Non-trivial Quadratic Approximations to Zero of a Family of Cubic Pisot Numbers

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 > 0, all but finitely many integer quadratics P satisfy |P (q)| ≥ C(q)− H(P )2 where H is the height function. (2) For all > 0 there exists a sequence of integer quadratics Pn(q) such that |Pn(q)| ≤ C(q) + H(Pn) . 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.