1 Introduction

Continued fractions give a representation for any real number by means of a sequence of integers, providing along the way rational approximations. In particular, they provide best approximations, i.e., the nth convergent of the continued fraction of a real number is closer to it than any other rational number with a smaller or equal denominator. 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 \((\alpha _0^{(1)}, \ldots , \alpha _0^{(m)})\) by means of m sequences of integers \(((a_n^{(1)})_{n\ge 0}, \ldots , (a_n^{(m)})_{n\ge 0})\) (finite or infinite) obtained by the Jacobi–Perron algorithm:

$$\begin{aligned} {\left\{ \begin{array}{ll} a_n^{(i)} = [\alpha _n^{(i)}], \qquad \qquad \qquad i = 1, ..., m, \\ \alpha _{n+1}^{(1)} = \frac{1}{\alpha _n^{(m)} - a_n^{(m)}}, \quad \quad n = 0, 1, 2, ...,\\ \alpha _{n+1}^{(i)} = \frac{\alpha _n^{(i-1)} - a_n^{(i-1)}}{\alpha _n^{(m)} - a_n^{(m)}}, \,\,\,i = 2, ..., m, \end{array}\right. } \end{aligned}$$
(1)

We shall write

$$\begin{aligned} (\alpha _0^{(1)}, \ldots , \alpha _0^{(m)}) = [(a_0^{(1)}, a_1^{(1)}, \ldots ), \ldots , (a_0^{(m)}, a_1^{(m)}, \ldots )]. \end{aligned}$$

The Jacobi–Perron algorithm has been widely studied concerning its periodicity and approximation properties. For instance, in [8, 9, 26, 35] the authors provided some classes of algebraic irrationalities whose expansion by the Jacobi–Perron algorithm becomes eventually periodic. In [30], a criterion of periodicity, involving linear recurrence sequences, is given. The periodicity of the Jacobi–Perron algorithm is also related to the study of Pisot numbers [19, 20]. Further studies on MCFs can be found in [2, 16, 29, 41].

Continued fractions for p-adic numbers were introduced in three different ways [10, 36, 37] and subsequently studied by several authors like [7, 11, 13, 21]. More recently they have been generalized to higher dimensions. In [31], the authors studied the fundamental properties of MCFs in \(\mathbb Q_p\), focusing on convergence properties and finite expansions, whereas in [32] further properties regarding finiteness and periodicity of the p-adic Jacobi–Perron algorithm have been proved.

The study of simultaneous approximations of real numbers is a very important topic in Diophantine approximation; classical and fundamental results can be found in [3, 15, 17, 18, 23]. Also in the p-adic setting simultaneous approximation has been investigated, e.g., in [1, 27, 39]. Some results can also be found regarding simultaneous approximations in \(\mathbb Q_p\), involving a p-adic number and its integral powers [12, 28]. Specific results regarding the case of a p-adic number and its square are investigated in [6, 40]. However, there are no applications of MCFs for providing simultaneous approximations of p-adic numbers and for studying the quality of such approximations.

MCFs have been deeply studied in this context for the real case, since they provide simultaneous rational approximations to real numbers. The quality of these simultaneous approximations has been studied in several works, such as [5, 14, 24, 33, 38], thus, it seems natural to exploit MCFs in \(\mathbb Q_p\) for approaching the problem of constructing simultaneous approximations to p-adic numbers. In this paper, we give a first study in this direction and we also investigate the relation between simultaneous approximations and algebraic dependence.

The paper is structured as follows. In Sect. 2, we introduce the notation and we give some basic definitions and properties. Section 3 is devoted to the study of the quality of the simultaneous approximations provided by p-adic MCFs. Finally, in Sect. 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. Secondly, 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 \(\mathbb Q\)-linearly dependent inputs.

2 Definitions and useful properties

In the following, we will focus on p-adic MCFs of dimension 2, i.e., using the notation of the previous section, we set \(m = 2\). Most of the results obtained in this paper can be adapted to any dimension \(m \ge 2\), but in the general case the notation is very annoying and possibly confusing. Hence, we now recall the p-adic Jacobi–Perron algorithm for the case \(m=2\); for more details see [31]. From now on, p will be an odd prime number.

Definition 1

The Browkin s-function \(s:\mathbb {Q}_p\longrightarrow \mathcal {Y} =\mathbb {Z}\left[ \frac{1}{p}\right] \cap \left( -\frac{p}{2},\frac{p}{2}\right) \) is defined by

$$\begin{aligned} s(\alpha )= \sum _{j=k}^0 x_jp^j, \end{aligned}$$

with \(\alpha \in \mathbb {Q}_p\) written as \(\alpha =\sum _{j=k}^\infty x_jp^j, k \in \mathbb {Z}\hbox {, and } x_j\in \mathbb {Z}\cap \left( -\frac{p}{2},\frac{p}{2}\right) .\)

Given \(\alpha , \beta \in \mathbb Q_p\), we get the corresponding MCF \((\alpha , \beta ) = [(a_0, a_1, \ldots ), (b_0, b_1, \ldots )]\) by the following iterative equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} a_k = s(\alpha _k), \\ b_k = s(\beta _k), \\ \alpha _{k+1} = \frac{1}{\beta _{k} - b_{k}}, \\ \beta _{k+1} = \frac{\alpha _k - a_k}{\beta _k - b_k}, \end{array}\right. } \end{aligned}$$

for \(k = 0, 1, \ldots \), with \(\alpha _0 = \alpha \) and \(\beta _0 = \beta \). Thus, the complete quotients satisfy the following relations:

$$\begin{aligned} \alpha _k = a_k + \frac{\beta _{k+1}}{\alpha _{k+1}}, \quad \beta _k = b_k + \frac{1}{\alpha _{k+1}} \end{aligned}$$

and, if the algorithm does not stop, then the initial values are represented by the following MCF:

$$\begin{aligned} \alpha =a_0+\frac{b_1+\frac{1}{a_2+\frac{b_3+\frac{1}{\ddots }}{a_3+\frac{\ddots }{\ddots }}}}{a_1+\frac{b_2+\frac{1}{a_3+\frac{\ddots }{\ddots }}}{a_2+\frac{b_3+\frac{1}{\ddots }}{a_3+\frac{\ddots }{\ddots }}}} \quad \text {and} \quad \beta =b_0+\frac{1}{a_1+\frac{b_2+\frac{1}{a_3+\frac{\ddots }{\ddots }}}{a_2+\frac{b_3+\frac{1}{\ddots }}{a_3+\frac{\ddots }{\ddots }}}}.\end{aligned}$$

