A composition law and refined notions of convergence for periodic continued fractions

Bradley W. Brock Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540-1966, USA [email protected] , Bruce W. Jordan Department of Mathematics, Baruch College, The City University of New York, One Bernard Baruch Way, New York, NY 10010-5526, USA [email protected] and Lawren Smithline Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540-1966, USA [email protected]

Abstract.

We define an equivalence relation on periodic continued fractions with partial quotients in a ring $\mathcal{O}\subseteq\mathbf{C}$ , a group law on these equivalence classes, and a map from these equivalence classes to matrices in $\operatorname{GL_{2}}(\mathcal{O})$ with determinant $\pm 1$ . We prove this group of equivalence classes is isomorphic to $\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ and study certain of its one- and two-dimensional representations.

For a periodic continued fraction with period $k$ , we give a refined description of the limits of the $k$ different $k$ -decimations of its sequence of convergents. We show that for a periodic continued fraction associated to a matrix with eigenvalues of different magnitudes, all $k$ of these limits exist in $\mathbb{P}^{1}(\mathbf{C})$ and a strict majority of them are equal.

Key words and phrases:

elementary matrices,

\operatorname{\operatorname{SL}_{2}}

2020 Mathematics Subject Classification:

Primary 11J70; Secondary 11G30

1. Introduction

Continued fractions with positive integer partial quotients are a staple of classical number theory, with an intimate connection to Euclid’s algorithm. Here, we consider the more general case of continued fractions with partial quotients in a subring $\mathcal{O}\ni 1$ of the complex numbers $\mathbf{C}$ , as in [BEJ21]. We define an equivalence relation on the set $\operatorname{\textup{{PCF}}}(\mathcal{O})$ of periodic continued fractions over $\mathcal{O}$ and a group law on the resulting equivalence classes $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ .

The theory of periodic continued fractions, unlike continued fractions in general, has connections to both arithmetic groups and diophantine equations. The work [BEJ21] relates the theory of $\operatorname{\textup{{PCF}}}(\mathcal{O})$ to the group $\operatorname{\operatorname{SL}_{2}}^{\pm}(\mathcal{O})$ , the subgroup of $\operatorname{GL_{2}}(\mathcal{O})$ consisting of matrices with determinant $\pm 1$ , and to diophantine problems on associated varieties. In this paper we further explore the relationship between $\operatorname{\textup{{PCF}}}(\mathcal{O})$ and the group $\operatorname{\operatorname{SL}_{2}}^{\pm}(\mathcal{O})$ . A major influence in our study is the 1966 paper [Coh66] by Cohn.

Section 2 sets out preliminaries related to actions of $2\times 2$ matrices. We consider the action of $\operatorname{GL_{2}}(\mathcal{O})$ by linear fractional transformations on the projective line $\mathbb{P}^{1}(\mathbf{C})$ . We also name subgroups of $\operatorname{GL_{2}}(\mathbf{C})$ determined by common eigenspaces. In Section 3, we introduce the concatentation binary operation $\star$ on the semigroup $\operatorname{\textup{{FCF}}}(\mathcal{O})$ of finite continued fractions over $\mathcal{O}$ . We develop an equivalence relation on $\operatorname{\textup{{FCF}}}(\mathcal{O})$ and show that the equivalence classes $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ form a group under $\star$ . We exhibit a map $M:\operatorname{\textup{{FCF}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{SL}_{2}}^{\pm}(\mathcal{O})$ in Proposition 3.12 arising from the algorithm for computing the value of a finite continued fraction. We show that $M$ is well defined on equivalence classes, giving a homomophism $\overline{M}:\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{SL}_{2}}^{\pm}(\mathcal{O})$ . We review results of Cohn [Coh66] that give information on $\ker(\overline{M})$ for certain rings $\mathcal{O}$ . Using this, we show how the standard amalgam presentation $\operatorname{\operatorname{SL}_{2}}(\mathbf{Z})\cong\mathbf{Z}/4\mathbf{Z}\ast_{\mathbf{Z}/2\mathbf{Z}}\mathbf{Z}/6\mathbf{Z}$ arises from presenting $\operatorname{\operatorname{SL}_{2}}(\mathbf{Z})$ as an explicit quotient of $\operatorname{\overline{\textup{{FCF}}}}(\mathbf{Z}).$

