Nontrivial 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), 47674779
MSC (2000):
Primary 11Y60, 11Y40
Published electronically:
July 24, 2003
MathSciNet review:
1997583
Fulltext PDF Free Access
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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
Furthermore, for all in this class of cubic Pisot numbers. What is surprising about this result is how precise it is, giving exact upper and lower bounds for these approximations.
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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:
http://dx.doi.org/10.1090/S0002994703033336
PII:
S 00029947(03)033336
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