We define the sequences of integers \((A_k)_{k \ge -2}\), \((B_k)_{k \ge -2}\), \((C_k)_{k \ge -2}\) of the numerators and denominators of the convergents, i.e.,

$$\begin{aligned}{}[(a_0, \ldots , a_n), (b_0, \ldots , b_n)] = \left( \frac{A_n}{C_n}, \frac{B_n}{C_n} \right) = (Q_n^{\alpha }, Q_n^\beta ) \end{aligned}$$
(2)

as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} A_{-2} = 0, \quad A_{-1} = 1, \quad A_0 = a_0, \\ B_{-2} = 1, \quad B_{-1} = 0, \quad B_0 = b_0, \\ C_{-2} = 0, \quad C_{-1} = 0, \quad C_0 = 1, \end{array}\right. } \quad {\left\{ \begin{array}{ll} A_n = a_n A_{n-1} + b_n A_{n-2} + A_{n-3}, \\ B_n = a_n B_{n-1} + b_n B_{n-2} + B_{n-3}, \\ C_n = a_n C_{n-1} + b_n C_{n-2} + C_{n-3}, \end{array}\right. }\end{aligned}$$
(3)

for any \(n \ge 1\). Then

$$\begin{aligned} \prod _{k=0}^n \begin{pmatrix} a_k &{} 1 &{} 0 \\ b_k &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \end{pmatrix} = \begin{pmatrix} A_n &{} A_{n-1} &{} A_{n-2} \\ B_n &{} B_{n-1} &{} B_{n-2} \\ C_n &{} C_{n-1} &{} C_{n-2} \end{pmatrix} \end{aligned}$$
(4)

for any \(n \ge 0\).

We define the sequences \((\tilde{A}_k)_{k \ge -1} = (A_k C_{k-1} - A_{k-1}C_k)\) and \((\tilde{B}_k)_{k \ge -1} = (B_k C_{k-1} - B_{k-1}C_k)\), arising from the difference between two consecutive convergents:

$$\begin{aligned} Q_n^\alpha - Q_{n-1}^\alpha = \frac{\tilde{A}_n}{C_n C_{n-1}}, \quad Q_n^\beta - Q_{n-1}^\beta = \frac{\tilde{B}_n}{C_n C_{n-1}}. \end{aligned}$$

The following relations hold true:

$$\begin{aligned} {\left\{ \begin{array}{ll} \tilde{A}_{-1} = 0, \quad \tilde{A}_{0} = -1, \quad \tilde{A}_1 = b_1, \\ \tilde{B}_{-1} = 0, \quad \tilde{B}_{0} = 0, \quad \tilde{B}_1 = 1, \end{array}\right. } \quad {\left\{ \begin{array}{ll} \tilde{A}_n = -b_n \tilde{A}_{n-1} - a_{n-1} \tilde{A}_{n-2} + \tilde{A}_{n-3}, \\ \tilde{B}_n = -b_n \tilde{B}_{n-1} - a_{n-1} \tilde{B}_{n-2} + \tilde{B}_{n-3}, \end{array}\right. } \end{aligned}$$
(5)

for any \(n \ge 2\). Indeed,

$$\begin{aligned} \tilde{A}_n= & {} A_n C_{n-1} - A_{n-1} C_n\\= & {} (a_n A_{n-1} +b_n A_{n-2} + A_{n-3}) C_{n-1} - A_{n-1}(a_n C_{n-1} + b_n C_{n-2} + C_{n-3})\\= & {} -b_n (A_{n-1} C_{n-2} - A_{n-2} C_{n-1}) + A_{n-3} (a_{n-1}C_{n-2}+b_{n-1}C_{n-3}+C_{n-4})\\&-C_{n-3}(a_{n-1}A_{n-2}+b_{n-1}A_{n-3}+A_{n-4})\\= & {} -b_n \tilde{A}_{n-1} - a_{n-1} \tilde{A}_{n-2} + \tilde{A}_{n-3}; \end{aligned}$$

similarly for the \(\tilde{B}_k\)s. From (4), we have

$$\begin{aligned} 1= & {} \det \begin{pmatrix} A_n &{} A_{n-1} &{} A_{n-2} \\ B_n &{} B_{n-1} &{} B_{n-2} \\ C_n &{} C_{n-1} &{} C_{n-2} \end{pmatrix}\\= & {} A_n B_{n-1} C_{n-2} - A_{n-1}B_nC_{n-2} - A_nB_{n-2}C_{n-1} + A_{n-2}B_nC_{n-1}\\&+ A_{n-1}B_{n-2}C_n - A_{n-2}B_{n-1} C_n, \end{aligned}$$

from which

$$\begin{aligned} \frac{1}{C_nC_{n-1}C_{n-2}} = (Q_n^\alpha - Q_{n-1}^\alpha )(Q_n^\beta - Q_{n-2}^\beta ) - (Q_n^\beta - Q_{n-1}^\beta )(Q_n^\alpha - Q_{n-2}^\alpha ). \end{aligned}$$

Moreover, since

$$\begin{aligned} \alpha = \frac{\alpha _n A_{n-1} + \beta _n A_{n-2} + A_{n-3}}{\alpha _n C_{n-1} + \beta _n C_{n-2} + C_{n-3}}\quad \mathrm{{and}} \quad \beta = \frac{\alpha _n B_{n-1} + \beta _n B_{n-2} + B_{n-3}}{\alpha _n C_{n-1} + \beta _n C_{n-2} + C_{n-3}}, \end{aligned}$$

we have

$$\begin{aligned}&(\alpha - Q_{n-1}^\alpha )(\beta - Q_{n-2}^\beta ) - (\beta - Q_{n-1}^\beta )(\alpha - Q_{n-2}^\alpha )\nonumber \\&\quad = \frac{1}{C_{n-1}C_{n-2}(\alpha _nC_{n-1}+\beta _nC_{n-2}+C_{n-3})}. \end{aligned}$$
(6)

Let us observe that the previous properties hold for general MCFs, while we now give some specific results regarding only p-adic MCFs. In the following we will use \(\nu _p(\cdot )\) for the p-adic valuation, \(|\cdot |_p\) for the p-adic norm and \(|\cdot |_\infty \) for the Euclidean norm. Moreover, we define

$$\begin{aligned} h_n = \nu _p\left( \frac{b_n}{a_n} \right) , \quad k_n = \nu _p\left( \frac{1}{a_n} \right) , \quad K_n = k_1 + \cdots + k_n \end{aligned}$$

for any \(n \ge 1\) and the sequences \((V_k^{\alpha })_{k \ge -2} = (C_k \alpha - A_k)_{k \ge -2}\), \((V_k^\beta )_{k \ge -2} = (C_k \beta - B_k)_{k \ge -2}\). We recall from [31] the following properties:

  • \(|a_n|_p>1\) and \(|b_n|_p<|a_n|_p \), for any \(n \ge 1\);

  • \(|a_n|_p = |\alpha _n|_p\), \(|b_n|_p = {\left\{ \begin{array}{ll} |\beta _n|_p \ \text { if } |\beta _n|_p \ge 1 \\ 0 \ \text { if } \ |\beta _n|_p <1 \end{array}\right. }\),   for any \(n \ge 1\);

  • \(\nu _p(C_n) = -K_n\), for any \(n \ge 1\);

  • \(\lim _{n \rightarrow +\infty } |V_n^{\alpha }|_p = \lim _{n \rightarrow +\infty } |V_n^{\beta }|_p = 0\).

3 The quality of the approximations of p-adic MCFs

In this section, we investigate how well the convergents of a bidimensional continued fraction approach their limit in \(\mathbb Q_p\).

