Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers
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- by Peter Borwein and Kevin G. Hare PDF
- Trans. Amer. Math. Soc. 355 (2003), 4767-4779
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:
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For all $\epsilon > 0$, all but finitely many integer quadratics $P$ satisfy \[ |P(q)| \geq \frac {C(q) - \epsilon }{H(P)^2}\] where $H$ is the height function.
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For all $\epsilon > 0$ there exists a sequence of integer quadratics $P_n(q)$ such that \[ |P_n(q)| \leq \frac {C(q) + \epsilon }{H(P_n)^2}.\]
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
- 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 - © Copyright 2003 copyright retained by the authors
- 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
- MathSciNet review: 1997583