# Continued fractions in the field of p-adic numbers

Giuliano Romeo

###### Abstract.

Continued fractions have a long history in number theory, especially in the
area of Diophantine approximation. The aim of this expository paper is to
survey the main results on the theory of $p$–adic continued fractions, i.e.
continued fractions defined over the field of $p$–adic numbers
$\mathbb{Q}_{p}$, which in the last years has recorded a considerable increase
of interest and research activity. We start from the very first definitions up
to the most recent developments and open problems.

Department of Mathematical Sciences Giuseppe Luigi Lagrange,

Politecnico of Torino, Corso Duca degli Abruzzi 24, Torino

<EMAIL_ADDRESS>MSC=11J70; 11D88; 11Y65; 12J25

###### Contents

  1. 1 Introduction
  2. 2 Some notation
  3. 3 Algorithms for $p$–adic continued fractions
  4. 4 Convergence in $\mathbb{Q}_{p}$
  5. 5 Continued fractions of rational numbers
  6. 6 Periodicity properties
  7. 7 Quality of the approximation
  8. 8 Transcendental $p$–adic continued fractions
  9. 9 Multidimensional $p$–adic continued fractions

##  1\. Introduction

Continued fractions are objects of the form

$a_{0}+\cfrac{b_{1}}{a_{1}+\cfrac{b_{2}}{a_{2}+\ddots}},$ (1)

and they have a long history in number theory, especially in the area of
Diophantine approximation. They have been extensively explored over the real
numbers, where they provide finite representations for rational numbers and
periodic representations for quadratic irrationals by means of integer
sequences (Lagrange’s Theorem). Moreover, the classical algorithm for
continued fractions in $\mathbb{R}$ provides, at each step, the best rational
approximation of irrational numbers. Because of their optimal properties, they
have been employed in various areas of mathematics and there exist several
generalizations of the classical theory of continued fractions. In 1940,
Mahler [55] raised the problem of defining a continued fraction algorithm
working in the field of $p$–adic numbers $\mathbb{Q}_{p}$. The field of
$p$–adic numbers is largely studied in algebraic number theory and it is
obtained as the completion of the field of rational numbers $\mathbb{Q}$ with
respect to the $p$–adic absolute value. The completion is the smallest field
containing $\mathbb{Q}$ in which all Cauchy sequences with respect to the
$p$–adic absolute value are convergent. The problem of defining a $p$–adic
continued fraction algorithm sharing all the optimal properties enjoyed by
classical continued fractions is still open. In particular, an algorithm
providing periodic representations for every $p$–adic quadratic irrational is
not known, i.e. a $p$–adic analogue of Lagrange’s Theorem does not exist yet.
The main problem is that there is not an intuitive satisfying definition for
the integer part of a $p$–adic number. There are at least two very natural
definitions, due to Schneider [81] and Ruban [75], both around 1970. The two
algorithms try to replicate the standard algorithm for real continued
fractions in two different senses, that we examine in Section 3. Historically,
another of the main algorithms for $p$–adic continued fractions is due to
Browkin [20]. His algorithm is very similar to Ruban’s one, but it has the
very important property of producing finite continued fractions for all
rational numbers. In fact, Ruban’s and Schneider’s continued fractions are not
always finite over $\mathbb{Q}$. Moreover, Schneider’s and Ruban’s algorithms
have been proved to be not periodic for every quadratic irrational [84, 31].
Although a proof has not been provided, Lagrange’s Theorem seems to fail also
for Browkin’s algorithm (see the numerical simulations in [57]). For these
reasons, the research of new definitions for expressing a $p$–adic number as a
continued fraction has been widely developed, particularly in the last few
years. Together with the design of new algorithms, the analysis of the
properties of the existent algorithms has been developed in several
directions, inspired by the numerous fields in which classical continued
fractions have been employed throughout the centuries.

The purpose of this survey is to give a concise overview of the main results,
developments and open problems in the theory of $p$–adic continued fractions.
In Section 2, we recall some notation for the classical theory of continued
fractions and $p$–adic numbers. In Section 3 we describe the various
algorithms that have been defined over the years, underlining their
motivations and their main properties. In Section 4, we discuss the $p$–adic
convergence of a continued fraction, which is the very first requirement for
the definition of an algorithm. In Section 5, we present the properties of the
expansions of rational numbers that, as already mentioned, are not always
finite. Section 6 contains all the main results related to the periodicity of
$p$–adic continued fractions and the main developments towards a $p$–adic
analogue of Lagrange’s Theorem. In Section 7, we analyze the properties of
approximation of continued fractions with respect to the $p$–adic absolute
value. In Section 8, we collect some recent results on the transcendence of
$p$–adic continued fractions, inspired by many famous transcendence criteria
for classical continued fractions. Finally, in Section 9 we introduce the
generalization of $p$–adic continued fractions to higher dimensions, in
analogy with Jacobi-Perron algorithm for real multidimensional continued
fractions.

##  2\. Some notation

In this section we recall some useful results from the theory of continued
fractions and the field of $p$–adic numbers and we fix some notation that we
are going to use throughout the survey. For more background on continued
fractions and $p$–adic numbers we refer the reader, respectively, to [63, 85]
and [36].

We call continued fraction an object of the form

$a_{0}+\cfrac{b_{1}}{a_{1}+\cfrac{b_{2}}{a_{2}+\ddots}},$ (2)

and we denote it by

$\begin{bmatrix}\ b_{1}&b_{2}&\ldots\ \\\ \ a_{0}&a_{1}&a_{2}&\ldots\ \\\
\end{bmatrix},$

where the coefficients $a_{i}$ and $b_{i}$ are elements in a field and the
expansion can be either finite or infinite. If $b_{i}=1$ for all $i$, the
continued fraction is called simple, that is

$a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\ddots}},$

and we denote it by $[a_{0},a_{1},a_{2},\ldots]$. The coefficients
$a_{0},a_{1},a_{2}\ldots$ are called partial quotients. For all
$n\in\mathbb{N}$, the rational number

$\frac{A_{n}}{B_{n}}=\begin{bmatrix}\ b_{1}&\ldots&b_{n-1}&b_{n}&\ \\\ \
a_{0}&a_{1}&\ldots&a_{n-1}&a_{n}\ \\\
\end{bmatrix}=a_{0}+\cfrac{b_{1}}{a_{1}+\ddots+\cfrac{b_{n-1}}{a_{n-1}+\cfrac{b_{n}}{a_{n}}}},$

corresponding to the continued fraction stopped at the $n$-th term, is called
$n$-th convergent of the continued fraction. The $A_{n}$’s are called partial
numerators and the $B_{n}$’s are called partial denominators. The sequences
$\\{A_{n}\\}_{n\in\mathbb{N}}$ and $\\{B_{n}\\}_{n\in\mathbb{N}}$ satisfy the
recursions

$\begin{cases}A_{0}=a_{0},\\\ A_{1}=a_{1}a_{0}+b_{1},\\\
A_{n}=a_{n}A_{n-1}+b_{n}A_{n-2},\ \ n\geq
2,\end{cases}\begin{cases}B_{0}=1,\\\ B_{1}=a_{1},\\\
B_{n}=a_{n}B_{n-1}+b_{n}B_{n-2},\ \ n\geq 2.\end{cases}$ (3)

The partial numerators and denominators of the convergents can be represented
also using the following matrix form. For all $n\in\mathbb{N}$,

$\begin{pmatrix}\ A_{n}&A_{n-1}\ \\\ \ B_{n}&B_{n-1}\
\end{pmatrix}=\begin{pmatrix}\ a_{0}&1\ \\\ \ 1&0\
\end{pmatrix}\begin{pmatrix}\ 1&0\ \\\ \ 0&b_{1}\
\end{pmatrix}\begin{pmatrix}\ a_{1}&1\ \\\ \ 1&0\
\end{pmatrix}\ldots\begin{pmatrix}\ 1&0\ \\\ \ 0&b_{n}\
\end{pmatrix}\begin{pmatrix}\ a_{n}&1\ \\\ \ 1&0\ \end{pmatrix}.$ (4)

Furthermore, for an infinite continued fraction representing an element
$\alpha$, we introduce the sequence of complete quotients
$\\{\alpha_{n}\\}_{n\in\mathbb{N}}$ defined, for all $n\in\mathbb{N}$, as

$\alpha_{n+1}=\frac{b_{n}}{\alpha_{n}-a_{n}},$

starting from $\alpha_{0}=\alpha$. We say that an infinite continued fraction
is periodic (or eventually periodic) if and only if there exist
$n_{0}\in\mathbb{N}$ and $k\geq 1$ such that $a_{n+k}=a_{n}$ $b_{n+k}=b_{n}$
for all $n\geq n_{0}$. In this case we call $k$ the period length and $n_{0}$
the pre-period length (or just period and pre-period where there is no risk of
ambiguity). If $n_{0}=0$, the continued fraction is said purely periodic and
the period starts from the begin, without a pre-period. The standard algorithm
to express a real number $\alpha$ through a simple continued fraction
$[a_{0},a_{1},\ldots]$ is the following, with $\alpha_{0}=\alpha$:

$\begin{cases}a_{n}=\lfloor\alpha_{n}\rfloor\\\
\alpha_{n+1}=\frac{1}{\alpha_{n}-a_{n}},\end{cases}$ (5)

where $\lfloor\alpha_{n}\rfloor$ denotes the integer part of $\alpha$, that is
the greatest integer $a_{n}\leq\alpha_{n}$. If $\alpha_{n}=a_{n}$ for some
$n\in\mathbb{N}$, then the algorithm terminates and the continued fraction is
finite.

In the following, let $p$ be an odd prime number and let us denote by
$v_{p}(\cdot)$ the $p$–adic valuation and by $|\cdot|$ and $|\cdot|_{p}$,
respectively, the Euclidean absolute value and the $p$–adic absolute value.
The field of $p$–adic numbers $\mathbb{Q}_{p}$ is the completion of
$\mathbb{Q}$ with respect to the $p$–adic absolute value, i.e. the smallest
field containing $\mathbb{Q}$ in which all the Cauchy sequences are
convergent. The field of $p$–adic numbers contains all the power series in $p$
with finite tail, that is:

$\mathbb{Q}_{p}=\left\\{\sum\limits_{n=-r}^{+\infty}c_{n}p^{n}\ \Big{|}\
r\in\mathbb{Z},\ c_{n}\in\mathbb{Z}/p\mathbb{Z}\right\\}.$ (6)

By construction $\mathbb{Q}\subset\mathbb{Q}_{p}$ and it is a known fact that
rational numbers correspond exactly to the set finite and eventually periodic
series. Moreover, we call $\mathbb{Z}_{p}$ the set of $p$–adic integers,

$\mathbb{Z}_{p}=\Bigg{\\{}\sum\limits_{n=0}^{+\infty}c_{n}p^{n}\ \Big{|}\
c_{n}\in\mathbb{Z}/p\mathbb{Z}\Bigg{\\}},$

that is the set of all $p$–adic numbers with non-negative valuation.

