On a continued fraction algorithm in finite extensions of $\mathbb{Q}_{p}$ and its metrical theory

Being an indispensable tool in number theory, especially in Diophantine approximation, the study of continued fractions has attracted many mathematicians over the years. The simple or classical continued fraction expansion of a real number $\alpha$ is an expression of the form

(1)

$\displaystyle\alpha=a_{0}+\frac{1}{\displaystyle{a_{1}+\frac{1}{\displaystyle{a_{2}+\frac{1}{\displaystyle{a_{3}+_{\ddots}}}}}}},$

which is also written as $\alpha=[a_{0};a_{1},a_{2},\dots]$ with $a_{i}$’s being natural numbers and $a_{i}>0$ for $i\geq 1$ (see [15] or [19] for more details). Here, $a_{i}$’s are called the partial quotients of the continued fraction expansion of $\alpha$. If $\frac{A_{n}}{B_{n}}=[a_{0};a_{1},\dots,a_{n}]$, then the rational numbers $\frac{A_{n}}{B_{n}}$ converges to $\alpha$, and $\frac{A_{n}}{B_{n}}$ is called the $n$th convergent to the continued fraction of $\alpha$. The classical continued fraction for real numbers has nice arithmetical properties such as rational numbers have finite continued fraction expansion; convergents are the best approximants among other rational numbers; quadratic irrationals have periodic continued fraction expansion expansions and vice versa, this fact is known as Lagrange’s theorem (see [19] for more details). The reader may look at [17] for various real continued fractions apart from the classical (simple) one. The starting of continued fraction theory for complex numbers goes back to $1887$ when A. Hurwitz ([16]) described the nearest integer continued fraction algorithm in the field of complex numbers, the partial quotients being elements of the ring of Gaussian integers. He also proved a version of lagrange’s theorem as well. See [11], [10] for more recent developments and a general approach to continued fraction theory in this setup.

It is quite natural to study continued fractions in the non-Archimedean setup as well. The reader is referred to [32] for a comprehensive introduction to the theory of continued fraction and its relation to Diophantine approximation in positive characteristics. See also the survey article [22] by Lasjaunias. For continued fraction in the field of Laurent series in one indeterminate over a finite field $\mathbb{F}_{q}$, viz. $\mathbb{F}_{q}((X^{-1}))$, there is a natural choice of a set for the set of partial quotients, viz. the polynomial ring $\mathbb{F}_{q}[X]$. The continued fraction in this setup is very well-behaved. For example, any element in $\mathbb{F}_{q}(X)$ has a finite continued fraction expansion, the convergents (which naturally belong to $\mathbb{F}_{q}(X)$) provide the best approximation, in fact it is true that if $\alpha\in\mathbb{F}_{q}((X^{-1}))$ and $\frac{P}{Q}\in\mathbb{F}_{q}(X)$ is such that $\left|\alpha-\frac{P}{Q}\right|<\frac{1}{|Q|^{2}}$, then $\frac{P}{Q}$ is a convergent from the continued fraction expansion of $\alpha$. A version of Lagrange’s theorem is true as well in this setup, see [32] or [22] for more details.

In $1940$, Mahler ([25]) initiated the study of continued fractions in the field of p-adic numbers. There are mainly two types of continued fractions in the field of $p$-adic numbers, one was introduced by Schneider ([33]) in $1968$, and the other was introduced by Ruban [31] in $1970$. Rational numbers need not have finite continued fraction expansion with respect to these algorithms. See [5] for rational numbers having infinite expansion with respect to Schneider’s algorithm. In fact, Wang [34] and Laohakosol [21] independently showed that a $p$-adic number $\alpha$ is rational if and only if the Ruban continued fraction expansion of $\alpha$ is either finite or periodic. In $1978$, Browkin ([3]) modified Ruban’s algorithm and proved that every rational number has a finite continued fraction expansion. Another desirable arithmetical property of any continued fraction is periodic expansion (the periodicity property) of quadratic irrational which is known as Lagrange’s theorem in the case of real numbers. In [4], Browkin modified his algorithm further and showed that $\sqrt{m}$ has periodic continued fraction expansion for certain positive integers for $p=5$. Though, the same is not true for larger values of $p$. So, Lagrange’s theorem is not true in this setup. See also references cited there in [4] for various work related to periodicity prior to Browkin’s ([4]) work in $2000$. Many research works have been done in recent times in which people have presented many modified algorithms to achieve the periodicity and other desirable properties of continued fractions in the field of $p$-adic numbers. See for example [9], [6],[8],[27], [26], and the references cited there in. See also the survey article by Romeo ([30]) for a comprehensive history of the development of the continued fraction theory in the field of $p$-adic numbers.

