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Some computations on the spectra of Pisot and Salem numbers

Authors: Peter Borwein and Kevin G. Hare
Journal: Math. Comp. 71 (2002), 767-780
MSC (2000): Primary 11Y60, 11Y40
Published electronically: November 14, 2001
MathSciNet review: 1885627
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Abstract: Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos, Joó and Komornik in 1990, is the determination of $l(q)$ for Pisot numbers $q$, where

\begin{displaymath}l(q) = \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^1 + \cdots + \epsilon_n q^n, \epsilon_i \in \{\pm 1, 0\}, y \neq 0).\end{displaymath}

Although the quantity $l(q)$ is known for some Pisot numbers $q$, there has been no general method for computing $l(q)$. This paper gives such an algorithm. With this algorithm, some properties of $l(q)$ and its generalizations are investigated.

A related question concerns the analogy of $l(q)$, denoted $a(q)$, where the coefficients are restricted to $\pm 1$; in particular, for which non-Pisot numbers is $a(q)$ nonzero? This paper finds an infinite class of Salem numbers where $a(q) \neq 0$.

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Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Kevin G. Hare
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Keywords: Pisot numbers, LLL, spectrum, beta numbers
Received by editor(s): April 12, 2000
Received by editor(s) in revised form: August 8, 2000
Published electronically: November 14, 2001
Additional Notes: Research of K.G. Hare supported by MITACS and by NSERC of Canada, and P. Borwein supported by MITACS and by NSERC of Canada.
Article copyright: © Copyright 2001 by the Authors

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