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 Q(p). 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 Q(p). We focus on the dimension two and study the quality of the simultaneous approximation to two p-adic numbers provided by p-adic MCFs, where p is an odd prime. Moreover, given algebraically dependent padic 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.