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

Abstract We show that the modified Jacobi–Perron algorithm gives the best simultaneous approximation to ( α , α 2 ) with α 3 + kα −1=0. We claim the following facts: (1) the limit set of {( q n (q n α−p n ), q n (q n α 2 −r n ) | n=1,2,…} become an ellipse, where ( p n , q n , r n ) is the n th convergent ( p n / q n , r n / q n ) of ( α , α 2 ) by the modified Jacobi–Perron algorithm, (2) the limit set of {( q (qα−p), q (qα 2 −r) | q∈ Z ,q>0} belongs to outside of the ellipse in (1).