}$ of Dufresnoy and Pisot, while the upper bound improves the obvious estimate $ \min {S^{(k)}} \leqslant k + 1$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0516454-5
Keywords: Pisot number, derived set
Article copyright: © Copyright 1979 American Mathematical Society

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