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

Abstract

Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of $p$--adic numbers ${\mathbb{Q}}_p$. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $\mathbb{R}$ by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in ${\mathbb{Q}}_p$. 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 $p$--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 $\mathbb{Q}$--linearly dependent inputs.