##  3\. Algorithms for $p$–adic continued fractions

In this section we present the main definitions for continued fractions in the
field of $p$–adic numbers that have appeared in literature. The first attempt
is to emulate Algorithm the standard algorithm (5) for real continued
fractions. However, the definition of integer part of a $p$–adic number
$\alpha\in\mathbb{Q}_{p}$ is not uniquely determined. In $\mathbb{R}$, the
integer part of a real number $\alpha$ is defined as the unique integer $b$
less than $\alpha$ for which $|\alpha-a|<1$. However, in $\mathbb{Q}_{p}$
there are infinitely many integers such that this happens, since there are
infinitely many $a\in\mathbb{Z}$ such that $p$ divides $\alpha-a$. One of the
first definitions of $p$-adic continued fractions is due to Schneider [81].
Let us observe that all the integers $a$ satisfying $0\leq|\alpha-a|_{p}<1$
are congruent modulo $p$. Therefore, it is meaningful to define the partial
quotients as the unique representatives $a_{n}$ of $a$ that lie inside
$\\{0,\ldots,p-1\\}$. Schneider followed this approach to provide a non-simple
continued fraction expansion

$a_{0}+\cfrac{b_{1}}{a_{1}+\cfrac{b_{2}}{a_{2}+\ddots}},$ (7)

for all $p$-adic integers. For $\alpha_{0}\in\mathbb{Z}_{p}$, with
representation

$\alpha_{0}=\sum\limits_{n=0}^{+\infty}c_{n}p^{n},\ \ \ \ \
c_{n}\in\\{0,\ldots,p-1\\},$

Schneider’s algorithm works as follows. The first partial quotient is
$a_{0}=c_{0}$. Then, for all $n\in\mathbb{N}$,

$\displaystyle\begin{cases}e_{n+1}=v_{p}(\alpha_{n}-a_{n})\\\
b_{n+1}=p^{e_{n+1}}\\\ \alpha_{n+1}=\dfrac{b_{n+1}}{\alpha_{n}-a_{n}}\\\
a_{n+1}=c_{0}^{(n+1)},\\\ \end{cases}$ (8)

where $c_{0}^{(n+1)}$ denotes the first term of the expansion of

$p^{e}\alpha_{n+1}=\sum\limits_{i=0}^{+\infty}c_{i}^{(n+1)}p^{i}.$

######  Example 1.

Let us give an idea on how Schneider’s algorithm (8) works to provide the
$5$–adic continued fraction of $\frac{2}{7}$. In $\mathbb{Q}_{5}$,

$\frac{2}{7}=1+2\cdot 5+5^{2}+\ldots,$

hence $a_{0}=2$ and $e_{1}=v_{p}(\alpha_{0}-a_{0})=v_{p}(2\cdot
5+5^{2}+\ldots)=1$. The next complete quotient is then

$\alpha_{1}=\frac{p^{e_{1}}}{\alpha_{0}-a_{0}}=\frac{5}{2\cdot
5+5^{2}+\ldots}=\frac{1}{2+5+\ldots}=3+3\cdot 5+4\cdot 5^{2}+\ldots.$

Hence, $a_{1}=3$ and the algorithm goes on as before. The expansion becomes
eventually periodic with $b_{n}=5$ for all $n$ and
$[a_{0},a_{1},\ldots]=[3,2,3,\overline{4}]$.

Ruban’s [75] and Browkin’s [20, 21] approach is to construct a simple
continued fraction by defining an intuitive $p$–adic analogue of the integer
part in $\mathbb{R}$. For any $p$-adic number
$\alpha=\sum\limits_{n=-r}^{+\infty}c_{n}p^{n}\in\mathbb{Q}_{p}$,
$c_{n}\in\\{0,\ldots,p-1\\}$, the integer part used by Ruban is

$\lfloor\alpha\rfloor_{p}=\sum\limits_{n=-r}^{0}c_{n}p^{n},$

and $\lfloor\alpha\rfloor_{p}=0$ if $r<0$. Notice that in this case
$\lfloor\alpha\rfloor_{p}$ is, in general, a rational number. Ruban’s
continued fractions are simple and the coefficients of the expansion can be
computed iteratively by the following algorithm, starting with
$\alpha_{0}=\alpha$.

$\displaystyle\begin{cases}a_{n}=\lfloor\alpha_{n}\rfloor_{p}\\\
\alpha_{n+1}=\frac{1}{\alpha_{n}-a_{n}}.\end{cases}$ (9)

If at some point $\alpha_{n}=a_{n}$, then the algorithm stops and
$\alpha=[a_{0},\ldots,a_{n}]$, i.e. $\alpha$ has finite Ruban’s continued
fraction. The choice of this integer part is, somehow, natural. In fact, in
the series $\sum\limits_{n=-r}^{+\infty}c_{n}p^{n}$, the positive powers of
$p$ have fractional $p$–adic absolute value and the floor function
$\lfloor\cdot\rfloor_{p}$ takes the part that has integral absolute value.

######  Example 2.

Let us compute Ruban’s expansion of $\alpha_{0}=-\frac{2}{5}$ in
$\mathbb{Q}_{7}$. The $7$–adic expansion of $\alpha_{0}$ is

$\alpha_{0}=1+4\cdot 7+5\cdot 7^{2}+\ldots,$

so that $a_{0}=\lfloor\alpha_{0}\rfloor_{p}=1$ and

$\alpha_{1}=\frac{1}{\alpha_{0}-a_{0}}=\frac{1}{-\frac{2}{5}-1}=-\frac{5}{7}=2\cdot
7^{-1}+6+6\cdot 7+\ldots.$

The second partial quotient is then

$a_{1}=\lfloor\alpha_{1}\rfloor_{p}=\frac{2}{7}+6=\frac{44}{7},$

and we compute the third complete quotient as

$\alpha_{2}=\frac{1}{\alpha_{1}-a_{1}}=\frac{1}{-\frac{5}{7}-\frac{44}{7}}=-\frac{1}{7}=6\cdot
7^{-1}+6+6\cdot 7+\ldots.$

Then

$a_{2}=\lfloor\alpha_{2}\rfloor_{p}=\frac{48}{7},$

and we find

$\alpha_{3}=\frac{1}{\alpha_{2}-a_{2}}=\frac{1}{-\frac{1}{7}-\frac{48}{7}}=-\frac{1}{7}=\alpha_{2}.$

We have found a repetition on the complete quotients, $\alpha_{3}=\alpha_{2}$,
so the continued fraction repeats from this point onward. It means that

$-\frac{2}{5}=\left[1,\frac{44}{7},\overline{\frac{48}{7}}\right],$

hence $-\frac{2}{5}$ has a periodic Ruban’s continued fraction in
$\mathbb{Q}_{7}$.

Browkin’s first algorithm [20] is very similar to Ruban’s algorithm, with the
exception that the representatives of $\mathbb{Z}/p\mathbb{Z}$ are chosen in
$\\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\\}$ instead of $\\{0,\ldots,p-1\\}$.
This small variation is fundamental since Browkin’s algorithm produces a
finite continued fraction for each rational number, while Ruban’s algorithm
can also be periodic over the rationals (more details are given in Section 5).
Given $\alpha=\sum\limits_{i=-r}^{+\infty}c_{i}p^{i}\in\mathbb{Q}_{p}$, with
$c_{i}\in\\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\\}$, Browkin defines the floor
function $s:\mathbb{Q}_{p}\rightarrow\mathbb{Q}$ as

$s(\alpha)=\sum\limits_{i=-r}^{0}c_{i}p^{i},$ (10)

and $s(\alpha)=0$ if $r<0$. Browkin’s first algorithm, that we call Browkin I,
work as follows. At the first step $\alpha_{0}=\alpha$ and, for all $n\geq 0$,

$\displaystyle\begin{cases}a_{n}=s(\alpha_{n})\\\
\alpha_{n+1}=\frac{1}{\alpha_{n}-a_{n}}.\end{cases}$ (11)

If at some point $\alpha_{n}=a_{n}$, then the algorithm stops and
$\alpha=[a_{0},\ldots,a_{n}]$, i.e. $\alpha$ has finite Browkin I continued
fraction.

######  Example 3.

Let us compute Browkin I continued fraction of $\alpha_{0}=-\frac{2}{5}$ in
$\mathbb{Q}_{7}$ and let us compare with Ruban’s expansion obtained in Example
2. For Browkin’s algorithm, we have to consider the $7$–adic expansion with
representatives in $\\{-3,\ldots,3\\}$, hence:

$\alpha_{0}=1-3\cdot 7-7^{2}+\ldots.$

Also in this case the first partial quotient is $a_{0}=s(\alpha_{0})=1$ and
then

$\alpha_{1}=\frac{1}{\alpha_{0}-a_{0}}=\frac{1}{-\frac{2}{5}-1}=-\frac{5}{7}=2\cdot
7^{-1}-1\ldots.$

The second partial quotient is then

$a_{1}=s(\alpha_{1})=\frac{2}{7}-1=-\frac{5}{7}=\alpha_{1}.$

Therefore,

$-\frac{2}{5}=\left[1,-\frac{5}{7}\right],$

hence $-\frac{2}{5}$ has a finite Browkin I continued fraction. The difference
with Ruban’s expansion of Example 2, which is periodic, relies only on the
choice of the representatives in $\\{-3,\ldots,3\\}$ instead of
$\\{0,\ldots,6\\}$.