Section 4 extends the structure on finite continued fractions to periodic continued fractions (PCFs). In Theorem 4.9, we show that this yields a natural equivalence relation on periodic continued fractions yielding a group of equivalence classes $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ isomorphic to $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ . Section 5 relates the group $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ to quantities defined in [BEJ21]. We review the fundamental matrix $E(P)\in\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ associated to a periodic continued fraction $P$ and apply the isomorphism of Theorem 4.9 to show that $E$ is well-defined on $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ , giving a homomophism $\overline{E}:\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{SL}_{2}}^{\pm}(\mathcal{O})$ . For $A\in\operatorname{GL_{2}}(\mathcal{O})$ we define subgroups $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)$ of $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ by requiring $\overline{E}(\overline{P})$ to be a linear combination of $A$ and the identity matrix $I$ . We exhibit characters corresponding to each eigenvalue of $A$ . These characters have values that are units in a quadratic extension of $\mathcal{O}$ . In Section 6, we give examples where $\mathcal{O}$ is a number ring.

In Section 7, we reformulate the PCF-convergence criteria of [BEJ21, Thm. 4.3] in parallel with the cases of convergence in Section 2.1. We show that when $E(P)$ has eigenvalues of different magnitudes (which implies quasiconvergence), the $k$ different $k$ -decimations of its sequence of convergents all converge in $\mathbb{P}^{1}(\mathbf{C})$ , and a strict majority agree. Unanimity is equivalent to convergence.

In Section 8, we demonstrate the sharpness of the result of Section 7. We give examples showing that the proportion of agreement of the $k$ different $k$ -decimations of the sequence of convergents of a quasiconvergent periodic continued fraction can be arbitrarily close to $1/2$ or arbitrarily close to and different from $1$ . We show that the equivalence relation $\sim$ on $\operatorname{\textup{{PCF}}}(\mathcal{O})$ respects quasiconvergence but does not respect convergence.

2. Actions of $2\times 2$ matrices

Let $\mathcal{O}\subseteq\mathbf{C}$ be a ring with $1$ . A finite continued fraction with partial quotients in $\mathcal{O}$ is an iterated quotient. It can be interpreted in terms of the action of $\operatorname{GL_{2}}(\mathcal{O})\leq\operatorname{GL_{2}}(\mathbf{C})$ on the projective line $\mathbb{P}^{1}(\mathbf{C})$ by linear fractional transformations which we review in this section. We also name certain subgroups of $\operatorname{GL_{2}}(\mathbf{C})$ determined by common eigenspaces.

2.1. Linear fractional transformations

Suppose

M=\begin{bmatrix}m_{11}&m_{12}\\ m_{21}&m_{22}\end{bmatrix}\in\operatorname{GL_{2}}(\mathbf{C})

(1)

and $\beta\in\mathbb{P}^{1}(\mathbf{C})$ . The matrix $M$ acts on $\beta$ by linear fractional transformation

M\beta=\frac{m_{11}\beta+m_{12}}{m_{21}\beta+m_{22}},

where we interpret $1/0=\infty\in\mathbb{P}^{1}(\mathbf{C})$ . For $\beta\in\mathbb{P}^{1}(\mathbf{C})$ , let

v(\beta)=\binom{\beta}{1}\text{ if }\beta\neq\infty\text{ and }v(\infty)=\binom{1}{0}.

