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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

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
Published electronically: July 24, 2003
MathSciNet review: 1997583
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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 $q$, with minimal polynomial $p$, such that $p(0) = -1$, and where $p$ has only one real root, then there exists a $C(q)$, explicitly given here, such that:

(1)
For all $\epsilon > 0$, all but finitely many integer quadratics $P$ satisfy

\begin{displaymath}\vert P(q)\vert \geq \frac{C(q) - \epsilon}{H(P)^2}\end{displaymath}

where $H$ is the height function.
(2)
For all $\epsilon > 0$ there exists a sequence of integer quadratics $P_n(q)$ such that

\begin{displaymath}\vert P_n(q)\vert \leq \frac{C(q) + \epsilon}{H(P_n)^2}.\end{displaymath}

Furthermore, $C(q) < 1$ for all $q$ 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/S0002-9947-03-03333-6
PII: S 0002-9947(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