In 2000, more than 20 years after the first algorithm, Browkin [21] defined
another floor function, that is similar to the first function $s$, but
excluding the constant term. For
$\alpha=\sum\limits_{i=-r}^{+\infty}c_{i}p^{i}\in\mathbb{Q}_{p}$, with
$c_{i}\in\\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\\}$, the second floor function
is the function $t:\mathbb{Q}_{p}\rightarrow\mathbb{Q}$, such that

$t(\alpha)=\sum\limits_{i=-r}^{-1}c_{i}p^{i},$

and $t(\alpha)=0$ if $r\leq 0$. The second algorithm, that we call Browkin II,
works on an input $\alpha$ as follows. At the first step $\alpha_{0}=\alpha$
and, for all $n\geq 0$,

$\displaystyle\begin{cases}a_{n}=s(\alpha_{n})\ \ \ \ \ &\textup{if}\ n\
\textup{even}\\\ a_{n}=t(\alpha_{n})&\textup{if}\ n\ \textup{odd}\
\textup{and}\ v_{p}(\alpha_{n}-t(\alpha_{n}))=0\\\
a_{n}=t(\alpha_{n})-sign(t(\alpha_{n}))&\textup{if}\ n\ \textup{odd}\
\textup{and}\ v_{p}(\alpha_{n}-t(\alpha_{n}))\neq 0\\\
\alpha_{n+1}=\frac{1}{\alpha_{n}-a_{n}}.\end{cases}$ (12)

If at some point $\alpha_{n}=a_{n}$, then the algorithm stops and
$\alpha=[a_{0},\ldots,a_{n}]$, i.e. $\alpha$ has finite Browkin II continued
fraction. The result of the alternation is that, in Browkin II, all the even
partial quotients are integers and all the odd partial quotients are
rationals. The use of the sign function is due to the convergence condition of
Proposition 8 proved by Browkin in [21].

######  Example 4.

Let us consider $\alpha_{0}=\frac{22}{7}\in\mathbb{Q}_{5}$ and let us compute
its Browkin II expansion. The $5$–adic expansion of $\alpha_{0}$ is

$\alpha_{0}=1-1\cdot 5+1\cdot 5^{2}+\ldots,$

so that $a_{0}=s(\alpha_{0})=1$ and

$\alpha_{1}=\frac{1}{\alpha_{0}-a_{0}}=\frac{1}{\frac{22}{7}-1}=\frac{7}{15}=-1\cdot
5^{-1}-1+2\cdot 5+\ldots.$

Now we apply the function $t$, thus obtaining
$b_{1}=t(\alpha_{1})=-\frac{1}{5}$ and

$\alpha_{2}=\frac{1}{\alpha_{1}-a_{1}}=\frac{3}{2}=-1-2\cdot 5-2\cdot
5^{2}+\ldots.$

At the next step we have $a_{2}=s(\alpha_{2})=-1$ and
$\alpha_{3}=\frac{2}{5}$. At this point, since
$v_{p}(\alpha_{3}-t(\alpha_{3}))>0$, then we use the sign function, obtaining:

$a_{3}=t(\alpha_{3})-sign(t(\alpha_{3}))=\frac{2}{5}-1=-\frac{3}{5}.$

Therefore $\alpha_{4}=s(\alpha_{4})=b_{4}=1$ and the expansion of is
$\frac{22}{7}=\left[1,-\frac{1}{5},-1,-\frac{3}{5},1\right]$.

It has been experimentally observed that Browkin II produces more periodic
expansions for quadratic irrationals than Browkin I, hence getting closer to a
$p$–adic analogue of Lagrange’s theorem (see, for example, [57]). For this
reason, Browkin II has been taken as a starting point for the definition of
some new algorithms with the aim of improving furthermore its properties of
periodicity. In [11], it is studied a a variant of Browkin II in which the
representatives are taken in $\\{0,\ldots,p-1\\}$ instead of
$\\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\\}$. This choice improves the
properties of approximation Browkin II and, unexpectedly, does not compromise
the finiteness of the continued fraction expansions of rational numbers. In
[59], it is proposed a $3$-steps generalization of Browkin II relying on the
convergence condition of Theorem 10 in Section 4. In order to satisfy that
convergence condition, a new integer part is defined, that acts on a $p$–adic
integer with zero valuation and leaves apart another $p$–adic integer with
zero valuation. For
$\alpha=\sum\limits_{n=0}^{+\infty}c_{i}p^{i}\in\mathbb{Q}_{p}$, with
$c_{i}\in\\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\\}$, the new integer part is

$\displaystyle u(\alpha)=\begin{cases}+1\ \ &\textup{if}\
c_{0}\in\Big{\\{}+2,\ldots,\dfrac{p-1}{2}\Big{\\}}\cup\\{-1\\}\\\
-1&\textup{if}\
c_{0}\in\Big{\\{}-\dfrac{p-1}{2},\ldots,-2\Big{\\}}\cup\\{+1\\}\\\ 0\ \
&\textup{if}\ c_{0}=0.\end{cases}$ (13)

In [59], two new algorithms have been defined by exploiting the three integer
parts $s,t,u$ in three different steps. For one of them, the authors proved
that all rational numbers have a finite continued fraction, hence becoming
more interesting than the other one. On input $\alpha_{0}\in\mathbb{Q}$ the
algorithm works as follows, for all $n\in\mathbb{N}$:

$\displaystyle\begin{cases}a_{n}=s(\alpha_{n})\ \ \ \ \ &\textup{if}\ n\equiv
0\bmod 3\\\ a_{n}=t(\alpha_{n})&\textup{if}\ n\equiv 1\bmod 3\ \textup{and}\
v_{p}(\alpha_{n}-t(\alpha_{n}))=0\\\
a_{n}=t(\alpha_{n})-sign(t(\alpha_{n}))&\textup{if}\ n\equiv 1\bmod 3\
\textup{and}\ v_{p}(\alpha_{n}-t(\alpha_{n}))\neq 0\\\
a_{n}=s(\alpha_{n})-u(\alpha_{n})&\textup{if}\ n\equiv 2\bmod 3\\\
\alpha_{n+1}=\frac{1}{\alpha_{n}-a_{n}}.\end{cases}$ (14)

If at some point $\alpha_{n}=a_{n}$, then the algorithm stops and
$\alpha=[a_{0},\ldots,a_{n}]$, i.e. $\alpha$ has a finite continued fraction.
Finally, in [57], the authors proposed a variant of Browkin II without the use
of the sign function. For all $\alpha_{0}\in\mathbb{Q}_{p}$, their algorithm
works as follows:

$\displaystyle\begin{cases}a_{n}=s(\alpha_{n})\ \ \ \ \ &\textup{if}\ n\
\textup{even}\\\ a_{n}=t(\alpha_{n})&\textup{if}\ n\ \textup{odd}\\\
\alpha_{n+1}=\frac{1}{\alpha_{n}-a_{n}}.\end{cases}$ (15)

If at some point $\alpha_{n}=a_{n}$, then the algorithm stops and
$\alpha=[a_{0},\ldots,a_{n}]$, i.e. $\alpha$ has a finite continued fraction.
This choice turns out to improve the periodicity properties of Browkin’s
second algorithm (more details are given in Section 6).

However, although all these algorithms improve on several aspects the known
methods to generate $p$–adic continued fractions, they experimentally seem
still far from an analogue of Lagrange’s Theorem and, in general, from a
satisfactory algorithm reproducing the same properties of classical continued
fractions in $\mathbb{R}$.

Very recently, in [30], the definition of $p$–adic continued fractions has
been generalized also to number fields, addressing some questions similar to
Rosen [72, 73] in the archimedean setting. The authors gave a general
definition of $\mathfrak{P}$–adic continued fractions for a prime ideal
$\mathfrak{P}$ of the ring of integers $\mathcal{O}_{K}$ of a number field
$K$. Moreover, they investigated their finiteness and periodicity properties,
focusing on a general number field $K$ and obtaining some more effective
results for the quadratic case. In [28], the construction of [30] has been
generalized to quaternion algebras over $\mathbb{Q}$ ramified at $p$, hence
including all the classical $p$–adic framework.

##  4\. Convergence in $\mathbb{Q}_{p}$

In the field of real numbers, every continued fraction with positive partial
quotients converges to an $\alpha\in\mathbb{R}$. This is not always the case
in $\mathbb{Q}_{p}$. The field of $p$–adic numbers is complete with respect to
the $p$–adic absolute value $|\cdot|_{p}$. Therefore, a sequence is convergent
in $\mathbb{Q}_{p}$ if and only if it is a Cauchy sequence. Then, a continued
fraction

$a_{0}+\cfrac{b_{1}}{a_{1}+\cfrac{b_{2}}{a_{2}+\ddots}},$

converges to a $p$–adic number if and only if the sequence of the convergents
$\\{\frac{A_{n}}{B_{n}}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence with respect
to $|\cdot|_{p}$. For a non-archimedean absolute value this is equivalent to
require that

$\lim\limits_{n\rightarrow+\infty}\left|\frac{A_{n+1}}{B_{n+1}}-\frac{A_{n}}{B_{n}}\right|_{p}=0.$

The latter quantity can be written as

$\left|\frac{A_{n+1}}{B_{n+1}}-\frac{A_{n}}{B_{n}}\right|_{p}=\left|\frac{(-1)^{n}}{B_{n}B_{n+1}}\right|_{p}=p^{v_{p}(B_{n}B_{n+1})},$
(16)

so that the continued fraction $[a_{0},a_{1},\ldots]$ converges to an element
of $\mathbb{Q}_{p}$ if and only if

$\lim\limits_{n\rightarrow+\infty}v_{p}(B_{n}B_{n+1})=-\infty.$

This is the minimum requirement in order to define meaningful continued
fractions in the field of $p$–adic numbers, since otherwise a continued
fraction does not represent any element of $\mathbb{Q}_{p}$. A proof of the
convergence of Schneider’s continued fractions can be found in [68] and it
exploits the matrix relations (4).

######  Remark 5.

In [20], Browkin proved that, for Browkin I,

$\begin{split}v_{p}(A_{n})&=v_{p}(a_{0})+v_{p}(a_{1})+\ldots+v_{p}(a_{n}),\\\
v_{p}(B_{n})&=v_{p}(a_{1})+v_{p}(a_{2})+\ldots+v_{p}(a_{n}),\end{split}$ (17)

or, equivalently,

$\begin{split}|A_{n}|_{p}&=|a_{0}|_{p}|a_{1}|_{p}\ldots|a_{n}|_{p},\\\
|B_{n}|_{p}&=|a_{1}|_{p}|a_{2}|_{p}\ldots|a_{n}|_{p}.\end{split}$ (18)

The proof is done by induction on $n$. Using a similar argument, it is not
hard to see that a sufficient condition for these equations to hold is having
$v_{p}(a_{n+1}B_{n})<v_{p}(B_{n-1})$ for all $n\in\mathbb{N}$.

The $p$–adic convergence strictly depends on the valuation of the partial
quotients, regardless of the representative of $\mathbb{Z}/p\mathbb{Z}$ that
are used. Therefore, the next proposition proves the convergence for both
Ruban’s and Browkin’s continued fractions.