It is quite natural to consider continued fraction in finite extensions of $\mathbb{Q}_{p}$ which is left-out in the above discussion while considering continued fractions in all locally compact fields. In this article, we consider canonical extensions of the algorithms of Ruban and Browkin for $\mathbb{Q}_{p}$ in its finite extensions. Given any finite (necessarily simple) extension $K$ of $\mathbb{Q}_{p}$, we consider this extension in two steps, viz. $K=L(\beta)$ with $K/L$ a totally ramified extension, and $L=\mathbb{Q}_{p}(\gamma)$ with $L/\mathbb{Q}_{p}$ an unramified extension (see next section for more details). In our algorithms, the partial quotients are elements from the set $\mathbb{Z}\left[\frac{1}{p}\right][\gamma,\beta]$. We show that any $\alpha\in K$ has a unique continued fraction expansion, and given any sequence of partial quotients $\{c_{i}\}_{i\geq 0}$, $[c_{0},c_{1},\ldots,c_{n}]$ converges to an element $\alpha$ of $K$. For a few small degree unramified extensions of the form $\mathbb{Q}_{p}(\gamma)$, we show that any element of $\mathbb{Q}(\gamma)$ has a finite continued fraction expansion.

In this article, we are also going to discuss the metrical theory of the associated continued fraction map. In the case of classical continued fraction for real numbers the Gauss map or the continued fraction map is defined as

$$T:(0,1)\rightarrow(0,1)$$

$$T(x)=\left\{\frac{1}{x}\right\},$$

where $\left\{\frac{1}{x}\right\}$ denotes the fractional part of $\frac{1}{x}$. It is well known that $T$ is ergodic with respect to the Gauss measure (see [14]). For the ergodicity of the continued fraction map for a more general class of continued fraction, see [18]. For complex continued fraction, the ergodicity of the maps associated with the nearest integer complex continued fractions over imaginary quadratic fields is discussed in a recent paper of Nakada et al. [13]. They, in fact, showed that the continued fraction map is exact (see Section $4$ for definition). See also some references in [13] for some earlier works related to the metrical theory of complex continued fractions.

In the non-Archimedean settings, Berthe and Nakada [2] proved the ergodicity of the continued fraction map in positive characteristic, and as an application they obtained various metrical results regarding the averages of partial quotients, average growth rates of the denominators of the convergents, etc. In [23], Lertchoosakul and Nair proved the exactness of the continued fraction map using which they could consider more general averages concerning the partial quotients and the growth rate of the denominator of the convergents. The quantitative version of the metric theory of the continued fraction map in this setup was considered by the same authors in a subsequent paper [24]. The reader is referred to [12] for quantitative metrical results concerning real continued fraction.

In this article, we discuss metrical theory of continued fractions in finite extension of $\mathbb{Q}_{p}$. We show that the associated continued fraction map is Haar measure preserving and exact. Then we obtain various metrical results analogous to the results of [23] concerning asymptotic behaviour of various quantities related to partial quotients, denominator of the convergents, etc. In these results, general averages using subsequence ergodic theory and moving averages are considered as done in [23].