(2)

The vector $v(\beta)$ is an eigenvector of $M$ if and only if $M\beta=\beta$ . For nonzero polynomial $Q=aX^{2}+bX+c\in\mathbf{C}[X]$ , let $\operatorname{Roots}(Q)$ be the multiset of zeros of $Q[x]$ in $\mathbb{P}^{1}(\mathbf{C})$ , with the convention that if $\deg(Q)=1$ , then $\infty$ is a simple root of $Q$ ; and if $\deg(Q)=0$ , then $0\neq Q$ has a double root at $\infty$ . When $Q=0$ , say $\operatorname{Roots}(Q)=\mathbb{P}^{1}(\mathbf{C})$ . For a $2\times 2$ matrix $M$ as above, set

\operatorname{Quad}(M)=m_{21}X^{2}+(m_{22}-m_{11})X-m_{12}.

(3)

We use $\operatorname{Roots}(M)$ to denote $\operatorname{Roots}(\operatorname{Quad}(M))$ . When $\beta\in\operatorname{Roots}(M)$ , $v(\beta)$ is an eigenvector of $M$ with eigenvalue

\lambda(\beta)\colonequals m_{21}\beta+m_{22}=m_{11}+m_{12}/\beta.

(4)

In particular, $\lambda(\infty)=m_{11}$ and $\lambda(0)=m_{22}$ .

Proposition 2.1.

Let $A$ , $B$ be $2\times 2$ matrices over $\mathcal{O}$ , $I$ be the $2\times 2$ identity matrix. Let $\kappa,\lambda\in\mathcal{O}$ be not both zero. There exists $\mu\in\mathcal{O}$ such that $\kappa B=\lambda A+\mu I$ if and only if $\kappa\operatorname{Quad}(B)=\lambda\operatorname{Quad}(A)$ .

Proof.

Suppose $\kappa,\lambda,\mu\in\mathcal{O}$ with $(\kappa,\lambda)\neq(0,0)$ such that $\kappa B=\lambda A+\mu I$ . The polynomial $\operatorname{Quad}(\lambda A+\mu I)$ is independent of $\mu$ . We have

\kappa\operatorname{Quad}(B)=\operatorname{Quad}(\kappa B)=\operatorname{Quad}(\lambda A+\mu I)=\operatorname{Quad}(\lambda A)=\lambda\operatorname{Quad}(A).

Conversely, suppose $\kappa\operatorname{Quad}(B)=\lambda\operatorname{Quad}(A)$ for $\kappa,\lambda\in\mathcal{O}$ , $(\kappa,\lambda)\neq(0,0)$ . Since $\operatorname{Quad}$ is linear map, $\operatorname{Quad}(\kappa B-\lambda A)=0$ . Thus, there is $\mu\in\mathcal{O}$ such that $\kappa B-\lambda A=\mu I$ . ∎

For a matrix $M$ , $\operatorname{Quad}(M)$ and the eigenvalues of $M$ govern the convergence behavior of the sequence $M^{n}\beta$ , $n\geq 0$ , for $\beta\in\mathbb{P}^{1}(\mathbf{C})$ .

Definition 2.2.

For any $x,y$ , let $\delta(x,y)$ denote the diagonal matrix

\delta(x,y)=\begin{bmatrix}x&0\\ 0&y\end{bmatrix}.

Proposition 2.3.

For $M\in\operatorname{GL_{2}}(\mathbf{C})$ , one of four mutually exclusive possibilities holds:

(a)

$\operatorname{Quad}(M)$ has one root $\hat{\beta}$ of multiplicity 2 and $\lim_{n\to\infty}M^{n}\beta=\hat{\beta}$ for all $\beta\in\mathbb{P}^{1}(\mathbf{C})$ .
(b)

$\operatorname{Quad}(M)=0$ . For all $\beta\in\mathbb{P}^{1}(\mathbf{C})$ , $M\beta=\beta$ and so $lim_{n\to\infty}M^{n}\beta=\beta$ .
(c)

$\operatorname{Quad}(M)$ has distinct roots $\beta_{+},\beta_{-}$ , and $M$ has corresponding eigenvalues $\lambda_{+}=\lambda(\beta_{+})$ , $\lambda_{-}=\lambda(\beta_{-})$ as in (4), with $|\lambda_{+}|>|\lambda_{-}|$ . For all $\beta\neq\beta_{-}\in\mathbb{P}^{1}(\mathbf{C})$ , $lim_{n\to\infty}M^{n}\beta=\beta_{+}$ . For $\beta=\beta_{-}$ , $lim_{n\to\infty}M^{n}\beta=\beta_{-}$ .
(d)

$\operatorname{Quad}(M)$ has distinct roots with $M$ having distinct eigenvalues of the same magnitude. For $\beta\in\operatorname{Roots}(M)$ , $\lim_{n\to\infty}M^{n}\beta=\beta$ . Otherwise, the sequence $M^{n}\beta$ diverges.

