On the finiteness and periodicity of the 𝑝-adic Jacobi–Perron algorithm

Abstract

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron to obtain periodic representations for algebraic irrationals, analogous to the case of simple continued fractions and quadratic irrationals. Continued fractions have been studied in the field ofpp-adic numbersQp\mathbb {Q}_p. MCFs have also been recently introduced inQp\mathbb {Q}_p, including, in particular, app-adic Jacobi–Perron algorithm. In this paper, we address two of the main features of this algorithm, namely its finiteness and periodicity. Regarding the finiteness of thepp-adic Jacobi–Perron algorithm, our results are obtained by exploiting properties of some auxiliary integer sequences. It is known that a finitepp-adic MCF representsQ\mathbb Q-linearly dependent numbers. However, we see that the converse is not always true and we prove that in this case infinitely many partial quotients of the MCF havepp-adic valuations equal to−1-1. Finally, we show that a periodic MCF of dimensionmmconverges to an algebraic irrational of degree less than or equal tom+1m+1; for the casem=2m=2, we are able to give some more detailed results.