Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers
Peter Borwein, Kevin G. Hare · Transactions of the American Mathematical Society · 2003
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 q , with minimal polynomial p p , such that p ( 0 ) = − 1 p(0) = -1 , and where p p has only one real root, then there exists a C ( q ) C(q) , explicitly given here, such that: For all ϵ > 0 \epsilon > 0 , all but finitely many integer quadratics P P satisfy \[ | P ( q ) | ≥ C ( q ) − ϵ H ( P ) 2 |P(q)| \geq \frac {C(q) - \epsilon }{H(P)^2} \] where H H is the height function. For all ϵ > 0 \epsilon > 0 there exists a sequence of integer quadratics P n ( q ) P_n(q) such that \[