Proof.

Every $M$ is conjugate to its Jordan normal form, so the result follows if it is so for $M$ in Jordan normal form. Let $\beta\in\mathbb{P}^{1}(\mathbf{C})$ . The design of cases (a)–(d) makes evident that exactly one of the following holds, where $\lambda_{1}$ , $\lambda_{2}$ are the eigenvalues of $M$ . In cases (a) and (b) below there is one eigenvalue $\lambda_{1}$ of multiplicity $2$ .

(a)

$M$ is defective, $M-\lambda_{1}I$ is nilpotent, and

$\lim_{n\to\infty}\begin{bmatrix}\lambda_{1}&1\\ 0&\lambda_{1}\end{bmatrix}^{n}\beta=\infty;$
(b)

$M=\lambda_{1}I$ , and

$\lim_{n\to\infty}\delta(\lambda_{1},\lambda_{1})^{n}\beta=\beta;$
(c)

$M=\delta(\lambda_{1},\lambda_{2}),|\lambda_{1}|>|\lambda_{2}|$ , and

$\lim_{n\to\infty}\delta(\lambda_{1},\lambda_{2})^{n}\beta=\begin{cases}\infty,\,\beta\neq 0,\\[3.61371pt] 0,\,\beta=0;\\ \end{cases}\\$

(d)

$M=\delta(\lambda_{1},\lambda_{2}),|\lambda_{1}|=|\lambda_{2}|$ , $\lambda_{1}\neq\lambda_{2}$ , and

\lim_{n\to\infty}\delta(\lambda_{1},\lambda_{2})^{n}\beta=\begin{cases}\beta,\textrm{ for }\beta\in\{0,\infty\},\\ \text{does not exist for }\beta\notin\{0,\infty\}.\end{cases}

∎

2.2. Subgroups of \excepttoc $\operatorname{GL_{2}}(\mathbf{C})$ \fortoc $\operatorname{GL_{2}}(\mathbf{C})$ determined by eigenspaces

The subsets of $\operatorname{GL_{2}}(\mathbf{C})$ with particular eigenvectors form subgroups of $\operatorname{GL_{2}}(\mathbf{C})$ .

Proposition 2.4.

Let $M\in\operatorname{GL_{2}}(\mathbf{C})$ . Every $\beta\in\mathbb{P}^{1}(\mathbf{C})$ satisfies $M\beta=\beta$ if and only if $M$ is a scalar multiple of the identity matrix $I$ .

Proof.

Write $M=[m_{ij}]_{1\leq i,j\leq 2}$ as in (1). If $M=m_{11}I$ , then every $\beta$ satisfies $M\beta=\beta$ . Conversely, if every $\beta$ satisfies $M\beta=\beta$ , then $v(0)$ and $v(\infty)$ are eigenvectors of $M$ and $M$ is diagonal. Since $v(1)$ is also an eigenvector, $M$ must be a scalar multiple of the identity. ∎

Proposition 2.5.

Let $0\neq L(X)\in\mathbf{C}[X]$ , $\deg(L)\leq 1$ , with root $\beta\in\mathbb{P}^{1}(\mathbf{C})$ . Set $T(\beta)=\{M\in\operatorname{GL_{2}}(\mathbf{C})\mid M\beta=\beta\}$ .