######  Proposition 6 ([20]).

Let an infinite sequence $a_{0},a_{1},\ldots\in\mathbb{Z}[\frac{1}{p}]$ such
that for all $n\in\mathbb{N}$, $v_{p}(a_{n})<0$. Then the continued fraction
$[a_{0},a_{1},\ldots]$ is convergent to a $p$-adic number.

######  Remark 7.

Proposition 6 and all the convergence results in this section contain
conditions that hold for all $n$. However, the results do not change if these
conditions hold for all $n\geq n_{0}\in\mathbb{N}$.

The proof of Proposition 6 is a simple induction which shows that, assuming
$v_{p}(a_{n})<0$ for all $n\in\mathbb{N}$, then
$\\{v_{p}(B_{n})\\}_{n\in\mathbb{N}}$ is a strictly decreasing sequence.
Therefore, $\\{v_{p}(B_{n}B_{n+1})\\}_{n\in\mathbb{N}}$ diverges to $-\infty$
and by the relation (16) the continued fraction converges in $\mathbb{Q}_{p}$.
However, requiring the sequence $\\{v_{p}(B_{n}B_{n+1})\\}_{n\in\mathbb{N}}$
strictly decreasing is equivalent to ask that $v_{p}(B_{n+1})<v_{p}(B_{n-1})$
for all $n\geq 1$. Therefore, the condition of Proposition 6 can be lightened
by allowing partial quotient at even steps to have null valuation. In fact,
the convergence of Browkin II relies on the following proposition.

######  Proposition 8 ([21]).

Let an infinite sequence $a_{0},a_{1},\ldots\in\mathbb{Z}[\frac{1}{p}]$ such
that for all $n\in\mathbb{N}$:

$\begin{cases}v_{p}(a_{2n})=0\\\ v_{p}(a_{2n+1})<0.\end{cases}$ (19)

Then the continued fraction $[a_{0},a_{1},\ldots]$ is convergent to a $p$-adic
number.

In both the proofs of Proposition 6 and Proposition 8, the sequence
$\\{v_{p}(B_{n}B_{n+1})\\}_{n\in\mathbb{N}}$ diverges to $-\infty$ because it
is a strictly decreasing sequence of integers. In [59], the strict decrease of
this sequence has been effectively characterized in terms of valuations of the
partial quotients.

######  Theorem 9 ([59]).

Let $a_{0},a_{1},\ldots\in\mathbb{Z}\left[\frac{1}{p}\right]$ be an infinite
sequence such that

$v_{p}(a_{n}a_{n+1})<0,$

for all $n>0$. Then the continued fraction $[a_{0},a_{1},\ldots]$ is
convergent to a $p$–adic number.

In the same paper, the authors explored some convergence conditions with
partial quotients not satisfying the hypothesis of 9. They proved the
following generalization of Proposition 8.

######  Theorem 10 ([59]).

Let $a_{0},a_{1},\ldots\in\mathbb{Q}_{p}$ such that, for all $n\in\mathbb{N}$:

$\begin{cases}v_{p}(a_{3n+1})<0\\\ v_{p}(a_{3n+2})=0\\\
v_{p}(a_{3n+3})=0.\end{cases}$

If $v_{p}(a_{3n+3}a_{3n+2}+1)=0$ for all $n\in\mathbb{N}$, then,

$v_{p}(B_{3n-2})=v_{p}(B_{3n-1})=v_{p}(B_{3n})>v_{p}(B_{3n+1}).$

Theorem 10 allows to define Algorithm (14), which satisfies both the
convergence condition and the additional restriction on partial quotients. In
[59], Theorem 10 has been also generalized to an arbitrary number of steps by
adding more hypotheses on the valuation of partial quotients.

##  5\. Continued fractions of rational numbers

In the field of real numbers, the rational numbers are characterized by finite
continued fractions. This is a natural consequence of the finiteness of the
Euclidean division algorithm. In fact, for all $a,b\in\mathbb{Z}$, $b\neq 0$,
the simple continued fraction of $\frac{a}{b}$ is obtained by iterating

$\frac{a}{b}=\frac{bq+r}{b}=q+\frac{r}{b}=q+\frac{1}{\frac{b}{r}},$ (20)

where $b=\lfloor a\rfloor$ and, hence, $|r|<|b|$. For $p$–adic versions of the
Euclidean algorithm, see [34, 47].

In [27], Bundschuh proved that Schneider’s continued fractions of rational
numbers are not always finite, but they can also be periodic. In [40], the
authors gave a combinatorial characterization of some non-terminating
expansions. Very recently, Pejkovic [65] proved the following effective
criterion for determining whether Schneider’s continued fraction of a rational
number terminates.

######  Theorem 11 ([65]).

Let $\alpha=\frac{a}{b}\in\mathbb{Q}$. Schneider’s continued fraction of
$\alpha$ either terminates or a period is detected within
$O(\log^{2}H(\alpha))$ steps, where
$H(\alpha)=H(\frac{a}{b})=\max\\{|a|,|b|\\}$.

For Ruban’s continued fractions, Laohakosol [49] proved the following, which
is the analogue of Bundschuh’s result.

######  Theorem 12 ([49]).

A $p$–adic number is rational if and only if either its Ruban’s continued
fraction terminates or it is eventually periodic with all partial quotients
equal to $p-\frac{1}{p}$ from a certain point onward.

######  Remark 13.

A $p$–adic number that has a finite Ruban’s continued fraction is, by
construction, rational and positive. Therefore, negative numbers can not have
a finite Ruban’s continued fractions. For example,

$\displaystyle-p=(p-1)p+(p-1)p^{2}+\ldots=\cfrac{1}{\frac{p-1}{p}+(p-1)+\cfrac{1}{\frac{p-1}{p}+(p-1)+\ddots}},$

that is, $-p=\left[\overline{p-\frac{1}{p}}\right]$.

In the proof of Theorem 12, Laohakosol showed that all the continued fractions
of rational numbers that are not finite, eventually have $-p$ as complete
quotient. A proof of the characterization of rational numbers through this
algorithm can be found also in [86]. Ruban’s continued fractions have been
deepened in more details by Capuano, Veneziano and Zannier [31]. In
particular, they provided an effective algorithm that determines in a finite
number of steps whether the continued fraction of a rational number terminates
or not. Moreover, in the non-terminating case, a negative complete quotients
appears in the expansion of $\alpha=\frac{a}{b}$ in at most
$\max\\{2,\frac{\log b}{\log p}\\}$ steps. Finally, they proved the following
result that, giving a rational number $\alpha$, establishes how its Ruban’s
continued fraction changes when varying $p$.

######  Proposition 14 ([31]).

Let $\alpha\in\mathbb{Q}$. Then:

  1. i)

If $\alpha<0$, then for every prime number $p$, Ruban’s continued fraction of
$\alpha$ does not terminate,

  2. ii)

If $\alpha\geq 0$ and $\alpha\in\mathbb{Z}$, there are only finitely many
prime numbers $p$ such that Ruban’s continued fraction of $\alpha$ does not
terminate,

  3. iii)

If $\alpha\geq 0$ and $\alpha\not\in\mathbb{Z}$, there are only finitely many
prime numbers $p$ such that Ruban’s continued fraction of $\alpha$ terminates.

The first definition of $p$–adic continued fractions that terminate for every
rational number is Browkin’s first algorithm (11). This is due to the choice
of the representatives in $\\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\\}$ instead
of $\\{0,\ldots,p-1\\}$. In this case, in fact, the Euclidean absolute value
of the partial quotients satisfies the following inequality:

$|s(\alpha)|=\left|\sum\limits_{n=-r}^{0}c_{n}p^{n}\right|\leq\frac{p-1}{2}\left|\sum\limits_{n=-r}^{0}p^{n}\right|<\frac{p}{2}.$
(21)

In order to give an idea of the proof of finiteness for Browkin’s algorithm,
let us write complete quotients of two consecutive steps

$\alpha_{n}=\frac{A_{n}}{p^{j}B_{n}},\ \
\alpha_{n+1}=\frac{A_{n+1}}{p^{k}B_{n+1}},$

with $j,k\geq 1$, $(A_{n},B_{n})=(A_{n+1},B_{n+1})=1$ and
$p\not|A_{n}B_{n}A_{n+1}B_{n+1}$. Then, by inequality (21) and the recursion

$\alpha_{n+1}=\frac{1}{\alpha_{n}-a_{n}},$

it is obtained, for all $n\in\mathbb{N}$,

$|B_{n+1}|<\frac{|A_{n}|}{2}+\frac{|B_{n}|}{2}.$ (22)

The latter inequality allows to prove that
$\\{|A_{n}|+2|B_{n}|\\}_{n\in\mathbb{N}}$ is a strictly decreasing sequence.
Since it is a sequence of natural numbers, then it is finite. It means that
there is only a finite number of complete quotients in Browkin’s expansion of
a rational number. Some results about the complexity of Browkin I continued
fractions for rational numbers can be found in [17]. The finiteness of
Browkin’s second algorithm (12) is less straightforward, hence Browkin left it
open in [21] as a conjecture. Most recently, Barbero, Cerruti and Murru [10]
proved it by providing an inequality similar to (21) also for the second floor
function $t$: for any $p$–adic number
$\alpha=\sum\limits_{n=-r}^{+\infty}c_{n}p^{n}$,

$\left|t(\alpha)\right|=\left|\sum\limits_{n=-r}^{-1}c_{n}p^{n}\right|<\frac{1}{2}.$
(23)

For the odd partial quotients $a_{2n+1}$, where the function $t$ is eventually
adjusted with the sign function,

$\left|a_{2n+1}\right|\leq 1-\frac{1}{p^{l}},$ (24)

where $l=-v_{p}(\alpha_{2n+1})$. Using (21) and (24), the authors provided two
inequalities similar to (22), one for the odd steps and one for the even steps
of the algorithm. This led to a strictly decreasing sequence of natural
numbers similar to the one obtained by Browkin. Also the $p$–adic continued
fractions algorithms defined in [57] and [59] terminate on rational inputs and
the ideas for the proofs are similar. The proof of finiteness for the
algorithm proposed in [11] is slightly different and it requires some further
remarks on the absolute values of the denominators of the convergents. In
fact, it has to handle with standard representatives in $\\{0,\ldots,p-1\\}$
and it is not possible to use inequalities (21) and (23).