For a prime number $p$, the field of $p$-adic numbers $\mathbb{Q}_{p}$ is the set of all Laurent series in $p$ of the form

$$\alpha=\sum\limits_{j\geq n_{0}}a_{j}p^{j},\text{ where }a_{j}\in\{0,1,\ldots,p-1\}\text{ and }n_{0}\in\mathbb{Z}.$$

The $p$-adic valuation $\mathfrak{v}_{p}$ on $\mathbb{Q}_{p}$ is defined as follows: if $\alpha=\sum_{j\geq n_{0}}a_{j}p^{j}$, then

$$\mathfrak{v}_{p}(\alpha):=\inf\,\{\,j\in\mathbb{Z}\ :\ a_{j}\neq 0\,\}.$$

Then the $p$-adic absolute value of $\alpha$ is given by

$$|\alpha|_{p}:=p^{-\mathfrak{v}_{p}(\alpha)}$$

when $\alpha\neq 0$, and $|0|_{p}=0$. The field of $p$-adic numbers is the completion of $\mathbb{Q}$ with respect to this absolute value. Let $K=\mathbb{Q}_{p}(\xi)$ be a finite extension of $\mathbb{Q}_{p}$ of degree $m$, i.e., $[\,K\,:\,\mathbb{Q}_{p}\,]=m$. We may then take $\mathfrak{B}=\{1,\xi,\ldots,\xi^{m-1}\}$ as a convenient vector space basis for $K$ over the field $\mathbb{Q}_{p}$. Otherwise said, any element $\mathfrak{b}\in K$ can be written uniquely as

$$\mathfrak{b}=b_{0}+b_{1}\xi+\cdots+b_{m-1}\xi^{m-1},\text{ where }b_{j}\in\mathbb{Q}_{p}\text{ for all }j.$$

Since every finite extension is an algebraic extension, we have for every $\mathfrak{b}\in K$ that there is some monic irreducible polynomial

$$g(x)=x^{n}+B_{1}x^{n-1}+\cdots+B_{n-1}x+B_{n}$$

of degree at most $m$ and coefficients $B_{j}\in\mathbb{Q}_{p}$ such that $g(\mathfrak{b})=0$ in $K$. The norm map for the finite field extension $K/\mathbb{Q}_{p}$ is then defined as

$$N_{K/\mathbb{Q}_{p}}(\mathfrak{b}):=(-1)^{n}B_{n}.$$

Our absolute value $|\cdot|_{p}$ on $\mathbb{Q}_{p}$ extends uniquely to $K$ in the following manner (see [20] for details):

$$|\mathfrak{b}|:=\big{|}N_{K/\mathbb{Q}_{p}}(\mathfrak{b})\big{|}_{p}^{\frac{1}{n}},\quad\mathfrak{b}\in K.$$

Let us choose an element $\pi\in K$ of maximal absolute value smaller than 1, say $0<\theta:=|\pi|<1$. Define

$$\mathcal{O}_{K}:=\{\,x\in K\ :\ |x|\leq 1\,\},\quad\mathfrak{m}_{K}:=\{\,x\in K\ :\ |x|<1\,\}$$

and

$$\mathcal{O}_{K}^{*}:=\{\,x\in K\ :\ |x|=1\,\}.$$

We have $\pi\mathcal{O}_{K}=\mathfrak{m}_{K}$, and the residue field $\overline{k}=\mathcal{O}_{K}/\mathfrak{m}_{K}$ is a finite extension of $\mathbb{F}_{p}$.

We also have that $\overline{k}=\mathbb{F}_{q}$, where $q=\#\,(\overline{k})=p^{f}$. This is because upto isomorphism there is exactly one finite field having $q$ elements.

For the uniformizer $\pi\in\mathfrak{m}_{K}$, we have $|\pi|^{e}=|p|$ thereby giving us

$$|\pi|=p^{-1/e}.$$

Any element $\alpha\in K$ can be represented as $\alpha=u\pi^{n}$ for some suitable $u\in\mathcal{O}^{*}_{K}$ and $n\in\mathbb{Z}$. Then,

$$|\alpha|=|\pi|^{n}=p^{-n/e}.$$

The integers $e$ and $f$ given above satisfy $ef=m$, where $m$ is the degree of the extension. We recall that  [29, Corollary 4-26] there exists an unramified subextension $L/\mathbb{Q}_{p}$ of degree $f$ such that $K/L$ is a totally ramified extension of degree $e$. Also, there exists a $\gamma\in L$ such that $L=\mathbb{Q}_{p}(\gamma)$ with $|\gamma|=1$ [20, § III.3]. (To boot, we may and do take $\gamma$ to be some primitive $(p^{f}-1)$-th root of unity)