(a)

The set $T(\beta)$ is a subgroup of $\operatorname{GL_{2}}(\mathbf{C})$ conjugate to the Borel subgroup of $\operatorname{GL_{2}}(\mathbf{C})$ consisting of upper triangular matrices.
(b)

For every $M\in T(\beta)$ , $L(X)$ divides $\operatorname{Quad}(M)$ . If $\deg L(X)=0$ , $L(X)|\operatorname{Quad}(M)$ means $\deg\operatorname{Quad}(M)\leq 1$ .
(c)

The group $T(\beta)$ has a multiplicative character $\lambda_{\beta}$ mapping $M$ to its $v(\beta)$ -eigenvalue.

Proof.

(a): The stabilizer $T(\infty)$ of $\infty\in\mathbb{P}^{1}(\mathbf{C})$ consists of the upper triangular matrices. Since $\operatorname{GL_{2}}(\mathbf{C})$ acts transitively on $\mathbb{P}^{1}(\mathbf{C})$ , $T(\beta)$ is a subgroup of $\operatorname{GL_{2}}(\mathbf{C})$ conjugate to $T(\infty)$ for every $\beta\in\mathbb{P}^{1}(\mathbf{C})$ .
(b): Suppose $M\in T(\beta)$ . Then $\beta\in\operatorname{Roots}(M)$ and $L(X)$ divides $\operatorname{Quad}(M)$ .
(c): Suppose $M,M^{\prime}\in T(\beta)$ . Then $MM^{\prime}\in T(\beta)$ and

\lambda_{\beta}(MM^{\prime})v(\beta)=MM^{\prime}v(\beta)=M\lambda_{\beta}(M^{\prime})v(\beta)=\lambda_{\beta}(M)\lambda_{\beta}(M^{\prime})v(\beta),

so $\lambda_{\beta}$ is a multiplicative character on $T(\beta)$ . ∎

Proposition 2.6.

Let $0\neq Q\in\mathbf{C}[X]$ with $\deg(Q)\leq 2$ . The set

G(Q)=\{M\in\operatorname{GL_{2}}(\mathbf{C})\mid\exists\lambda\in\mathbf{C}\text{ such that }\operatorname{Quad}(M)=\lambda Q\}

is a group. When $Q$ is not a square,

G(Q)=T(\beta)\cap T(\beta^{*})\text{ with }\{\beta,\beta^{*}\}=\operatorname{Roots}(Q),\,\,\beta\neq\beta^{\ast},

and $G(Q)$ is conjugate to the group of diagonal matrices in $\operatorname{GL_{2}}(\mathbf{C})$ . When $Q$ is a square in $\mathbf{C}[X]$ , $G(Q)$ is conjugate to the the subgroup of upper triangular matrices in $\operatorname{GL_{2}}(\mathbf{C})$ with equal diagonal entries.

Proof.

When $Q$ is not a square, $\operatorname{Roots}(Q)=\{\beta,\beta^{*}\}$ , $\beta\neq\beta^{*}$ . In this case, $M\in G(Q)$ is equivalent to $M$ having eigenvectors $v(\beta)$ and $v(\beta^{*})$ . Hence, $G(Q)=T(\beta)\cap T(\beta^{*})$ . A change of basis mapping $\beta$ to $\infty$ and $\beta^{*}$ to $0$ conjugates $G(Q)$ to $G(x)$ , the group of diagonal matrices.

When $Q$ is a square, $\operatorname{Roots}(Q)$ has one element $\beta$ of multiplicity $2$ .. Every $M\in G(Q)$ is either a scalar multiple of $I$ or defective. A change of basis mapping $\beta$ to $\infty$ conjugates $G(Q)$ to $G(1)$ . A matrix in $G(1)$ is either a scalar multiple of $I$ or upper triangular with equal diagonal entries and not diagonal. ∎

3. Finite continued fractions

Let $\mathcal{O}\subseteq\mathbf{C}$ be a ring containing 1. We define the set $\operatorname{\textup{{FCF}}}(\mathcal{O})$ of finite continued fractions with partial quotients in $\mathcal{O}$ , and an operation $\star$ making $\operatorname{\textup{{FCF}}}(\mathcal{O})$ a semigroup. We give an equivalence relation on $\operatorname{\textup{{FCF}}}(\mathcal{O})$ and show that the equivalence classes $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ form a group.

