On simultaneous approximation to (α,α2) with α3+kα−1=0 Original Research Article

We show that the modified Jacobi–Perron algorithm gives the best simultaneous approximation to (α,α2) with α3+kα−1=0. We claim the following facts: the limit set of image become an ellipse, where (pn,qn,rn) is the nth convergent (pn/qn,rn/qn) of (α,α2) by the modified Jacobi–Perron algorithm, the limit set of image belongs to outside of the ellipse in (1).