The metric balls in $K$ will have radius $\frac{s}{e}$ for $s\in\mathbb{Z}$. More precisely, for $\alpha\in K$ and $s\in\mathbb{Z}$, let

$$B\,(\,\alpha,\,p^{\frac{s}{e}}\,)\ =\ \{\,x\in K\ :\ |x-\alpha|<p^{\frac{s}{e}}\,\}$$

be the ball around $\alpha$ of radius $p^{\frac{s}{e}}$. Let $\mu$ denote the Haar measure on the local field $K$ (see Chapter $4$ of [29] for existence of Haar measure) and it is normalized in such a way that $\mu\,\big{(}\,B(0,1)\,\big{)}=1$. Note that

$$B(0,p^{s})\ =\ p^{-s}B(0,1)\ =\ \{\,p^{-s}x\ :\ x\in B(0,1)\,\}$$

and, therefore,

(8)		$\displaystyle\mu\,\big{(}\,B(0,p^{s})\,\big{)}$	$\displaystyle=\mu\big{(}\,p^{-s}B(0,1)\,\big{)}=\!\mod_{K}\left(\frac{1}{p^{s}}\right)\mu(B(0,1))$
		$\displaystyle=\!\mod_{\mathbb{Q}_{p}}\left(\frac{1}{p^{s}}\right)^{m}\ =\ p^{sm}$

by [29, Proposition 4-13]. Next two technical lemmas are useful for computing measure of various metric balls inside $K$.

Now we are in a position to calculate the measure of any ball around zero inside $K$.

As $Z$ is a uniformly discrete set, we can count the number of elements inside a set of elements with a fixed absolute value. This counting will be useful in some of the subsequent sections. Let $Y$ be the set given by

$$Y:=\left\{\sum_{j=0}^{f-1}b_{j}\gamma^{j}:b_{j}\in X\right\}.$$

For $y\in Y$, $|y|\leq p^{t}$ for some $t\in{\mathbb{N}}\cup\{0\}$ if and only if $|b_{j}|<p^{t+1}$ for all $j=0,\ldots,f-1$. Then it follows that for each $n\geq 1$,

	$\displaystyle\#\{y\in Y:|y|=p^{n}\}$	$\displaystyle=\#\{y\in Y:|y|\leq p^{n}\}-\#\{y\in Y:|y|<p^{n}\}$
		$\displaystyle=\big{(}p^{n+1}\big{)}^{f}-\big{(}p^{n}\big{)}^{f}$
		$\displaystyle=p^{fn}(p^{f}-1).$

Let us denote by $Z^{*}$ the set of all $c\in Z$ such that $|c|>1$. Also let $s$ be a positive integer and $a\in\{0,\ldots,e-1\}$. Writing an element $c\in Z^{*}$ as

$$c=\sum_{i=0}^{e-1}\sum_{j=0}^{f-1}b_{i,j}\gamma^{j}\beta^{i}\ :\ \ b_{i,j}\in X,$$

note that if $|b_{i,j}|\geq p^{s+1}$ for some $0\leq i\leq a\ \&\ 0\leq j<f$ or $|b_{i,j}|\geq p^{s}$ for some $a<i<e\ \&\ 0\leq j<f$, then the set $\{c\in Z^{*}:|c|\leq p^{s+\frac{a}{e}}\}=\phi$. Then it follows that

	$\displaystyle\#\left\{c\in Z^{*}:|c|=p^{s+\frac{a}{e}}\right\}$	$\displaystyle=\#\left\{c\in Z^{*}:|c|\leq p^{s+\frac{a}{e}}\right\}-\#\left\{c\in Z^{*}:|c|\leq p^{s+\frac{a-1}{e}}\right\}$
		$\displaystyle=\big{(}p^{s+1}\big{)}^{f(a+1)}\big{(}p^{s}\big{)}^{f(e-(a+1))}\,-\,\big{(}p^{s+1}\big{)}^{af}\big{(}p^{s}\big{)}^{f(e-a)}$
		$\displaystyle=p^{f(a+es)}\big{(}p^{f}-1\big{)}.$