Let $c_{i}\in\mathbf{C}$ , $1\leq i\leq n$ .

Definition 3.1.

A finite continued fraction (FCF) $F=[c_{1},c_{2},\ldots,c_{n}]$ is the formal expression

c_{1}+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}+\cfrac{1}{c_{4}+\cfrac{1}{\raisebox{-11.99998pt}{$\ddots\quad$}\raisebox{-23.99997pt}{$c_{n-1}+\cfrac{1}{c_{n}}$}}}}}.

(5)

We say that the FCF $F=[c_{1},\ldots,c_{n}]$ has length $\operatorname{length}(F)=n$ .

The elements $c_{i}$ are called the partial quotients. For $k\leq n$ , the convergent $\mathcal{C}_{k}(F)$ of $F$ is the evaluation of the finite continued fraction $[c_{1},c_{2},\ldots,c_{k}]$ , where we interpret $1/0$ as $\infty\in\mathbb{P}^{1}(\mathbf{C})$ . We distinguish between the formal object $F=[c_{1},c_{2},\ldots c_{k}]$ and its value $\hat{F}\colonequals\mathcal{C}_{k}(F)\in\mathbb{P}^{1}(\mathbf{C})$ . There is a simple rule to iteratively compute $\mathcal{C}_{k}(F)$ as a ratio $p_{k}/q_{k}$ . Let

D(x)=\begin{bmatrix}x&1\\ 1&0\end{bmatrix}.

Let $I$ be the $2\times 2$ identity matrix and $F=[c_{1},\ldots,c_{n}]$ be a finite continued fraction as in (5). Define $p_{0},q_{0},p_{-1},q_{-1}$ by the matrix equation

\begin{bmatrix}p_{0}&p_{-1}\\ q_{0}&q_{-1}\end{bmatrix}=I.

Recursively define $p_{k}=p_{k}(F)$ , $q_{k}=q_{k}(F)$ for $n\geq k\geq 1$ by

\begin{bmatrix}p_{k}&p_{k-1}\\ q_{k}&q_{k-1}\end{bmatrix}=\begin{bmatrix}p_{k-1}&p_{k-2}\\ q_{k-1}&q_{k-2}\end{bmatrix}D(c_{k})=D(c_{1})\cdots D(c_{k}).

The numerators $p_{k}$ and denominators $q_{k}$ with $\mathcal{C}_{k}(F)=p_{k}(F)/q_{k}(F)$ can be written in terms of continuant polynomials (see, e.g., [BEJ21, p. 385]).

An $\mathcal{O}$ -FCF is a FCF $[c_{1},\ldots,c_{n}]$ having all partial quotients $c_{i}\in\mathcal{O}$ .

3.1. An equivalence relation on finite continued fractions

Definition 3.2.

The semigroup $\operatorname{\textup{{FCF}}}(\mathcal{O})$ is the set $\{F\mid F\text{ is an $\mathcal{O}$-FCF}\}$ with the concatenation operation $\star$ :

[c_{1},\ldots,c_{n}]\star[c^{\prime}_{1},\ldots,c^{\prime}_{n^{\prime}}]=[c_{1},\ldots,c_{n}][c^{\prime}_{1},\ldots,c^{\prime}_{n^{\prime}}]\colonequals[c_{1},\ldots,c_{n},c_{1}^{\prime},\ldots,c_{n^{\prime}}].

Equivalently, $\operatorname{\textup{{FCF}}}(\mathcal{O})$ is the semigroup of the set of words in formal symbols $\mathbf{D}(x)$ , for $x\in\mathcal{O}$ , with the two descriptions identified by

[x_{1},\ldots,x_{n}]\leftrightarrow\mathbf{D}(x_{1})\cdots\mathbf{D}(x_{n}).

Definition 3.3.