3.1 The rate of convergence

In the first instance, we give some results about the rate of convergence of the real sequences \(|V_n^\alpha |_p\) and \(|V_n^\beta |_p\).

Theorem 1

Let \([(a_0, a_1, \ldots ), (b_0, b_1, \ldots )]\) be the p-adic MCF expansion of \((\alpha , \beta ) \in \mathbb Q_p^2\), then

$$\begin{aligned} \nu _p(\alpha - Q_n^{\alpha }) \ge K_n + \left\lfloor \frac{n+2}{2} \right\rfloor , \quad \nu _p(\beta - Q_n^{\beta }) \ge K_n + \left\lfloor \frac{n+3}{2} \right\rfloor , \end{aligned}$$

for \(n \ge -2\)

Proof

We will prove by induction that

$$\begin{aligned} \nu _p\left( Q_{n+1}^\alpha - Q_n^\alpha \right) = \nu _p\left( \frac{\tilde{A}_{n+1}}{C_{n+1}C_n} \right) \ge -\nu _p(C_n) + \left\lfloor \frac{n+2}{2} \right\rfloor , \end{aligned}$$

i.e., we have to prove that

$$\begin{aligned} \nu _p(\tilde{A}_{n+1}) \ge \nu _p(C_{n+1}) + \left\lfloor \frac{n+2}{2} \right\rfloor \end{aligned}$$

for any \(n \ge -2\). We can observe that

$$\begin{aligned} \nu _p(\tilde{A}_{-1}) = \nu _p(C_{-1}) = \infty , \end{aligned}$$

for \(n=-2\), and

$$\begin{aligned} \nu _p(\tilde{A}_0) = \nu _p(\tilde{C}_0) = 0, \end{aligned}$$

for \(n=-1\). Moreover,

$$\begin{aligned} \nu _p(\tilde{A}_1) = \nu _p(b_1), \quad \nu _p(C_1) = \nu _p(a_1), \end{aligned}$$

for \(n=0\), and we know that \(\nu _p(b_1) > \nu _p(a_1)\). Now, we proceed by induction, supposing that it is true \(\nu _p(\tilde{A}_n) \ge \nu _p(C_n) + \lfloor \frac{n+1}{2} \rfloor \). Consider

$$\begin{aligned} \nu _p(\tilde{A}_{n+1})&= \nu _p(-b_{n+1} \tilde{A}_n - a_n \tilde{A}_{n-1} + \tilde{A}_{n-2})\\&\ge \inf \{ \nu _p(b_{n+1}\tilde{A}_n), \nu _p(a_n \tilde{A}_{n-1}), \nu _p(\tilde{A}_{n-2}) \}, \end{aligned}$$

by the inductive hypothesis we have

$$\begin{aligned} \nu _p(b_{n+1}\tilde{A}_n)&\ge \nu _p(b_{n+1}) + \nu _p(C_n)+ \left\lfloor \frac{n+1}{2} \right\rfloor \\&\ge \nu _p(a_{n+1}) + 1 + \nu _p(C_n) + \left\lfloor \frac{n+1}{2} \right\rfloor \ge \nu _p(C_{n+1}) + \left\lfloor \frac{n+2}{2} \right\rfloor . \end{aligned}$$

Similarly,

$$\begin{aligned} \nu _p(a_n \tilde{A}_{n-1})&\ge \nu _p(a_n) + \nu _p(C_{n-1}) + \left\lfloor \frac{n}{2} \right\rfloor \ge \nu _p(C_{n+1}) + \left\lfloor \frac{n}{2} \right\rfloor + 1\\&= \nu _p(C_{n+1}) + \left\lfloor \frac{n+2}{2} \right\rfloor \end{aligned}$$

and

$$\begin{aligned} \nu _p(\tilde{A}_{n-2})&\ge \nu _p(C_{n-2}) + \left\lfloor \frac{n-1}{2} \right\rfloor \ge \nu _p(C_{n+1}) + \left\lfloor \frac{n-1}{2} \right\rfloor + 3\\&\ge \nu _p(C_{n+1}) + \left\lfloor \frac{n+2}{2} \right\rfloor . \end{aligned}$$

Thus, we also have \(\nu _p(Q_{n+k}^\alpha - Q_n^\alpha ) \ge -\nu _p(C_n) + \left\lfloor \frac{n+2}{2} \right\rfloor \) and for \(k \rightarrow \infty \), we have \(\nu _p(\alpha - Q_n^\alpha ) \ge -\nu _p(C_n) + \lfloor \frac{n+2}{2} \rfloor \).

Similar arguments hold for proving \(\nu _p(Q_{n+1}^\beta - Q_n^\beta ) \ge -\nu _p(C_n) + \lfloor \frac{n+3}{2} \rfloor \), i.e., for proving \(\nu _p(\tilde{B}_{n+1}) \ge \nu _p(C_{n+1}) +\left\lfloor \frac{n+3}{2} \right\rfloor \). We just check the basis of the induction:

$$\begin{aligned} \nu _p(\tilde{B}_{-1}) = \infty , \quad \nu _p(\tilde{B}_0) = \infty , \quad \nu _p(\tilde{B}_1) = 0 \end{aligned}$$

and

$$\begin{aligned} \nu _p(C_{-1}) = \infty , \quad \nu _p(C_0) = 0, \quad \nu _p(C_1) = \nu _p(a_1) < 0. \end{aligned}$$

\(\square \)

Corollary 1

Let \([(a_0, a_1, \ldots ), (b_0, b_1, \ldots )]\) be the p-adic MCF expansion of \((\alpha , \beta ) \in \mathbb Q_p^2\), then

$$\begin{aligned} \nu _p(V_n^\alpha ) \ge \left\lfloor \frac{n+2}{2} \right\rfloor , \quad \nu _p(V_n^\beta ) \ge \left\lfloor \frac{n+3}{2} \right\rfloor , \end{aligned}$$

so that

$$\begin{aligned} \min \{\nu _p(V_n^\alpha ), \nu _p(V_n^\beta )\}\ge \left\lfloor \frac{n+2}{2} \rfloor =\lfloor \frac{n}{2} \right\rfloor +1. \end{aligned}$$

Remark 1

In the real case, given \((\alpha , \beta ) = [(a_0, a_1, \ldots ), (b_0, b_1, \ldots )]\) it is well known that

$$\begin{aligned} \left| \alpha - \frac{A_n}{C_n}\right| _\infty< \frac{1}{|C_n|_\infty }, \quad \left| \beta - \frac{B_n}{C_n}\right| _\infty < \frac{1}{|C_n|_\infty }. \end{aligned}$$

In the p-adic case, a stronger result holds, indeed from the previous theorem we have

$$\begin{aligned} \left| \alpha - \frac{A_n}{C_n}\right| _p< \frac{1}{k|C_n|_p}, \quad \left| \beta - \frac{B_n}{C_n}\right| _p < \frac{1}{k|C_n|_p} \end{aligned}$$

where \(k = k(n)\) tends to infinity.