##  6\. Periodicity properties

The continued fraction expansion of a real number is eventually periodic if
and only it is a quadratic irrational number, by Lagrange’s Theorem. Also for
$p$–adic continued fractions, periodicity is often related to quadratic
irrationals. In fact, a periodic continued fraction can be regarded as the
root of a quadratic polynomial. However, we have seen in the previous section
that periodic $p$–adic continued fractions can correspond also to rational
numbers. On the other hand, none of the existent algorithms has been proved to
produce a periodic continued fractions for all quadratic irrationals in
$\mathbb{Q}_{p}$. Therefore, none of the two implications of Lagrange’s
Theorem is true in general for $p$–adic continued fractions. The latter is one
of the most challenging open problems in this research area. This section is
devoted to the main results regarding the periodicity of $p$–adic continued
fractions up to the most recent developments.

######  Remark 15.

Periodic continued fractions, when not representing a rational number,
converge to some irrational number $\alpha\in\mathbb{Q}_{p}$ that is quadratic
over $\mathbb{Q}$, i.e. it is a root of an irreducible polynomial
$f(x)\in\mathbb{Q}[x]$ of degree $2$. We denote by $\overline{\alpha}$ the
conjugate of $\alpha$, that is the other root of $f(x)$ over $\mathbb{Q}$.
Moreover, when we write $\sqrt{D}$ as a root of $x^{2}-D$ over
$\mathbb{Q}_{p}$, it must be clear which of the two roots we are meaning. In
the case of standard representatives in $\\{0,\ldots,p-1\\}$, we choose the
root that in its $p$–adic expansion has the smaller first representative,
while in the case of representatives in
$\\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\\}$ we take the one with positive
first representative. We denote the other root by $-\sqrt{D}$.

Let $\alpha_{0}=\frac{P_{0}+\sqrt{D}}{Q_{0}}\in\mathbb{Q}_{p}$, where $D$ is a
non-square integer that is a quadratic residue modulo $p$ and
$P_{0},Q_{0}\in\mathbb{Q}$. For all $n\in\mathbb{N}$, the complete quotients
of the continued fraction expansion of $\alpha$ can be written as

$\alpha_{n}=\frac{P_{n}+\sqrt{D}}{Q_{n}},$ (25)

where $P_{n},Q_{n}\in\mathbb{Q}$. Similarly as in the real framework, the
sequences $P_{n}$ and $Q_{n}$, for all $n\in\mathbb{N}$ can be computed
recursively, starting from $P_{0}$ and $Q_{0}$, by

$P_{n+1}=a_{n}Q_{n}-P_{n},\ \ \ \ \ \ \ Q_{n+1}=\frac{D-P^{2}_{n+1}}{Q_{n}},$

where $a_{n}$ is the $n$-th partial quotient of $\alpha_{0}$.

######  Remark 16.

A famous result of Galois [35] states that the continued fraction expansion of
a quadratic irrational $\alpha\in\mathbb{R}$ is purely periodic, i.e. periodic
without a pre-periodic part, if and only if it is reduced, that is $\alpha>1$
and its conjugate $-1<\overline{\alpha}<0$. One of technique to prove
Lagrange’s Theorem [48] in $\mathbb{R}$ consists in showing that the expansion
of any quadratic irrational eventually reaches a reduced complete quotients,
hence it starts to be periodic. Equivalently, it is possible to prove that
there are only finitely many choices for the values of $P_{n}$ and $Q_{n}$ in
(25) for the expansions of a quadratic irrational. Therefore there is a finite
number of complete quotients and the continued fractions must become periodic
at some point.

One of the first works on the periodicity of $p$–adic continued fractions is
due to Bundschuh [27]. He studied Schneider’s continued fractions and
suggested that Lagrange’s Theorem fails for this algorithm through some
numerical computations, although he did not prove it. Some years later, de
Weger [88] proved the following criterion for the non-periodicity of
Schneider’s continued fractions.

######  Proposition 17 ([88]).

Let $P_{n}$ and $Q_{n}$ as in (25). Then, if for some $n$ the signs of $P_{n}$
and $Q_{n}$ are different and $P_{n+1}^{2}>c$, Schneider’s continued fraction
of $\sqrt{c}\in\mathbb{Z}_{p}$ is not periodic. In particular,
$\sqrt{c}\in\mathbb{Z}_{p}$ is never periodic for $c<0$.

In [87], the same author approached the periodicity of $p$–adic continued
fractions from another point of view, i.e. by associating a sequence of
approximation lattices to every $p$–adic number. In the spirit of Lagrange’s
Theorem, de Weger proved that the sequence of approximation lattices attached
to $\alpha\in\mathbb{Z}_{p}$ becomes periodic if and only if $\alpha$ is a
quadratic irrational. However, this method is not effective for the
construction of a periodic continued fraction for a given $p$–adic quadratic
irrational. Few years later, Tilborghs [84] determined an algorithm to detect
the non-periodicity of Schneider’s continued fraction in a finite number of
steps. Becker [12] then showed the following result on the length of the pre-
periods.

######  Proposition 18 ([12]).

Let $\alpha\in\mathbb{Z}_{p}$ be quadratic over $\mathbb{Q}$ with periodic
Schneider’s continued fraction. Then, if $p$ does not divide the discriminant
of $\alpha$, the pre-period length is at most $1$.

A fairly complete survey on the periodicity of Schneider’s continued fractions
is contained in [68]. More recently, this algorithm has been studied also from
other points of view and in more generality in [37, 38, 41].

For Ruban’s algorithm, using an argument similar to de Weger, Ooto [64] showed
that not all quadratic irrationals in $\mathbb{Q}_{p}$ have a periodic Ruban’s
continued fraction. In particular, he proved the following result, giving some
sufficient conditions to have a non-periodic $p$–adic continued fractions.

######  Proposition 19 ([64]).

Let $\alpha=\sqrt{D}\in\mathbb{Q}_{p}$ and the sequences $P_{n},Q_{n}$ as in
(25) for some $n\in\mathbb{N}$, $P_{n}Q_{n}\leq 0$ and $P^{2}_{n+1}>D$, then
the Ruban’s continued fraction of $\alpha$ is not periodic.

The idea of the proof of Proposition 19 is to show that, in these hypotheses,
the sequence $P_{n}$ has strictly increasing absolute value from some point
onward, hence it can not be periodic. From the latter proposition it easily
follows that, for example, $\sqrt{D}$ can not have a periodic continued
fraction if $D<0$. In fact, in this case, $P_{0}Q_{0}=0$ and $P_{1}\geq 0>D$
(in analogy to Proposition 17 for Schneider’s algorithm).
In [31], Capuano, Veneziano and Zannier made a more extensive analysis of
Ruban’s algorithm. In particular, they provided an effective criterion for
determining in a finite number of step whether the expansion of a quadratic
irrational becomes periodic. They proved the following result.

######  Theorem 20 ([31]).

Let $\alpha\in\mathbb{Q}_{p}$ be a quadratic irrational over $\mathbb{Q}$.
Then, Ruban’s continued fraction of $\alpha$ is periodic if and only if there
exists a unique real embedding $j:\mathbb{Q}(\alpha)\rightarrow\mathbb{R}$
such that the image of each complete quotient $\alpha_{n}$ under the map $j$
is positive. Moreover, there exists an effectively computable constant
$N_{\alpha}$ with the property that, either exists $n\leq N_{\alpha}$ such
that $\alpha_{n}$ does not have a positive real embedding, therefore the
expansion is not periodic, or $\alpha_{n_{1}}=\alpha_{n_{1}}$ for
$n_{1}<n_{2}\leq N_{\alpha}$, hence the expansion is periodic.

The idea behind the proof of Theorem 20 is similar to the proof of Lagrange’s
Theorem pointed out in Remark 16. First of all, they proved an analogue of
Galois’ Theorem for classical continued fractions, finding a necessary
condition for purely periodic continued fractions. This result is similar to
Theorem 21 for Browkin I, since it is not affected by a different choice of
representatives. Then, the authors used this necessary condition on the
$p$–adic norm to prove that there are only finitely many quadratic irrational
$\alpha_{n}$ of the form (25) such that $\alpha_{n}$ has a purely periodic
Ruban’s continued fractions. Moreover, they showed that having two negative
embeddings for some complete quotient $\alpha_{n}$, is a sufficient condition
for the non-periodicity of the expansion. Finally, they effectively computed a
constant $N_{\alpha}$ such that, for $n\leq N_{\alpha}$, either one of the
$\alpha_{n}$ has a purely periodic Ruban’s continued fractions or it has two
negative embeddings.

The situation is more complicated for Browkin’s algorithms (11) and (12),
where the problem of deciding whether the $p$–adic continued fraction of a
quadratic irrational is eventually periodic is still open. However, some
numerical simulations suggest that this is not always the case (see, for
example, [10, 11, 20, 21, 29, 57]). As we have seen in Section 5, for these
two algorithms finite continued fractions characterize rational numbers.
Therefore, all periodic continued fractions correspond to irrational elements
that are quadratic over $\mathbb{Q}$. The first systematic study of the
periodicity of Browkin I has been performed by Bedocchi in [13, 14, 15, 16].
The first results concern purely periodic continued fractions.

######  Theorem 21 ([13]).

Let $\alpha\in\mathbb{Q}_{p}$ having a periodic Browkin I continued fraction
expansion. Then the expansion is purely periodic if and only if

$|\alpha|_{p}>1,\quad|\overline{\alpha}|_{p}<1.$

This result is the analogue of Galois’ Theorem for classical continued
fractions and it is similar both in the statement and the proof. Moreover, he
characterized the possible lengths of pre-periods for periodic Browkin I
continued fractions.

######  Proposition 22 ([13]).

Let $D\in\mathbb{Z}$ not a perfect square, such that
$\sqrt{D}\in\mathbb{Q}_{p}$. If the Browkin I expansion of $\sqrt{D}$ is
periodic, then the pre-period is

$\displaystyle\begin{cases}2\ \ \ \ \ &D\not\equiv 4\mod 8\ \text{when}\
p=2\\\ 3\ \ \ \ \ &otherwise.\end{cases}$

Then, in [14, 15, 16], Bedocchi focused on the possible lengths of the periods
for square roots of integers. We collect all the results in the next
proposition

######  Proposition 23 ([14, 15, 16]).

For Browkin I, the following statements are true.

  1. i)

There are no periodic square roots of integers with period of length $1$;

  2. ii)

For every odd integer $d$, there are only finitely many square roots of
integers that are periodic with period of length $d$;

  3. iii)

There exist infinitely many square roots of integers that are periodic with
period of length $2$, $4$ and $6$.

In view of the results of Proposition 23, Bedocchi conjectured that, for all
$h$ even, there exist infinitely many square roots of integers that are
periodic and have period length $h$. This conjecture is still open. However,
for all the length that are powers of $2$, it has been proved by Capuano,
Murru and Terracini in [29].

