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

DOI:
https://doi.org/10.1090/S0025-5718-01-01336-9

Published electronically:
November 14, 2001

MathSciNet review:
1885627

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 for Pisot numbers , where

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

A related question concerns the analogy of , denoted , where the coefficients are restricted to ; in particular, for which non-Pisot numbers is nonzero? This paper finds an infinite class of Salem numbers where .

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Additional Information

**Peter Borwein**

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

Email:
pborwein@math.sfu.ca

**Kevin G. Hare**

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

Email:
kghare@cecm.math.sfu.ca

DOI:
https://doi.org/10.1090/S0025-5718-01-01336-9

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