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

2019/06/23 by Murru, Nadir, Terracini, Lea
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1906.09570

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.

Related

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• Continued fractions in the field of p-adic numbers