######  Theorem 24 ([29]).

For every $n,k\geq 1$, there are infinitely many $D\in\mathbb{Z}$, with
$p\not|D$, such that the Browkin I expansion of $p^{k}\sqrt{D}$ is periodic
with period of length $2^{n}$.

The proof of Theorem 24 exploits a new definition of a particular class of
Browkin’s continued fractions, that they call nice, for which they proved the
following theorem.

######  Theorem 25 ([29]).

Let $[a_{0},a_{1},\ldots,a_{t-1}]$ be a nice Browkin’s continued fractions,
for some $t\in\mathbb{N}$. Then there are infinitely many Browkin’s partial
quotients $a_{t}$ such that the continued fraction

$[a_{0},\overline{a_{1},\ldots,a_{t-1},a_{t},a_{t-1},\ldots,a_{1},2a_{0}}]$

converges to a quadratic irrational number of the form
$\frac{1}{p^{e}\sqrt{D}}$, for some $D\in\mathbb{Z}$ not a perfect square.

Basically, using Theorem 25, starting with a nice sequence
$[a_{0},a_{1},\ldots,a_{t-1}]$ it is possible to provide an infinite family of
integers $D$ such that the Browkin’s continued fraction of $p^{e}\sqrt{D}$ is
periodic with period $2t$. They left the following conjecture, together with
other problems about nice continued fractions.

######  Conjecture 26.

For every $t\geq 1$ there exist a nice Browkin’s continued fraction of length
$t$, except when $t=1$ and $p=3$.

By Theorem 25, solving the latter conjecture implies the existence of
infinitely many square roots of integers that have periodic Browkin I
expansion with period $2t$, for any $t$. Thus, it would give a positive answer
to Bedocchi’s conjecture. Moreover, in the same paper, the authors deepened
other aspects of the periodicity of Browkin I. The aim was to use some
arguments similar to those of [31] in order to prove an effective criterion to
predict the periodicity and the non-periodicity of Browkin I. Unfortunately,
Theorem 20 strongly depends on the fact that a periodic Ruban’s continued
fraction can always be embedded in the real numbers. This is not always true
for Browkin’s continued fractions, hence it is not a necessary condition for
periodicity. On the other hand, having two negative embeddings is not a
sufficient condition for non-periodicity (see [29] for more details). This
important difference between Ruban’s and Browkin’s continued fractions is
again due to the choice of the (possibly negative) representatives in Browkin
I. For these reasons, there is not an “easy” way to prove non-periodicity in
Browkin I up to now. In particular, no quadratic irrational has been proved to
have non-periodic Browkin I continued fraction, although it is largely
believed to fail Lagrange’s Theorem (see, for example, the experimental
computations in [57]). The SageMath code that we have developed for [57] is
publicly available111https://github.com/giulianoromeont/p-adic-continued-
fractions and contains the implementation of the main algorithms presented in
Section 3.

In [1], the authors followed another approach to find periodic $p$–adic
continued fractions for all the square roots of integers in $\mathbb{Q}_{p}$.
They did not focus on a specific algorithm and they used Rédei rational
functions [71] to construct a periodic continued fraction converging to
$\sqrt{D}$ simultaneously in the real and the $p$–adic field. It is possible
to notice that, for any integer $z$,

$\displaystyle\sqrt{D}$
$\displaystyle=z+(\sqrt{D}-z)=z+\frac{1}{\frac{1}{\sqrt{D}-z}}=z+\frac{1}{\frac{\sqrt{D}+z}{D-z^{2}}}=$
$\displaystyle=z+\frac{1}{\frac{2z}{D-z^{2}}+\frac{\sqrt{D}-z}{D-z^{2}}}=z+\frac{1}{\frac{2z}{D-z^{2}}+\frac{1}{z+\sqrt{D}}},$

hence

$\sqrt{D}=\left[z,\overline{\frac{2z}{d-z^{2}},2z}\right],$ (26)

even if it does not coincide with its expansion with the standard algorithm
(see [66]). Rédei rational functions are known to converge to $\sqrt{D}$ in
$\mathbb{R}$ (see, for example, [52]). In [1], the authors proved that Rédei
functions converge to $\sqrt{D}$ also in $\mathbb{Q}_{p}$, so that (26) is
true also for $p$–adic numbers. This construction has been generalized in [10]
in order to manage any quadratic irrational, not only square roots of
integers. The authors proposed a generalization of Rédei rational functions to
provide a periodic continued fraction for $\alpha$ root of the polynomial
$x^{2}+hx-d$, where $h,d\in\mathbb{Z}$. The expansion is

$\alpha=\left[z,-\overline{\frac{h+2z}{z^{2}+hz-d},h+2z}\right],$ (27)

that yields a periodic representation for $\alpha$ both in $\mathbb{R}$ and
$\mathbb{Q}_{p}$. This result solves the problem of expressing every $p$–adic
quadratic irrational number as a periodic continued fraction, i.e. through a
periodic sequence of rational numbers. However, it is not equivalent to
Lagrange’s Theorem, since the expansion is not obtained by a specific
algorithm. In fact, in order to find the expansion (27) for $\alpha$, we need
from the begin the knowledge that it is a quadratic irrational number and, in
addition, of its characteristic polynomial over $\mathbb{Q}$. In the same
paper, the authors characterized the cases when Browkin II provides a periodic
expansion of the form (27).

######  Theorem 27 ([10]).

Given $\alpha\not\in\mathbb{Q}$ root of the polynomial $x^{2}+hx-d$, with
$h,d\in\mathbb{Z}$, Browkin II produces the $p$–adic continued fraction

$\alpha=\left[z,-\overline{\frac{h+2z}{p},h+2z}\right],$ (28)

if and only if

$1\leq|z|\leq\frac{p-1}{2},\ \ \ 1\leq|h+2z|\leq\frac{p-1}{2},$

for $z$ such that $z^{2}+hz-d=p$.

The authors left open the problem of finding an actual algorithm that, for all
quadratic irrationals in $\mathbb{Q}_{p}$, provides periodic representations
of the form (27). Recently, the properties of periodicity of Browkin’s second
algorithm (12) have been considered. In [58], an approach similar to Bedocchi
in [13, 14, 15, 16] is employed to characterize purely periodic continued
fractions and to study the lengths of pre-periods and periods for periodic
expansion. The situation is more complicated than Browkin I, because of the
presence of the sign function in (12) and it is possible to obtain only some
partial results. In fact, following an argument similar to the proof of
Theorem 21, the condition on the $p$–adic absolute values of $\alpha$ and its
conjugate are only necessary.

######  Theorem 28 ([13]).

If $\alpha\in\mathbb{Q}_{p}$ has a purely periodic continued fraction
expansion

$\alpha=[\overline{a_{0},\ldots,a_{k-1}}],$

with Browkin II, then

$|\alpha|_{p}=1,\ \ |\overline{\alpha}|_{p}<1.$

Conversely, if $\alpha$ has a periodic Browkin II expansion

$\alpha=[a_{0},a_{1},\ldots,a_{h-1},\overline{a_{h},\ldots,a_{h+k-1}}],$

and $|\alpha|_{p}=1,|\overline{\alpha}|_{p}<1$, then the preperiod length can
not be odd.

Theorem 28 can not be improved, since there are counterexamples of periodic
continued fractions for elements that satisfy hypotheses, but are not purely
periodic. Moreover, in the spirit of Bedocchi’s Proposition 23, some results
have obtained on the possible pre-periods and periods of Browkin II
expansions.

######  Proposition 29 ([58]).

The following results hold for Browkin II:

  1. i)

If $\sqrt{D}\in\mathbb{Q}_{p}$ has a periodic continued fraction, then the
pre-period has length either $1$ or even;

  2. ii)

There are infinitely many $\sqrt{D}\in\mathbb{Q}_{p}$ having a periodic
$p$–adic expansion with period $4$.

Supported by experimental computations, the authors left the following
conjecture, which is the exact analogue of Bedocchi’s conjecture for Browkin’s
second algorithm.

######  Conjecture 30 ([58]).

For all even $h\in\mathbb{Z}$, there exist infinitely many $\sqrt{D}$,
$D\in\mathbb{Z}$ not perfect square, such that Browkin II continued fraction
of $\sqrt{D}$ is periodic with period of length $h$.

In [57], the authors proved some further results on the periodicity Browkin
II, underlining some issues arising from the unpredictability of the
appearance of the sign function in the algorithm. Algorithm (15), introduced
in [57], improves Browkin II from both a theoretical and an experimental point
of view.

######  Theorem 31 ([57]).

The following results hold for Algorithm (15):

  1. i)

If $\alpha\in\mathbb{Q}_{p}$ has a periodic continued fraction, then it is
purely periodic if and only if $|\alpha|_{p}\geq 1$ and
$|\overline{\alpha}|_{p}<1$;

  2. ii)

If $\sqrt{D}$ has a periodic continued fraction, then the pre-period is $1$
for $v_{p}(\sqrt{D})\leq 0$ and $2$ for $v_{p}(\sqrt{D})>0$

Theorem 31 establishes that an analogue of Galois’ Theorem is true for
Algorithm (15), unlike Theorem 28 for Browkin II. Moreover, the second part is
very similar to what happens in $\mathbb{R}$, where the pre-periods of square
roots of integers is always $1$.

##  7\. Quality of the approximation

Classical continued fractions are one of the most powerful tools to produce
rational approximations of real numbers. In fact, the convergents provide at
each step the best approximation of real numbers, in the sense of the
following proposition.

######  Proposition 32.

Let $\alpha\in\mathbb{R}$ and $\\{\frac{A_{n}}{B_{n}}\\}_{n\in\mathbb{N}}$ the
sequence of convergents of $\alpha$. Then

$|B_{n}\alpha-A_{n}|\leq|B\alpha-A|,$

for all $n\in\mathbb{N}$ and $A,B$ integers such that $0<B\leq B_{n}$.

######  Remark 33.

In other words, Proposition 32 tells us that there is no rational number with
denominator minor than $B_{n}$ that approximates $\alpha$ better than
$\frac{A_{n}}{B_{n}}$ and, fixing $B_{n}$ as denominator, the numerator
$A_{n}$ is the best possible. Therefore, the approximations provided by the
convergents of a continued fraction are the best approximations without
increasing the denominator (of course, with a bigger denominator, it is
possible to obtain better approximations).

