Some computations on the spectra of Pisot and Salem numbers

Borwein, Peter; Hare, Kevin

doi:10.1090/S0025-5718-01-01336-9

Some computations on the spectra of Pisot and Salem numbers
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by Peter Borwein and Kevin G. Hare PDF

Math. Comp. 71 (2002), 767-780

Abstract:

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdős, Joó and Komornik in 1990, is 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 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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