Let $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ be the set $\operatorname{\textup{{FCF}}}(\mathcal{O})$ modulo the equivalence relation $\sim$ generated by the relations

\mathbf{D}(x)\mathbf{D}(0)\mathbf{D}(y)=\mathbf{D}(x+y),\ x,y\in\mathcal{O}.

(6)

For $F=[c_{1},\ldots,c_{n}]\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ denote by $\overline{F}=|\![c_{1},\ldots,c_{n}]\!|\in\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ the equivalence class containing $F$ . For a word $\mathbf{w}$ in the $\mathbf{D}(x)$ ’s denote by $\overline{\mathbf{w}}$ the equivalence class containing $\mathbf{w}$ .

Proposition 3.4.

(a)

If $F_{1},G_{1},F_{2},G_{2}\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ with $\overline{F}_{1}=\overline{F}_{2}$ and $\overline{G}_{1}=\overline{G}_{2}$ , then

$\overline{F_{1}\star G_{1}}=\overline{F_{2}\star G_{2}}\in\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O}).$
(b)

The binary operation $\star$ on $\operatorname{\textup{{FCF}}}(\mathcal{O})$ induces a well-defined binary operation $\star$ on $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ , making $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ a semigroup.
(c)

The semigroup $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ is a group with identity $|\![0,0]\!|\leftrightarrow\overline{\mathbf{D}}(0)^{2}$ .
(d)

The element $\overline{\mathbf{D}}(x)\in\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ has inverse $\overline{\mathbf{D}}(x)^{-1}=\overline{\mathbf{D}}(0)\overline{\mathbf{D}}(-x)\overline{\mathbf{D}}(0)$ .

Proof.

(a) is immediate, and (b) is implied by (a).
(c): Substituting $y=0$ into relation (6) shows that the class of $\mathbf{D}(0)^{2}$ is the identity in $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ . The equality

\mathbf{D}(x)\mathbf{D}(0)\mathbf{D}(-x)\mathbf{D}(0)=\mathbf{D}(0)^{2},\ x\in\mathcal{O},

exhibits the inverse $\overline{\mathbf{D}}(x)^{-1}=\overline{\mathbf{D}}(0)\overline{\mathbf{D}}(-x)\overline{\mathbf{D}}(0)$ . ∎

Definition 3.5.

Define $\mathbf{J},\mathbf{U}(x),\mathbf{L}(x)\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ for $x\in\mathcal{O}$ by $\mathbf{J}\colonequals\mathbf{D}(0)$ , $\mathbf{U}(x)\colonequals\mathbf{D}(x)\mathbf{D}(0)$ , and $\mathbf{L}(x)\colonequals\mathbf{D}(0)\mathbf{D}(x)$ . We consider the group $\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ , where $\mathcal{O}$ is viewed as an additive group and $j\in\mathbf{Z}/2\mathbf{Z}$ is a generator. Let $W_{\mathsf{F}}:\operatorname{\textup{{FCF}}}(\mathcal{O})\rightarrow\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ be the surjective map of semigroups

W_{\mathsf{F}}:\mathbf{D}(x)\mapsto xj\in\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}.

In particular, note that $W_{\mathsf{F}}(\mathbf{J})=j\in\mathbf{Z}/2\mathbf{Z}$ , $W_{\mathsf{F}}(\mathbf{U}(x))=x\in\mathcal{O}$ , and $W_{\mathsf{F}}(\mathbf{L}(x)))=jxj$ .

Proposition 3.6.

The morphism of semigroups $W_{\mathsf{F}}:\operatorname{\textup{{FCF}}}(\mathcal{O})\rightarrow\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ of Definition 3.5 induces an isomorphism of groups $\overline{W}_{\mathsf{F}}:\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ .

Proof.