In this section we analyze the quality of the approximations provided by
continued fractions in $\mathbb{Q}_{p}$. First of all, let us give a more
precise shape to the approximations given by the convergents of a $p$–adic
continued fraction. Let $\alpha\in\mathbb{Q}_{p}$ such that
$\alpha=[a_{0},a_{1},\ldots]$. We can write
$\alpha=[a_{0},a_{1},\ldots,a_{n},\alpha_{n+1}]$, so that, by (3),

$\alpha=\frac{\alpha_{n+1}A_{n}+A_{n-1}}{\alpha_{n+1}B_{n}+B_{n-1}}.$

Therefore, the difference between $\alpha$ and its $n$-th convergent is

$\alpha-\frac{A_{n}}{B_{n}}=\frac{\alpha_{n+1}A_{n}+A_{n-1}}{\alpha_{n+1}B_{n}+B_{n-1}}-\frac{A_{n}}{B_{n}}=\frac{(-1)^{n}}{(\alpha_{n+1}B_{n}+B_{n-1})B_{n}}.$

Since $v_{p}(\alpha_{n+1})=v_{p}(a_{n+1})$, then
$v_{p}(\alpha_{n+1}B_{n})<v_{p}(B_{n-1})$, hence

$v_{p}(\alpha_{n+1}B_{n}+B_{n-1})=v_{p}(\alpha_{n+1}B_{n})=v_{p}(a_{n+1}B_{n})=v_{p}(B_{n+1}).$

The valuation of the difference is then

$v_{p}\left(\alpha-\frac{A_{n}}{B_{n}}\right)=-v_{p}(B_{n}B_{n+1}),$ (29)

that is,

$\left|\alpha-\frac{A_{n}}{B_{n}}\right|_{p}=p^{v_{p}(B_{n}B_{n+1})}.$ (30)

By Remark 5, we know that, for all $n\in\mathbb{N}$,

$v_{p}(B_{n})=v_{p}(a_{0})+v_{p}(a_{1})+\ldots+v_{p}(a_{n}).$

Therefore, by (29) and (30), the quality of approximation of $\alpha$ depends
on the size of the valuation of the denominators $v_{p}(B_{n})$. In fact, the
algorithm that better approximates a $p$–adic number $\alpha$ is the one where
$v_{p}(B_{n})$ becomes “very negative” rapidly.

######  Remark 34.

By Proposition 6 and Proposition 8, we know that Browkin I decreases the
valuation of $B_{n}$ at each step, while Browkin II only at odd steps.
Therefore, at each step $n\in\mathbb{N}$, we expect Browkin I to produce
better approximations than Browkin II.

Some computational results on the quality of approximations for square roots
of integers by means of Algorithms (11), (12) and (15) can be found in [57]
and they empirically confirm the insight of Remark 34.

In [55], Mahler attempted to obtain a sequence of best approximations for a
$p$–adic number $\alpha$, similarly to the sequence of convergents for real
continued fractions. Let $\Phi(X,Y)$ be a reduced positive definite quadratic
form of determinant $-1$ (this is in analogy with the determinant of the
matrices appearing in (4)). Mahler defined an algorithm to provide a sequence
of pairs of integers $\\{(A_{n},B_{n})\\}_{n\in\mathbb{N}}$, such that, for
all $n\in\mathbb{N}$,

$|A_{n}-B_{n}\alpha|_{p}\leq\frac{1}{p^{n}},$ (31)

and

$|A_{n}-B_{n}\alpha|_{p}<|A-B\alpha|_{p},$

for all the other pairs $(A,B)$ with $0<\Phi(A,B)<\Phi(A_{n},B_{n})$. Notice
that this result is very similar to Proposition 32 for real continued
fractions, i.e. it tries to reproduce the idea of best approximations also in
the $p$–adic framework. In [32], Deanin focused on Mahler’s approach and
studied the periodicity of the sequence $(A_{n},B_{n})$ by varying the
function $\Phi$. As already mentioned in Section 6, de Weger [87] studied
rational approximations in $\mathbb{Q}_{p}$ by associating to each $p$–adic
number a sequence of approximation lattices. For all $n\in\mathbb{N}$, the
$n$-th approximation lattice of $\alpha\in\mathbb{Q}_{p}$ is

$\Gamma_{n}=\\{(A,B)\in\mathbb{Z}^{2}:|B\alpha-A|_{p}\leq p^{-n}\\},$

i.e. the lattice containing all the pairs of integers satisfying (31). In this
context, de Weger was able to prove a form of Hurwitz’s Theorem [42], together
with other known results of Diophantine approximation, over $\mathbb{Q}_{p}$.
In [24, 26, 25, 8], Schneider’s $p$–adic continued fractions are employed for
the study of simultaneous uniform approximations for a $p$–adic number and its
powers. These problems are similar to those addressed in the real framework in
[23, 51, 8] (see also [22] for a survey on the topic). Simultaneous
approximations of two $p$–adic numbers have been studied also in [62] using
multidimensional continued fractions. In fact, by definition, continued
fractions in two dimensions provide simultaneous approximations by means of
two sequences of integers (more details are given in Section 9, which is
devoted to the study of multidimensional $p$-adic continued fractions).

##  8\. Transcendental $p$–adic continued fractions

Continued fractions have been proved to be very efficient for the construction
of transcendental numbers. Over the real numbers, Khinchin [46] addressed the
problem of continued fraction expansions of algebraic irrational numbers which
are not quadratic over $\mathbb{Q}$. Liouville [53] was the first to construct
transcendental numbers by using continued fractions with unbounded partial
quotients. Transcendental continued fractions with bounded partial quotients
have been defined by Maillet [56] and Baker [9]. An extensive analysis has
been lately performed in [2, 3, 4, 5], where the authors showed several
transcendence criteria and provided many examples of families of
transcendental continued fractions. One of the most powerful tool is the use
of Roth’s Theorem [74] and its subsequent improvements, in particular the
Schmidt’s Subspace Theorem [82]. Let us recall Roth’s Theorem in one of its
simplest forms.

######  Theorem 35 (Roth’s Theorem).

Let $\alpha$ be an algebraic irrational over $\mathbb{Q}$ and let
$\epsilon>0$. Then there exist only finitely many $(a,b)\in\mathbb{Z}^{2}$,
$b>0$, such that

$\left|\alpha-\frac{a}{b}\right|<\frac{1}{b^{2+\epsilon}}.$

Roth’s Theorem tells us that an algebraic irrational number admits only
finitely many “good” approximations. Therefore, it is possible to prove the
transcendence of a real number by constructing infinitely many “good enough”
rational approximations. In [80], Schlickewei proved the following $p$–adic
version of the Subspace Theorem.

######  Theorem 36 ([80]).

Let $\mathbf{x}=(x_{1},\ldots,x_{n})\in\mathbb{Q}^{n}$ and let us define

$\|\mathbf{x}\|_{\infty}=\max\\{|x_{i}|,i=1,\ldots,n\\},\ \
\|\mathbf{x}\|_{p}=\max\\{|x_{i}|_{p},i=1,\ldots,n\\}.$

Let $L_{1,\infty}(\mathbf{x}),\ldots,L_{n,\infty}(\mathbf{x})$ and
$L_{1,p}(\mathbf{x}),\ldots,L_{n,p}(\mathbf{x})$ be, respectively, real and
$p$–adic linearly independent linear forms with algebraic coefficients and let
$\epsilon>0$. Then, the solutions $\mathbf{x}\in\mathbb{Z}^{n}$ of

$\prod\limits_{i=1}^{n}(|L_{i,\infty}(\mathbf{x})|\cdot|L_{1,p}(\mathbf{x})|_{p})\leq\frac{1}{\|\mathbf{x}\|_{\infty}^{\epsilon}},$

lie in finitely many proper subspaces of $\mathbb{Q}^{n}$.

Theorem 36 has been employed in [18] to prove a transcendence result for the
$p$–adic Thue-Morse continued fractions, inspired by Queffélec’s result over
the real numbers [69]. In [64], Ooto studied transcendental Ruban’s $p$–adic
continued fractions, performing an analysis similar to Baker [9]. Baker’s
result for continued fractions in $\mathbb{R}$ relies on the definition of
quasi-periodic continued fractions, i.e. continued fractions where blocks of
partial quotients repeat several times.

######  Definition 37.

Let $\\{n_{i}\\}_{i\in\mathbb{N}}$, $\\{\lambda_{i}\\}_{i\in\mathbb{N}}$ and
$\\{k_{i}\\}_{i\in\mathbb{N}}$ be sequences of positive integers. If
$n_{i+1}\geq n_{i}+\lambda_{i}k_{i}$ and $a_{m+k_{i}}=a_{m}$ for all
$m=n_{i},\ldots,n_{i}+(\lambda_{i}-1)k_{i}-1$, then the continued fraction
$[a_{0},a_{1},\ldots]$ is called quasi-periodic.

The main result of [64] is the following transcendence criterion for quasi-
periodic Ruban’s continued fractions.

######  Theorem 38 ([64]).

Let $[a_{0},a_{1},\ldots]$ be a quasi-periodic Ruban’s continued fraction
which is not periodic and let $A\geq p$. Let
$a_{n_{i}}=\ldots=a_{n_{i}+k_{i}-1}=p-p^{-1}$ for infinitely many $i$, where
the $n_{i}$ and $k_{i}$ are as in Definition 37. Then, if

$\liminf\limits_{i\rightarrow+\infty}\frac{\lambda_{i}}{n_{i}}>B,$

where $B=\frac{2\log A}{\log p}-1$, then Ruban’s continued fraction
$[a_{0},a_{1},\ldots]$ is transcendental.

Other results on the transcendence of Ruban’s continued fractions are obtained
in [6, 7], exploiting Theorem 36 and using a method similar to [5] for the
real case. A study of the transcendence Ruban’s continued fraction has been
performed also in [33, 50, 86, 88], using unbounded partial quotients.
Finally, very recently, a transcendence criterion has been proved in [54] for
Browkin I. We collected the main results in the following theorem, which
contains the $p$–adic analogues of the transcendence criteria proved by Baker
in [9] and by Adamczewski and Bugeaud in [4].

