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Some Computations on Pisot and Salem Numbers

Borwein, Peter and Hare, Kevin G. (2000) Some Computations on Pisot and Salem Numbers. [Preprint]

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      Abstract

      Properties of Pisot numbers have long been of interest.One line of questioning, initiated by Erd{\H{o}}s, Jo{\'o} and Komornik in 1990 \cite{ErdosJooKomornik90} has been the determination of $l(q)$ for Pisot numbers $q$; where $l(q) = \inf(|y|: y = \epsilon_0 + \epsilon_1 q^1 + \cdots + \epsilon_n q^n, \epsilon_i \in \{\pm 1, 0\}, y \neq 0)$.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 is concerning the analogy of $l(q)$, denoted $a(q)$ where the coefficients are restricted to $\pm 1$; in particular for which non-Pisot numbers is $a(q)$ non-zero.This paper finds an infinite class of Salem numbers where $a(q) \neq 0$.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: Pisot numbers, LLL, spectrum, beta numbers
      Subjects: 11-xx Number theory > 11Yxx Computational number theory
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 28 Nov 2003
      Last Modified: 21 Apr 2010 11:13
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/225

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