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. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced over the real field by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in the p-adic setting. 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 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.