######  Theorem 39 ([54]).

Let $\alpha=[0,a_{1},a_{2},\ldots]$ be a non-periodic Browkin I $p$–adic
continued fraction. Let $\\{n_{i}\\}_{i\in\mathbb{N}}$,
$\\{\lambda_{i}\\}_{i\in\mathbb{N}}$ and $\\{k_{i}\\}_{i\in\mathbb{N}}$ be
such that $n_{i+1}\geq n_{i}+\lambda_{i}k_{i}$, with $k_{i}$ bounded, and
suppose there exist $\delta>0$ and $B$ such that $\lambda_{i}>(B+\delta)n_{i}$
for all $i$. Assume that one of the following holds:

  1. i)

the sequence $\\{|a_{n}|_{p}\\}_{n\geq 1}$ is bounded and there are infinitely
many subsequences

$a_{n_{i}}=a_{n_{i}+1}=\ldots=a_{n_{i}+k_{i}-1}=p^{-1},$

  2. ii)

the sequence $\\{|a_{n}|_{p}\\}_{n\geq 1}$ begins with arbitrarily long
palindromes,

where the $n_{i}$ and $k_{i}$ are as in Definition 37. Then $\alpha$ is
transcendental.

##  9\. Multidimensional $p$–adic continued fractions

The theory of continued fractions has been extended in several senses. One of
the most famous generalization is given by multidimensional continued
fractions. Motivated by a question of Hermite [39], Jacobi [43] introduced a
continued fraction algorithm with the purpose of providing periodic
representations for cubic irrationals, in the spirit of Lagrange’s Theorem for
quadratic irrationals. Perron [67] generalized Jacobi’s approach in order to
handle algebraic irrationalities of any degree. Hermite’s problem for
multidimensional continued fractions is still open. We refer the interested
readers to [19, 83] for the general theory of multidimensional continued
fractions and to [44, 45] for some of the most recent developments on the
Hermite’s problem. One of the first work on multidimensional continued
fractions in the field of $p$–adic numbers is due to Ruban [76, 77, 78], who
studied some of the ergodic properties of Jacobi-Perron algorithm in
$\mathbb{Q}_{p}$. In the very last years, other multidimensional continued
fractions algorithms have been defined in $\mathbb{Q}_{p}$. In particular, in
[60] and [79] the authors introduced two different multidimensional $p$–adic
continued fraction algorithms trying to generalize, respectively, Browkin’s
and Schneider’s approach (see also [70]). In [61], the authors deepened the
study of the algorithm from [60], by exploring its properties of finiteness.

The classical Jacobi-Perron algorithm for multidimensional continued fraction
works on a $m$-tuple of real numbers
$(\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)})$ and provides their
representations through an $m$-tuple of integer sequences
$(\\{a_{n}^{(1)}\\}_{n\in\mathbb{N}},\ldots,\\{a_{n}^{(m)}\\}_{n\in\mathbb{N}})$,
generated by the following algorithm, for all $n\in\mathbb{N}$:

$\displaystyle\begin{cases}a_{n}^{(i)}=\lfloor\alpha_{n}^{(i)}\rfloor,\ \
i=1,\ldots,m\\\ \alpha_{n+1}^{(i)}=\frac{1}{\alpha_{n}^{(m)}-a_{n}^{(m)}}\\\
\alpha_{n+1}^{(i)}=\frac{\alpha_{n}^{(i-1)}-a_{n}^{(i-1)}}{\alpha_{n}^{(m)}-a_{n}^{(m)}},\
\ i=2,\ldots,m.\end{cases}$

Therefore, the Jacobi-Perron algorithm express every real number of the
$m$-tuple $(\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)})$ as a continued fraction
by

$\displaystyle\begin{cases}\alpha_{n}^{(i-1)}=a_{n}^{(i-1)}+\frac{\alpha_{n+1}^{(i)}}{\alpha_{n+1}^{(1)}}\\\
\alpha_{n}^{(m)}=a_{n}^{(m)}+\frac{1}{\alpha_{n+1}^{(1)}}.\end{cases}$

Also for multidimensional continued fractions it is possible to introduce the
convergents $\frac{A_{n}^{(i)}}{A_{n}^{(m+1)}}$, where the quantities
$A_{n}^{(i)}$ are defined as follows, for all $n\in\mathbb{N}$ and for all
$i=1,\ldots,m$:

$\displaystyle\begin{cases}A_{-j}^{(i)}=\delta_{ij}\\\ A_{0}=a_{0}^{(i)}\\\
A_{n}^{(i)}=\sum\limits_{j=1}^{m+1}a_{n}^{(j)}A_{n-j}^{i},\end{cases}$

with the convention that $a_{n}^{(m+1)}=1$ for all $n\in\mathbb{N}$. The
$p$–adic Jacobi-Perron algorithm presented in [60] uses Browkin’s $s$ function
(10) in place of the floor function of the classical algorithm. Hence, it
works as follows, on an input $m$-tuple of $p$–adic numbers
$(\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)})$:

$\displaystyle\begin{cases}a_{n}^{(i)}=s(\alpha_{n}^{(i)}),\ \ i=1,\ldots,m\\\
\alpha_{n+1}^{(i)}=\frac{1}{\alpha_{n}^{(m)}-a_{n}^{(m)}},\\\
\alpha_{n+1}^{(i)}=\frac{\alpha_{n}^{(i-1)}-a_{n}^{(i-1)}}{\alpha_{n}^{(m)}-a_{n}^{(m)}},\
\ i=2,\ldots,m.\end{cases}$ (32)

The first results in [60] concern the $p$–adic convergence of a generic
multidimensional continued fractions. Some conditions on the $p$–adic absolute
value similar to those seen in Section 4 are necessary in order to guarantee
convergence.

######  Proposition 40 ([60]).

Let
$(\\{a_{n}^{(1)}\\}_{n\in\mathbb{N}},\ldots,\\{a_{n}^{(m)}\\}_{n\in\mathbb{N}})$
be the sequences of partial quotients of a multidimensional $p$–adic continued
fraction with, for all $n\in\mathbb{N}$,

$\displaystyle\begin{cases}v_{p}(a_{n}^{(1)})\leq 0\\\
v_{p}(a_{n}^{(1)})<v_{p}(a_{n}^{(i)}),\ \ \ i=2,\ldots,m+1.\end{cases}$

Then, for all $i=1,\ldots,m$, each of the sequences
$\left(\frac{A_{n}^{(i)}}{A_{n}^{(m+1)}}\right)_{n\in\mathbb{N}}$ converges to
a $p$–adic number.

When the $p$–adic Jacobi-Perron converges to an $m$-tuple of $p$–adic numbers

$(\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)}),$

then it is also strong convergent, that is,

$\lim_{n\rightarrow+\infty}|A_{n}^{(i)}-\alpha_{0}^{(i)}A_{n}^{(m+1)}|_{p}=0,$

for all $i=1,\ldots,m$. This is not the case in $\mathbb{R}$ where strong
convergence is not guaranteed in general. Unlike classical continued fractions
in one dimension, finite multidimensional continued fractions do not
characterize rational numbers.

######  Proposition 41 ([60]).

If the $p$–adic Jacobi-Perron algorithm (32) stops in a finite number of steps
on input $(\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)})\in\mathbb{Q}_{p}^{m}$,
then $1,\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)}$ are $\mathbb{Q}$-linearly
dependent.

######  Theorem 42 ([60]).

If $(\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)})\in\mathbb{Q}^{m}$, then the
$p$–adic Jacobi-Perron algorithm (32) stops in a finite number of steps.

The conditions of both the previous results are only sufficient and not
necessary. In fact, there are $\mathbb{Q}$-linearly dependent inputs for which
the algorithm is not finite but periodic, and also $\mathbb{Q}$-linearly
dependent inputs having finite continued fraction although they are not all
rationals (see [60] for examples). Moreover, in [61] the same authors
characterized some of the $\mathbb{Q}$-linearly dependent inputs giving rise
to finite continued fractions.

######  Theorem 43 ([61]).

Let $1,\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)}$ be $\mathbb{Q}$-linearly
dependent $p$–adic numbers, with

$v_{p}(a_{n}^{(j)})-v_{p}(a_{n}^{(1)})\geq j-1,$

for $j=3,\ldots,m+1$ and $n$ sufficiently large. Then the Jacobi-Perron
algorithm stops in finitely many steps on input
$(\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)})$.

Therefore, the study of the expansion of $\mathbb{Q}$-linearly dependent
$p$–adic numbers by means of the $p$–adic Jacobi-Perron algorithm is not
complete. However, it has not been observed an $m$-tuple of
$\mathbb{Q}$-linearly dependent $p$–adic numbers for which the continued
fraction is infinite and not periodic. Therefore, in [60] the authors left the
following (still open) conjecture.

######  Conjecture 44.

Let $1,\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)}$ be $\mathbb{Q}$-linearly
dependent $p$–adic numbers. Then the Jacobi-Perron algorithm on input
$(\alpha_{0}^{(1)},\ldots,\alpha_{0}^{(m)})$ is either finite or periodic.

For the periodicity of the $p$–adic Jacobi-Perron algorithm, the following
result holds, similarly to multidimensional continued fractions in
$\mathbb{R}$.

######  Theorem 45 ([61]).

A periodic $m$-dimensional continued fractions obtained by the $p$–adic
Jacobi-Perron algorithm (32), converges to an $m$-tuple of algebraic
irrationalities of degree less or equal than $m+1$.

In [62], the same authors explored the properties of approximation of the
Jacobi-Perron algorithm in two dimensions, i.e. for the simultaneous
approximation of two $p$–adic numbers. They studied the rate of convergence
and gave some precise results for length of the finite continued fraction of
two rational numbers, following the technique of the proof of finiteness of
Theorem 42. Moreover, they focused on the study of algebraic dependence of two
$p$–adic numbers. In particular, they proved the following sufficient
condition to ensure the finiteness of the $p$–adic Jacobi-Perron algorithm on
$\mathbb{Q}$-linearly dependent inputs.

######  Theorem 46 ([62]).

Let $\alpha,\beta\in\mathbb{Q}_{p}$ and let
$\left(\frac{A_{n}}{C_{n}},\frac{B_{n}}{C_{n}}\right)$ be, for all
$n\in\mathbb{N}$, the sequences of convergents of the multidimensional
continued fraction representing $(\alpha,\beta)$. Moreover, let us define

$M_{n}=\max\\{|A_{n}|_{\infty},|B_{n}|_{\infty},|C_{n}|_{\infty}\\},\ \ \
U_{n}=\max\left\\{\left|\alpha-\frac{A_{n}}{C_{n}}\right|,\left|\alpha-\frac{B_{n}}{C_{n}}\right|\right\\}.$

Then, if

$\lim\limits_{n\rightarrow+\infty}U_{n}\cdot M_{n}=0,$

either $1,\alpha,\beta$ are $\mathbb{Q}$-linearly independent or the expansion
of $(\alpha,\beta)$ is finite.

Finally, they found some conditions to define a $p$–adic multidimensional
continued fraction converging to algebraically independent numbers.

## Acknowledgments

The author is a member of GNSAGA of INdAM.

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