On the distribution of certain Pisot numbers
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Date
2012
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Selsevier
Abstract
A Pisot number θ is said to be simple if the beta-expansion of its fractional part, in base θ, is finite.
Let P be the set of such numbers, and S \ P be the complement of P in the set S of Pisot numbers. We show several results about the derived sets of P and of S \P. A Pisot number θ, 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 θ. The set, say X, of such numbers has been defined by Boyd (1993) [5], and is contained in S \ 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.
⃝⃝c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
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Keywords
Beta-expansion, Pisot numbers, Beta-numbers