Now, let $a\in\{1,\ldots,e-1\}$. By a similar argument, we also obtain

$$\#\{c\in Z^{*}:|c|=p^{\frac{a}{e}}\}=p^{af}(p^{f}-1).$$

Hence, for all $n\in{\mathbb{N}}$,

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$$\#\big{\{}c\in Z^{*}:|c|=p^{\frac{n}{e}}\big{\}}=p^{fn}\big{(}p^{f}-1\big{)}.$$

Now we describe a continued fraction algorithm for elements in any finite extension $K$ of $\mathbb{Q}_{p}$. This algorithm generalizes Ruban’s algorithm of $\mathbb{Q}_{p}$ (for $X=X_{1}$), as well as Browkin’s algorithm (for $X=X_{2}$). Our algorithm is a very natural extension of Ruban’s and Browkin’s algorithm with partial quotients coming from the set $Z$.

The $p$-adic floor function for Ruban’s algorithm is a function from $\mathbb{Q}_{p}$ to $X$ defined as follows: for $\alpha=\sum\limits_{j\geq n_{0}}a_{j}p^{j}\in\mathbb{Q}_{p}$ with $a_{j}\in\{0,1,\ldots,p-1\}$ for $X=X_{1}$ and $a_{j}\in\{-\frac{p-1}{2},\ldots,0,\ldots,\frac{p-1}{2}\}$ for $X=X_{2}$,

$$\ \lfloor\alpha\rfloor_{p}=\left\{\begin{array}[]{rcl}\displaystyle{\sum\limits_{j=n_{0}}^{0}a_{j}p^{j}},&\mbox{if}&\mathfrak{v}_{p}(\alpha)\leq 0\ (\text{or}\ |\alpha|_{p}\geq 1)\\ 0\ \ \ ,&\mbox{}&\text{otherwise.}\end{array}\right.$$

Using this floor function, we define a floor function on $K$ which is a function from $K$ to $Z$, as follows:

$$\text{For}\ \alpha=\sum_{i=0}^{e-1}\sum_{j=0}^{f-1}b_{i,j}\gamma^{j}\beta^{i},\ b_{i,j}\in\mathbb{Q}_{p},$$

(10)

$$\lfloor\alpha\rfloor=\sum_{i=0}^{e-1}\sum_{j=0}^{f-1}\lfloor b_{i,j}\rfloor_{p}\ \gamma^{j}\beta^{i}.$$

It is easy to see that

(11)

$$|\alpha-\lfloor\alpha\rfloor|\leq\frac{1}{p^{\frac{1}{e}}}<1$$

for any $\alpha\in K$.

Following the existing literature, we call an expression of the form

$$\displaystyle{c_{0}+\frac{1}{\displaystyle{c_{1}+\frac{1}{\ddots}}}}$$

with $c_{j}\in Z$ and $c_{j}\in Z^{*}$ for $j\geq 1$, a continued fraction which is also written as $[c_{0};c_{1},\ldots]$. It is a finite continued fraction if the sequence $(c_{j})$ is a finite one, otherwise, it is an infinite continued fraction. We call $c_{j}$’s the partial quotients of the continued fraction. We write,

$$[c_{0};c_{1},\ldots,c_{n}]=\frac{s_{n}}{t_{n}},$$

with $s_{n}$, $t_{n}\in\mathbb{Q}(\gamma,\beta)$, and call it the $n$th convergent of the continued fraction $[c_{0};c_{1},\ldots]$. It is easy to see that the sequence $(s_{n})$ and $(t_{n})$ satisfy the following recurrence relations:

$$s_{n}=c_{n}s_{n-1}+s_{n-2}\ \text{and}\ t_{n}=c_{n}t_{n-1}+t_{n-2},\ n\geq 2,$$

with $s_{0}=c_{0}$, $t_{0}=1$, $s_{1}=c_{0}c_{1}+1$, $t_{1}=c_{1}$. The numerator and denominator of the convergents also satisfy

(12)

$$t_{n}s_{n-1}-s_{n}t_{n-1}=(-1)^{n},\ n\geq 0.$$

Now we discuss the convergence properties of the continued fraction in our setup.