On optimal simultaneous rational approximation to (omega, omega(2))(tau) with omega being some kind of cubic algebraic function

摘要

It is shown that each rational approximant to (omega, omega(2))(tau) given by the Jacobi-Perron algorithm (JPA) or modified Jacobi-Perron algorithm (MJPA) is optimal, where omega is an algebraic function (a formal Laurent series over a finite field) satisfying omega(3) + k omega - 1 = 0 or omega(3) + kd omega - d = 0. A result similar to the main result of Ito et al. [On simultaneous approximation to (alpha, alpha(2)) with alpha(3) + k alpha - 1 = 0, J. Number Theory 99 (2003) 255-283] is obtained. (c) 2007 Elsevier Inc. All rights reserved.