On the other hand formula (6) implies

$$\begin{aligned} \nu _p(V_{n-1}^\alpha V_{n-2}^\beta -V_{n-1}^\beta V_{n-2}^\alpha )=K_n, \end{aligned}$$

which provides an upper bound for the p-adic valuation of the \(V_n\)’s, namely

$$\begin{aligned} \min \{\nu _p(V_n^\alpha ), \nu _p(V_n^\beta )\} + \min \{\nu _p(V_{n-1}^\alpha ), \nu _p(V_{n-1}^\beta )\}\le K_{n+1}. \end{aligned}$$
(7)

This shows that the lower bound for \(\min \{\nu _p(V_n^\alpha ), \nu _p(V_n^\beta )\} \) provided by Corollary 1 is optimal, in the sense that it is reached in some cases:

Example 1

Consider an infinite MCF such that \(\nu _p(a_n)=-1\) for every \(n\ge 1\). Then \(K_n=n\) for every n, so that by Corollary 1 and formula (7) we get

$$\begin{aligned} \min \{\nu _p(V_n^\alpha ), \nu _p(V_n^\beta )\} + \min \{\nu _p(V_{n-1}^\alpha ), \nu _p(V_{n-1}^\beta )\}=n+1 \end{aligned}$$

so that \(\min \{\nu _p(V_n^\alpha ), \nu _p(V_n^\beta )\}= \left\lfloor \frac{n}{2} \right\rfloor +1\) for every \(n\ge 1\).

However, in many other cases the bound provided by Corollary 1 can be improved, as stated by the following propositions:

Proposition 1

Let \((\ell _n)_{n\ge 0}\) be a sequence of natural numbers \(>0\); put \(\ell _{-1}=\ell _{-2}=0\) and define \(f(n)=\sum _{j=0}^n \ell _n\). Let \([(a_0, a_1, \ldots ), (b_0, b_1, \ldots )]\) be an infinite p-adic MCF satisfying \(h_{n+1}\ge \ell _{n}, k_{n+1}\ge \ell _n+\ell _{n-1}\) for \(n\ge 0\). Then for every \(n\in \mathbb {N}\)

$$\begin{aligned} \min \{\nu _p(V_n^\alpha ), \nu _p(V_n^\beta )\}\ge f(n). \end{aligned}$$

Proof

For \(n\ge 1\), either \(\nu _p(\beta _n)>0\), \(b_n=0\), \(\nu _p\left( \frac{\beta _{n}}{\alpha _{n}}\right) >k_{n}\) or \(\nu _p(\beta _n)=\nu _p(b_n)\le 0\), \(\nu _p\left( \frac{\beta _{n}}{\alpha _{n}}\right) =\nu _p\left( \frac{b_{n}}{a_{n}}\right) =h_n\). In any case \(\nu _p\left( \frac{\beta _{n+1}}{\alpha _{n+1}}\right) \ge \ell _{n}\), for \(n\ge 0\). Let \(V_n\) be either \(V_n^\alpha \) or \(V_n^\beta \). From the formula

$$\begin{aligned} V_n=-\frac{\beta _{n+1}}{\alpha _{n+1}} V_{n-1}-\frac{1}{\alpha _{n+1}} V_{n-2}, \end{aligned}$$
(8)

we get for \(n\ge 0\)

$$\begin{aligned} \frac{V_n}{p^{f(n)}}=\mu _n\frac{V_{n-1}}{p^{f(n-1)}}+\nu _n\frac{V_{n-2}}{p^{f(n-2)}} \end{aligned}$$

where

$$\begin{aligned} \mu _n =-\frac{\beta _{n+1}}{\alpha _{n+1}}\cdot \frac{1}{p^{\ell _n}},\quad \nu _n = -\frac{1}{\alpha _{n+1}}\cdot \frac{1}{p^{\ell _n+\ell _{n-1}}} \in \mathbb {Z}_p. \end{aligned}$$

Since \(V_{-1}, V_{-2}\in \mathbb {Z}_p\) we obtain by induction \(\frac{V_n}{p^{f(n)}}\in \mathbb {Z}_p\). \(\square \)

Corollary 2

Let \(f:\mathbb {N}\rightarrow \mathbb {N}\) be any function. There are infinitely many \((\alpha ,\beta )\in \mathbb {Q}_p^2\) satisfying

$$\begin{aligned} \min \{\nu _p(V_n^\alpha ), \nu _p(V_n^\beta )\}\ge f(n). \end{aligned}$$

Proof

Of course we can assume f(n) strictly increasing, so that \(f(n)=\sum _{j=0}^n \ell _n\) with \(\ell _n\in \mathbb {N}, \ell _n>0\); the proof follows from Proposition 1 by observing that there are infinitely many p-adic MCF satisfying \(h_{n+1}\ge \ell _{n}, k_{n+1}\ge \ell _n+\ell _{n-1}\) for \(n\ge 0\). \(\square \)

We would like to investigate in which sense and to which extent the approximations given by p-adic convergents may be considered “good approximations.”Observe that the Browking s-function is locally constant, hence so is the function \(\mathbb {Q}_p^2\rightarrow \mathbb {Q}^2\) associating to a pair \((\alpha ,\beta )\) of its nth convergents \((Q_n^\alpha ,Q_n^\beta )\) (where this function is defined). Therefore every \((\alpha ,\beta )\in \mathbb {Q}_p^2\) having a MCF of length \(\ge n\) has a neighborhood U such that every \((\alpha ',\beta ')\in U\) has the same k-convergents as \((\alpha ,\beta )\) for \(k\le n\). The following proposition will provide an explicit radius for this neighborhood.

Proposition 2

Let \((\alpha , \beta )\in \mathbb {Q}_p^2\) be such that the associated MCF \([(a_0, a_1, \ldots ), (b_0, b_1, \ldots )]\) has length \(\ge n\). Let \((\alpha ', \beta ')\in \mathbb {Q}_p^2\). If \(\max \{|\alpha - \alpha '|_p, |\beta - \beta '|_p\} < \frac{1}{p^{2K_n}}\), then the MCF \([(a_0', a_1', \ldots ), (b_0', b_1', \ldots )]\) associated to \((\alpha ', \beta ')\) has length \(\ge n\) and \(a_i = a_i'\), \(b_i = b_i'\), for \(i = 0, \ldots , n\).

