Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers

Authors:
Peter Borwein and Kevin G. Hare

Journal:
Trans. Amer. Math. Soc. **355** (2003), 4767-4779

MSC (2000):
Primary 11Y60, 11Y40

DOI:
https://doi.org/10.1090/S0002-9947-03-03333-6

Published electronically:
July 24, 2003

MathSciNet review:
1997583

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number , with minimal polynomial , such that , and where has only one real root, then there exists a , explicitly given here, such that:

- (1)
- For all , all but finitely many integer quadratics satisfy

where is the height function. - (2)
- For all there exists a sequence of integer quadratics such that

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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@cecm.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/S0002-9947-03-03333-6

Keywords:
Pisot numbers,
continued fraction,
quadratic approximation

Received by editor(s):
March 1, 2001

Published electronically:
July 24, 2003

Additional Notes:
The first author was supported by MITACS and by NSERC of Canada

The research of the second author was supported by MITACS and by NSERC of Canada

Article copyright:
© Copyright 2003
copyright retained by the authors