Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions

Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of p–adic numbers Qp. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in R by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in Qp. We focus on the dimension two and study the quality of the simultaneous approximation to two p-adic numbers provided by padic MCFs, where p is an odd prime. Moreover, given algebraically dependent p–adic numbers, we see when infinitely many simultaneous approximations satisfy the same algebraic relation. This also allows to give a condition that ensures the finiteness of the p–adic Jacobi–Perron algorithm when it processes some kinds of Q–linearly dependent inputs.