Proof

Notice that \( \frac{1}{p^{2K_n}}= \frac{1}{|C_{n}|^2_p}\). We prove the thesis by induction on n. The claim is certainly true for \(n=0\), since in general

$$\begin{aligned} |x-y|_p < 1 \Leftrightarrow s(x) = s(y). \end{aligned}$$

Suppose now \(n\ge 1\), and \(\max \{|\alpha - \alpha '|_p, |\beta - \beta '|_p\} < \frac{1}{|C_{n+1}|^2_p}\). By the case \(n=0\) we have \(a'_0=a_0\), \(b'_0=b_0\). Moreover, we observe that our hypothesis implies \(|\beta -\beta '|<\frac{1}{|a_1|_p}=|\beta -b_0|_p\). By the properties of the non-Archimedean norm, we have

$$\begin{aligned} \frac{1}{|a'_1|_p} =|\beta '-b_0|_p=\max \{|\beta '-\beta |_p,|\beta -b_0|_p\}=|\beta -b_0|_p=\frac{1}{|a_1|_p}, \end{aligned}$$

so that \(|a_1|_p=|a'_1|_p\). We have

$$\begin{aligned} |\alpha _1-\alpha '_1| = \left| \frac{1}{\beta -b_0} -\frac{1}{\beta '-b_0}\right| _p =|a_1|^2_p |\beta -\beta '|_p <\prod _{j=2}^{n+1} \frac{1}{|a_j|_p^2} = \frac{1}{|C_n^{(1)}|^2_p}, \end{aligned}$$
(9)

where \(C_n^{(1)}\) is the nth denominator of the convergents of the MCF expansion of \((\alpha _1, \beta _1)\). Moreover,

$$\begin{aligned} |\beta _1-\beta '_1|_p&= \left| \alpha _1 (\alpha -a_0) -\alpha '_1(\alpha '-a_0)\right| _p = |(\alpha -a_0)(\alpha _1-\alpha '_1)+\alpha '_1(\alpha -\alpha ')|_p\nonumber \\&\le \max \{|(\alpha -a_0)(\alpha _1-\alpha '_1)|,|a_1|_p|(\alpha -\alpha ')|_p \} < \frac{1}{|C_n^{(1)}|^2_p}. \end{aligned}$$
(10)

Thus, by inductive hypothesis we have \(a_i=a'_i, b_i=b'_i\) for \(i=1,\ldots , n+1\). \(\square \)

Unfortunately, in general the pair \((Q_n^\alpha ,Q_n^\beta )\) does not lie in the p-adic ball centered in \((\alpha ,\beta ) \) and having radius \(\frac{1}{p^{2K_n}}\), as Example 1 shows. The next proposition gives a constructive sufficient condition ensuring this property.

Proposition 3

Consider an infinite MCF such that \(k_{n+1}> k_n+k_{n-1}\) and \(h_n> k_{n-1}\) for \(n\ge 2\). Then for every \(n\in \mathbb {N}\), \(\max \{|\alpha - Q_n^\alpha |_p, |\beta - Q_n^\beta |_p\} < \frac{1}{p^{2K_n}}\).

Proof

It is a consequence of Proposition 1 . \(\square \)

3.2 Diophantine study

In this section, we want to relate the rate of approximation of the convergents of a p-adic MCF to the Euclidean size of its numerators and denominators. First, we give a bound on this size.

Lemma 1

Let \((a_n)_{n\in \mathbb {N}} \) be a sequence of real numbers, such that there exists \(m\in \mathbb {N}\), \(c_0,\ldots , c_m\) positive real numbers such that \(c_m>0\) and

$$\begin{aligned} |a_{n+m+1}|_\infty < c_{m}|a_{n+m}|_\infty +c_{m-1}|a_{n+m-1}|_\infty +\cdots + c_{0}|a_n|_\infty . \end{aligned}$$

Let \(\tilde{x}\) be the (unique, by the Cartesian rule of signs) positive real root of the polynomial

$$\begin{aligned} f(X)=X^{m+1}-c_mX^m-\cdots - c_1X-c_0 \end{aligned}$$
(11)

and let \(M\ge \max \{|a_0|_\infty , \frac{|a_1|_\infty }{\tilde{x}},\ldots ,\frac{|a_m|_\infty }{\tilde{x}^m}\}.\) Then \( |a_n|_\infty \le M\tilde{x}^n\) for every \(n\in \mathbb {N}\).

Proof

The proof is straightforward by induction on n. \(\square \)

Notice that \(f(0)=-c_0< 0\), so that \(\tilde{x}>0\), more precisely

$$\begin{aligned} \tilde{x} =c_m +\frac{c_{m-1}}{\tilde{x}} + \frac{c_{m-2}}{\tilde{x}^2}+\cdots +\frac{c_{0}}{\tilde{x}^m}, \end{aligned}$$

which implies \(c_m<\tilde{x}\). Put \(C=\sum _{i=0}^m |c_i|_\infty \), if \(C<1\), then \(f(1)=1-C>0\), so that \(0<\tilde{x} <1\), and we can conclude that \(c_m<\tilde{x}<1.\) In the following, \(\tilde{x}_p\) will be the real root of the polynomial

$$\begin{aligned} X^3-\frac{1}{2} X^2 -\frac{1}{2p} X -\frac{1}{p^3} \end{aligned}$$

so that \(\frac{1}{2}<\tilde{x}_p< 1\) and \(\lim _{p \rightarrow \infty } \tilde{x}_p = \frac{1}{2}\) in \(\mathbb R\). By specializing to the case of p-adic MCF we can apply the previous Lemma to the sequences \((A_n)\), \((B_n)\), \((C_n)\) as in (3) with \(m = 2\). Considering that \(|a_n|_\infty , |b_n|_\infty < \frac{p}{2}\), for every \(n \ge 0\), in this special case the role of the polynomial f(X) of the Lemma is played by

$$\begin{aligned} X^3-\frac{p}{2} X^2-\frac{p}{2} X-1 \end{aligned}$$

whose real root is \(p\tilde{x}_p\). Thus, we obtain the following proposition.

Proposition 4

Given the sequences \((A_n)\), \((B_n)\), \((C_n)\) as in (3), there exists \(H>0\) such that

$$\begin{aligned} \max \{|A_n|_\infty ,|B_n|_\infty , |C_n|_\infty \} \le H(p \tilde{x}_p)^n, \end{aligned}$$

for every \(n\in \mathbb {N}\) and in particular

$$\begin{aligned} \max \{|A_n|_\infty ,|B_n|_\infty , |C_n|_\infty \} =o(p^n). \end{aligned}$$

Proposition 5

Let \(\varvec{\alpha }=(\alpha ,\beta )\in \mathbb {Q}^2\), and write

$$\begin{aligned} \alpha =\frac{x_0}{z_0}, \quad \quad \beta =\frac{y_0}{z_0} \end{aligned}$$

with \(z_0\in \mathbb {Z}\) and \(x_0,y_0\in \mathbb {Z}\left[ \frac{1}{p}\right] \).

