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

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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 nα−p n), q n (q nα 2−r n) | n=1,2,…} become an ellipse, where ( p n , q n , r n ) is the nth 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).