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

Authors:	Peter Borwein Kevin G Hare

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

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 $\epsilon > 0$ , all but finitely many integer quadratics satisfy $\begin{displaymath}\vert P(q)\vert \geq \frac{C(q) - \epsilon}{H(P)^2}\end{displaymath}$ where is the height function. (2) For all $\epsilon > 0$ there exists a sequence of integer quadratics such that $\begin{displaymath}\vert P_n(q)\vert \leq \frac{C(q) + \epsilon}{H(P_n)^2}.\end{displaymath}$ 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.

Keywords:	Pisot numbers continued fraction quadratic approximation

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