The p-adic Jacobi–Perron algorithm applied to \(\varvec{\alpha }\) stops in a number of steps bounded by \(-\frac{\log (M)}{\log (\tilde{x}_p)} \) where

$$\begin{aligned} M&=\max \left\{ |z_0|_\infty , \frac{1}{p} |y_0|_\infty +\frac{1}{2} |z_0|_\infty , \frac{1}{p^2}|x_0|_\infty +\frac{1}{2p} |y_0|_\infty +\left( \frac{1}{2p}+\frac{1}{4}\right) |z_0|_\infty \right\} \nonumber \\&\le \max \left\{ |z_0|_\infty , \frac{1}{2} (|y_0|_\infty +|z_0|_\infty ), \frac{1}{4} (|x_0|_\infty +|y_0|_\infty +|z_0|_\infty )\right\} \end{aligned}$$
(12)
$$\begin{aligned}&\le |x_0|_\infty +|y_0|_\infty +|z_0|_\infty \end{aligned}$$
(13)
$$\begin{aligned}&\le 3\max \{ |x_0|_\infty ,|y_0|_\infty ,|z_0|_\infty \}. \end{aligned}$$
(14)

Proof

The proof is the same as [31, Theorem 5], but we take into account the number of steps. The p-adic JP algorithm produces the sequence of complete quotients \((\varvec{\alpha _n})_{n\ge 0}\), where

$$\begin{aligned} \varvec{\alpha _n}=(\alpha _n,\beta _n)\in \mathbb {Q}^m, \quad \quad \alpha _n=\frac{x_n}{z_n}, \quad \quad \beta _n=\frac{y_n}{z_n}, \end{aligned}$$

and \(x_n,y_n,z_n\) are generated by the following rules:

$$\begin{aligned} \left\{ \begin{array}{lll} x_n &{}=&{} a_nz_n+y_{n+1},\\ y_n &{}=&{} b_nz_n+z_{n+1},\\ z_n &{}=&{} x_{n+1} \end{array} \right. \end{aligned}$$

with \(a_n,b_n\in \mathcal {Y}\), \(|y_{n+1}|_p,|z_{n+1}|_p<|z_n|_p\). Then \(\frac{z_n}{p^n}\in \mathbb {Z}\) and from the formula

$$\begin{aligned} z_{n+1}=z_{n-2}-a_{n-1}z_{n-1}-b_nz_n \end{aligned}$$

we get by Lemma 1

$$\begin{aligned} \frac{|z_{n+1}|_\infty }{p^{n+1}}&< \frac{1}{2} \frac{|z_{n}|_\infty }{p^{n}} + \frac{1}{2p} \frac{|z_{n-1}|_\infty }{p^{n-1}}+\frac{1}{p^3} \frac{|z_{n-2}|_\infty }{p^{n-2}}\\&< M'\tilde{x}_p^n, \end{aligned}$$

where

$$\begin{aligned} M'=\max \left\{ \frac{|z_{2}|_\infty }{p^{2}},\frac{|z_{1}|_\infty }{p}, {|z_{0}|_\infty }\right\} . \end{aligned}$$

Then \(z_{n+1}=0\) when \(\tilde{x}_p^n \le \frac{1}{M'}\). We have

$$\begin{aligned} \frac{|z_{1}|_\infty }{p}&= \frac{1}{p} |y_0-b_0z_0|_\infty< \frac{1}{p} {|y_0|_\infty } + \frac{1}{2} |z_0|_\infty \\ \frac{|z_{2}|_\infty }{p^2}&= \frac{1}{p^2} |y_1-b_1z_1|_\infty \\&= \frac{1}{p^2} |(x_0-a_0z_0)-b_1(y_0-b_0z_0)|_\infty \\&< \frac{1}{p^2} {|x_0|_\infty }+\frac{1}{2p}{|y_0|_\infty } +\left( \frac{1}{2p} + \frac{1}{4} \right) {|z_0|_\infty }. \end{aligned}$$

Therefore \(M'<M\), so that

$$\begin{aligned} z_{n+1}=0&\quad \hbox { for } \tilde{x}_p^n \le \frac{1}{M}\nonumber \\&\hbox { that is for }\quad n \ge -\frac{\log (M)}{\log (\tilde{x}_p)}. \end{aligned}$$
(15)

Inequalities (12) and (14) are straightforward. \(\square \)

Corollary 3

Let \(t,u\in \mathbb {Z}[\frac{1}{p}]\) and \(v\in \mathbb {Z}\) such that the p-adic MCF for \((\frac{t}{v}, \frac{u}{v})\) has length \(\ge n+1\). Then

$$\begin{aligned} \max \{|t|_\infty , |u|_\infty , |v|_\infty \} \ge \frac{1}{3\tilde{x}_p^n}. \end{aligned}$$

Proof

With the notation of the proof of Proposition 5, we have \(z_{n+1}\not =0\), then the claim follows from by (14) and (15). \(\square \)

Corollary 4

Let \((\alpha ,\beta )\in \mathbb {Q}_p^2\) be a pair having a p-adic MCF expansion of length \(\ge n+1\). Then

$$\begin{aligned} \max \{|A_n|_\infty , |B_n|_\infty , |C_n|_\infty \} \ge \frac{1}{3 p^{K_n} \tilde{x}_p^n}. \end{aligned}$$

Proof

If we set \(t=p^{K_n}A_n, u=p^{K_n}B_n, v=p^{K_n}C_n\), then the hypothesis of Corollary 3 is fulfilled. \(\square \)

The following theorem establishes an explicit lower bound for the Euclidean length of a pair of rational numbers which is a “good approximation”of a p-adic pair w.r.t the corresponding \(K_n\).

Theorem 2

Let \((\alpha , \beta )\in \mathbb {Q}_p^2\) be a pair having a p-adic MCF expansion of length \(\ge n+1\). Let \((\frac{t}{v}, \frac{u}{v})\in \mathbb {Q}^2\) with \(t,u\in \mathbb {Z}[\frac{1}{p}]\), \(v\in \mathbb {Z}\), and assume \(\max \left\{ \left| \alpha - \frac{t}{v}\right| _p,\left| \beta - \frac{u}{v}\right| _p\right\} < \frac{1}{p^{2K_{n+1}}}\); then \(\max \{|t|_\infty , |u|_\infty , |v|_\infty \}\ge \frac{1}{3\tilde{x}_p^n}\).

Proof

By Proposition 2 the pair \((\frac{t}{v}, \frac{u}{v})\) has the same MCF expansion as \((\alpha ,\beta )\) up to \(n+1\). The claim follows from Corollary 3. \(\square \)

4 Results related to algebraic dependence

4.1 A p-adic Liouville-type theorem on algebraic dependence

The quality of rational approximations to real numbers is related to their algebraic dependence. Indeed, if it is possible to find infinitely many good approximations to a m-tuple of real numbers, then they are algebraically independent, see, e.g., [4]. Similar results also hold for the p-adic numbers [25]. In the following theorem, we prove a new result of this kind and then we apply it to p-adic MCFs.

Lemma 2

Let C be a non-zero integer number, then

$$\begin{aligned} |C|_p\ge \frac{1}{|C|_\infty }. \end{aligned}$$

Proof

The result follows from \(p^{\nu _p(C)}\le |C|_\infty \) and \(|C|_p= \frac{1}{p^{\nu _p(C)}}\). \(\square \)

The following result is a variant of [25, Theorem 3].

