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

Nadir Murru,Lea Terracini

doi:10.1007/s11139-021-00466-z

Nadir Murru, Lea Terracini

Open Access

https://doi.org/10.1007/s11139-021-00466-z

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Journal: The Ramanujan Journal	Publication Date: Jul 17, 2021
Citations: 1	License type: open-access

Affiliation: University of Trento, University of Turin

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.

Highlights

Continued fractions give a representation for any real number by means of a sequence of integers, providing along the way rational approximations
Multidimensional continued fractions (MCFs) are a generalization of classical continued fractions introduced by Jacobi [22] and Perron [34] in an attempt to answer a question posed by Hermite about a possible generalization of the Lagrange theorem for continued fractions to other algebraic irrationalities
A MCF is a representation of a m-tuple of real numbers (α0(1), . . . , α0(m)) by means of m sequences of integers ((an(1))n≥0, . . . , (an(m))n≥0) obtained by the Jacobi–Perron algorithm:

Summary

Introduction

Continued fractions give a representation for any real number by means of a sequence of integers, providing along the way rational approximations. 4, we focus on algebraically dependent pairs of p-adic numbers; firstly we find a condition on the quality of approximation under which a sequence of simultaneous rational approximations satisfies the same algebraic relation. We apply this result to MCFs and deduce a condition that ensures the finiteness of the p-adic Jacobi–Perron algorithm when it processes some kinds of Q-linearly dependent inputs

Definitions and useful properties

The rate of convergence

Diophantine study

A p-adic Liouville-type theorem on algebraic dependence

Some consequences on linear dependence

A class of fast convergent p-adic MCFs