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 of $p$-adic numbers $\mathbb {Q}_p$. MCFs have also been recently introduced in $\mathbb {Q}_p$, including, in particular, a $p$-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 the $p$-adic Jacobi–Perron algorithm, our results are obtained by exploiting properties of some auxiliary integer sequences. It is known that a finite $p$-adic MCF represents $\mathbb Q$-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 have $p$-adic valuations equal to $-1$. Finally, we show that a periodic MCF of dimension $m$ converges to an algebraic irrational of degree less than or equal to $m+1$; for the case $m=2$, we are able to give some more detailed results.