Theorem 3

Given \(\alpha ,\beta \in \mathbb {Q}_p\setminus \mathbb {Q}\) such that \(F(\alpha ,\beta )=0\), for \(F(X,Y)\in \mathbb {Z}[X,Y]\) non-zero polynomial with minimal total degree D, let \((t_n)_{n \ge 0}, (u_n)_{n \ge 0}, (v_n)_{n \ge 0}\) be sequences of integers such that \(v_n\not =0\) for every \(n\in \mathbb {N}\) and

$$\begin{aligned} \lim _{n \rightarrow \infty }\frac{u_n}{v_n} = \alpha ,\quad \lim _{n \rightarrow \infty } \frac{t_n}{v_n}=\beta \end{aligned}$$
(16)

in \(\mathbb Q_p\). Consider \(M_n =\max \{|t_n|_\infty ,|u_n|_\infty ,|v_n|_\infty \}\) and

$$\begin{aligned} U_n = \max \left\{ \left| \alpha - \frac{t_n}{v_n}\right| _p, \left| \beta - \frac{u_n}{v_n}\right| _p \right\} ; \end{aligned}$$

if

$$\begin{aligned} \lim _{n \rightarrow \infty } U_n\cdot M_n^D = 0 \end{aligned}$$
(17)

in \(\mathbb R\), then \(F\left( \frac{t_n}{v_n},\frac{u_n}{v_n}\right) =0\) for \(n\gg 0\).

Proof

We observe that \( v_n^D \cdot F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \in \mathbb {Z}\) and

$$\begin{aligned} \left| v_n^D \cdot F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \right| _\infty \le KM_n^D, \end{aligned}$$
(18)

where K is the sum of the Euclidean absolute values of the coefficients of F(X, Y). Therefore, if \(F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \) is not zero, we have

$$\begin{aligned} \left| F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \right| _p \ge \left| v_n^D\cdot F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \right| _p\ge \frac{1}{\left| v_n^D \cdot F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \right| _\infty }\ge \frac{1}{KM_n^D} \end{aligned}$$
(19)

by Lemma 2 and (18). On the other hand, we can write

$$\begin{aligned} F(X,Y)=\sum _{i,j}A_{ij}(X-\alpha )^i(Y-\beta )^j \end{aligned}$$

with \(A_{ij}\in \mathbb {Q}_p\). We have

$$\begin{aligned} A_{10}=\frac{ \partial F}{\partial X}(\alpha , \beta ), \quad A_{01}=\frac{ \partial F}{\partial Y}(\alpha , \beta ), \end{aligned}$$

if \(A_{10}=0\) and \(\frac{ \partial F}{\partial X}(X, Y)\not =0\) then the latter polynomial would give an algebraic dependence relation between \(\alpha \) and \(\beta \) of total degree \(\le D-1\), therefore \(\frac{ \partial F}{\partial X}(X, Y)=0\), that is X does not appear in F(X, Y). Analogously \(A_{01}=0\) implies that \(\frac{ \partial F}{\partial Y}(X, Y)=0\), that is Y does not appear in F(X, Y). It follows that if \(A_{ij}\not =0\) for some \(i >0\) then \(A_{10}\not =0\); and if \(A_{ij}\not =0\) for some \(j >0\) then \(A_{01}\not =0\). Hence, it is easy to see that for every i, j such that \(i+j>1\) and \(n\gg 0\)

$$\begin{aligned}&\nu _p\left( A_{ij}\left( \frac{t_n}{v_n}-\alpha \right) ^i\left( \frac{u_n}{v_n}-\beta \right) ^j\right) \\&\quad > \min \left\{ \nu _p\left( A_{10}\left( \frac{t_n}{v_n}-\alpha \right) \right) , \nu _p\left( A_{01}\left( \frac{u_n}{v_n}-\beta \right) \right) \right\} . \end{aligned}$$

Therefore for \(n\gg 0\), we obtain

$$\begin{aligned} \left| F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \right| _p&\le \max _{ij}\left\{ \left| A_{ij}\left( \frac{t_n}{v_n}-\alpha \right) ^i\left( \frac{u_n}{v_n}-\beta \right) ^j\right| \right\} \nonumber \\&= \max \left\{ \left| A_{01}\left( \frac{t_n}{v_n}-\alpha \right) \right| _p,\left| A_{10}\left( \frac{u_n}{v_n}-\beta \right) \right| _p\right\} \le H \cdot U_n, \end{aligned}$$
(20)

for \(H=\max \{|A_{01}|_p, |A_{10}|_p\}\). Putting together equations (19) and (20), we get

$$\begin{aligned} frac 1 {KM_n^D} \le \left| F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \right| _p \le H\cdot U_n, \end{aligned}$$

for every n such that \(F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \not =0\). This implies that there exists \(C>0\) such that if \(F\left( \frac{t_n}{v_n}, \frac{u_n}{v_n}\right) \not =0\) then \(U_n\cdot M_n>C\). Then hypothesis (17) proves the claim. \(\square \)

Remark 2

We shall apply Theorem 3 with

$$\begin{aligned} \frac{u_n}{v_n} = Q_n^{\alpha }, \quad \frac{t_n}{v_n} = Q_n^{\beta }, \end{aligned}$$

with \(u_n,t_n,v_n\in \mathbb {Z}\) coprime. Set \(\delta =\max \{0,-v(\alpha ),-v(\beta )\},\) then for \(n\gg 0\), we have

$$\begin{aligned} (t_n,u_n, v_n)=p^{K_n+\delta }(A_n,B_n,C_n). \end{aligned}$$

Consequently, if \(M_n=\max \{|t_n|_\infty ,|u_n|_\infty ,|v_n|_\infty \}\) then by Proposition 4 there exists \(H>0\) such that

$$\begin{aligned} M_n\le Hp^{K_n+n}\tilde{x}_p^n =o( p^{n+K_n}). \end{aligned}$$
(21)

4.2 Some consequences on linear dependence

We specialize Theorem 3 to the case \(D=1\), i.e., when we have linear dependence. In [31], the authors proved that if the p-adic Jacobi–Perron algorithm stops in a finite number of steps, then the initial values are \(\mathbb Q\)-linearly dependent. Further results about linear dependence and p-adic MCFs can be found in [32], where it is conjectured that if we start the p-adic Jacobi–Perron algorithm with a m-tuple of \(\mathbb Q\)-linearly dependent numbers, then the algorithm is finite or periodic. Here, exploiting the previous results, we can give a condition that ensures the finiteness of the p-adic Jacobi–Perron algorithm when it processes certain \(\mathbb Q\)-linearly dependent inputs.

Theorem 4

Given \(\alpha , \beta \in \mathbb Q_p\), consider

$$\begin{aligned} M_n = \max \{ |A_n|_\infty , |B_n|_\infty , |C_n|_\infty \}, \quad U_n = \max \left\{ \left| \alpha - \frac{A_n}{C_n} \right| _p, \left| \beta - \frac{B_n}{C_n} \right| _p\right\} , \end{aligned}$$

where \((A_n)\), \((B_n)\), \((C_n)\) are the sequences of numerators and denominators of convergents of the MCF representing \((\alpha , \beta )\). If

$$\begin{aligned} \lim _{n \rightarrow \infty } U_n\cdot M_n = 0, \end{aligned}$$

then either \(\alpha ,\beta ,1\) are linearly independent over \(\mathbb {Q}\) or the p-adic MCF expansion of \((\alpha ,\beta )\) is finite.

