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

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 \[ |P(q)| \geq \frac {C(q) - \epsilon }{H(P)^2}\] where $H$ is the height function.

2. 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.