Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers |
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Authors: | Peter Borwein Kevin G Hare |
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Institution: | Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 ; Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 |
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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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Keywords: | Pisot numbers continued fraction quadratic approximation |
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