Proof

Assume that the p-adic MCF for \((\alpha ,\beta )\) is not finite, then the sequence \((Q_n^\alpha , Q_n^\beta )\) p-adically converges to \((\alpha ,\beta )\) by [31, Proposition 3] and \((\alpha ,\beta )\not \in \mathbb {Q}^2\), by Proposition 5. Suppose that \(A\alpha +B\beta +C=0\) for some \(A,B,C\in \mathbb {Q}\) not all zero. We define the sequence \(S_n=AA_{n-1}+BB_{n-1}+CC_{n-1}\); Theorem 3 implies that \(S_n=0\) for n sufficiently large. Furthermore, it is straightforward to see that \( S_n=AV_{n-1}^\alpha +B V_{n-1}^\beta \) (see also [32]) and by Corollary 1 we should have \(V_n^\alpha =V_n^\beta =0\), which is a contradiction, as \((\alpha ,\beta )\not \in \mathbb {Q}^2\), since we have supposed the MCF for \((\alpha ,\beta )\) is not finite. \(\square \)

Remark 3

Theorem 4 is an improvement of a result implicitly contained in [32, Proposition 10], namely that if \((1,\alpha ,\beta )\) are linearly dependent over \(\mathbb {Q}\) and there is a constant \(K>0\) such that

$$\begin{aligned} \max \{|V_n^\alpha |_p,|V_n^\beta |_p\}\le \frac{K}{p^n},\end{aligned}$$
(22)

then the p-adic Jacobi–Perron algorithm stops in finitely many steps when applied to \((\alpha ,\beta )\). In fact (22) implies

$$\begin{aligned} U_n\cdot p^{K_n}\le \frac{K}{p^n}, \end{aligned}$$

so that, by (21),

$$\begin{aligned} U_n\cdot M_n\le H\tilde{x}_p^n U_np^{K_n+n}\le KH \tilde{x}_p^n\buildrel \infty \over \rightarrow 0\hbox { for } n\rightarrow \infty . \end{aligned}$$

4.3 A class of fast convergent p-adic MCFs

Finally, we see some conditions on the partial quotients that produce MCFs converging to algebraically independent numbers or having convergents that satisfy an algebraic relation.

Lemma 3

Given a MCF \([(a_0, a_1, \ldots ), (b_0, b_1, \ldots )]\) such that

$$\begin{aligned}&k_{n+1} \ge (D-1)(k_n+k_{n-1})+2D;\hbox { and } \end{aligned}$$
(23)
$$\begin{aligned}&h_{n+1}\ge (D-1)k_n+D \end{aligned}$$
(24)

for \(n \gg 0\) and \(D \ge 1\), then there exists \(C\in \mathbb {R}\) such that

$$\begin{aligned} \min \{v_p(V_n^\alpha ),v_p(V_n^\beta )\}\ge (D-1)K_n+Dn +C\quad \hbox { for } n\in \mathbb {N}. \end{aligned}$$

Proof

The argument is the same as in the proof of Proposition 1. In fact, if conditions (23) and (24) hold for every \(n\ge 0\) then the claim directly follows from Proposition 1 by putting \(\ell _n=(D-1)k_n+D\). In any case hypotheses (23) and (24) imply that

$$\begin{aligned} v_p\left( \frac{\beta _{n+1}}{\alpha _{n+1}}\right) \ge (D-1)k_n+D \quad \hbox { for } n\gg 0. \end{aligned}$$
(25)

Let \(V_n\) be one of \(V_n^\alpha \), \(V_n^\beta \). From the formula (8) we get for \(n\gg 0\)

$$\begin{aligned} \frac{V_n}{p^{(D-1)K_n+Dn}}= \mu _n\frac{V_{n-1}}{p^{(D-1)K_{n-1}+D(n-1)}}+\nu _n\frac{V_{n-2}}{p^{(D-2)K_{n-1}+D(n-2)}}, \end{aligned}$$
$$\begin{aligned} \mu _n = -\frac{\beta _{n+1}}{\alpha _{n+1}}\frac{1}{p^{(D-1)k_n+D}},\quad \qquad \nu _n = -\frac{1}{\alpha _{n+1}}\frac{1}{p^{(D-1)(k_n+k_{n-1})+2D}}. \end{aligned}$$

By (23) and (25) there exists \(n_0\) such that \(\mu _n,\nu _n\in \mathbb {Z}_p\) for \(n > n_0\). Then

$$\begin{aligned} \left| \frac{V_n}{p^{(D-1)K_n+Dn}}\right| _p\le \max \left\{ \left| \frac{V_i}{p^{(D-1)K_i+Di}}\right| _p, i=0,\ldots , n_0\right\} \hbox { for } n>n_0 \end{aligned}$$

so that the claim follows by setting

$$\begin{aligned} C=\min \left\{ v_p\left( \frac{V_i^\alpha }{p^{(D-1)K_i+Di}}\right) ,v_p\left( \frac{V_i^\beta }{p^{(D-1)K_i+Di}}\right) ,i=0 ,\ldots , n_0\right\} . \end{aligned}$$

\(\square \)

Theorem 5

Assume that \(\alpha ,\beta \) are algebraically dependent and let \(F(X,Y)\in \mathbb {Q}[X,Y]\) be a non-zero polynomial of minimum total degree D such that \(F(\alpha ,\beta )=0\). If the MCF expansion of \((\alpha , \beta )\) satisfies conditions (23) and (24), then \(F(Q_n^\alpha ,Q_n^\beta )=0\) for \(n\gg 0\).

Proof

Let \(M_n,U_n\) be as in Theorem 3. For \(n\gg 0\) and a suitable constant \(C>0\) we have

$$\begin{aligned} U_n&=\frac{1}{p^{K_n}} \max \left\{ | V_n^\alpha |_p, |V_n^\beta |_p \right\} \\&\le \frac{C}{p^{D(K_n+n)}}\hbox { by Lemma }3 \end{aligned}$$

so that \(\lim _{n\rightarrow \infty } U_n\cdot M_n=0\) by (21). Then the claim follows from Theorem 3. \(\square \)

Theorem 6

Given \((\alpha , \beta ) = [(a_0, a_1, \ldots ), (b_0, b_1, \ldots )]\) such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{k_n}{k_{n-1}+k_{n-2}} = \infty , \quad \lim _{n \rightarrow \infty } \frac{h_n}{k_{n-1}} = \infty \end{aligned}$$

in \(\mathbb R\), then either \(\alpha ,\beta \) are algebraically independent or there exists a non-zero polynomial \(F(X,Y)\in \mathbb {Q}[X,Y]\) such that \(F(Q_n^\alpha ,Q_n^\beta )=0\) for \(n\gg 0\).

Proof

Let \(D>0\). For \(n\gg _D 0\) we have

$$\begin{aligned} k_n&\ge D(k_{n-1}+k_{n-2})\ge (D-1) D(k_{n-1}+k_{n-2})+2D,\hbox { and }\\ h_n&\ge D k_{n-1}\ge (D-1)k_{n-1} +D. \end{aligned}$$

Then the claim follows from Theorem 5. \(\square \)

Remark 4

By Faltings theorem, an algebraic curve having infinitely many rational points must have genus 0 or 1. This is a strong condition on polynomials \(F(X,Y)\in \mathbb {Q}[X,Y]\) such that \(F(Q_n^\alpha ,Q_n^\beta )=0\), for \(n\gg 0\).