Verify that $W_{\mathsf{F}}(\mathbf{D}(x)\mathbf{D}(0)\mathbf{D}(y))=W_{\mathsf{F}}(\mathbf{D}(x+y))$ for $x,y\in\mathcal{O}$ . Hence $W_{\mathsf{F}}$ induces a well-defined map on equivalence classes $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})=\operatorname{\textup{{FCF}}}(\mathcal{O})/\sim$ by the Definition 3.3 of $\sim$ . To see that $\overline{W}_{\mathsf{F}}$ is an isomorphism, verify that its inverse is given by

\overline{W}_{\mathsf{F}}^{\,-1}:\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}\longrightarrow\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\quad\text{with}\quad\overline{W}_{\mathsf{F}}^{\,-1}(j)=\overline{\mathbf{J}}\text{ and }\overline{W}_{\mathsf{F}}^{\,-1}(x)=\overline{\mathbf{U}}(x)\text{ for }x\in\mathcal{O}.\qed

Remark 3.7.

For $F=[c_{1},\ldots,c_{n}]\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ , set $F^{\ast}\colonequals[0,-c_{n},\ldots,-c_{1},0]$ . Observe that $\overline{F^{\ast}}=\overline{F}^{\,-1}\in\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ :

|\![c_{1},\ldots,c_{n}]\!|\star|\![0,-c_{n},\ldots,-c_{1},0]\!|=|\![c_{1},\ldots,c_{n}]\!||\![0,-c_{n},\ldots,-c_{1},0]\!|=|\![0,0]\!|.

This formula for $\overline{F}^{\,-1}$ also follows from Proposition 3.4(d).

Definition 3.8.

The finite continued fraction $F=[c_{1},\ldots,c_{n}]$ is reduced if it has no interior zeros: $c_{i}\neq 0$ for $2\leq i\leq n-1$ .

Proposition 3.9.

For $F\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ , there is a unique reduced $F_{\mathrm{red}}\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ such that $F_{\mathrm{red}}\sim F.$

Proof.

Given $F\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ , iteratively apply relation (6) to remove interior zeros to produce a reduced $F_{\textrm{red}}\sim F$ .

On the other hand, suppose $F_{\textrm{red}},F^{\prime}_{\textrm{red}}\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ are both reduced and $\overline{F}_{\textrm{red}}=\overline{F^{\prime}}_{\textrm{red}}$ . Then $\overline{W}_{\mathsf{F}}(\overline{F}_{\textrm{red}})=\overline{W}_{\mathsf{F}}(\overline{F^{\prime}}_{\textrm{red}})\in\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ . Since they are reduced, both $\overline{F}_{\textrm{red}}$ and $\overline{F^{\prime}}_{\textrm{red}}$ map under $\overline{W}_{\mathsf{F}}$ to words in $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\cong\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ , each literally expressed without spurious insertions of the identity, and they are equal. Since $\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ is a free product, these words are the same and $F^{\prime}_{\textrm{red}}=F_{\textrm{red}}$ . ∎

Definition 3.10.

For any $\overline{F}\in\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ , the normal form of $\overline{F}$ is the unique reduced representative $F_{\textrm{red}}\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ of the equivalence class $\overline{F}$ .

Remark 3.11.

Two elements $\overline{F},\overline{F}^{\prime}\in\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ are equal if and only if they have the same normal forms in $\operatorname{\textup{{FCF}}}(\mathcal{O})$ :

\overline{F}=\overline{F}^{\prime}\Longleftrightarrow F_{\textrm{red}}=F^{\prime}_{\textrm{red}}\in\operatorname{\textup{{FCF}}}(\mathcal{O}).

In practice one computes $F_{\mathrm{red}}$ for $F\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ by the algorithm of repeatedly applying relation (6) until all interior zeros are eliminated. For example, consider

F=[0,-2,0,2,0,3,0,5]\in\operatorname{\textup{{FCF}}}(\mathbf{Z}).

We have

F=[0,-2,0,2,0,3,0,5]\sim[0,0,0,3,0,5]\sim[0,3,0,5]\sim[0,8],

and hence $F_{\mathrm{red}}=[0,8]\in\operatorname{\textup{{FCF}}}(\mathbf{Z})$ with $\overline{F}=|\![0,-2,0,2,0,3,0,5]\!|=|\![0,8]\!|=\overline{F}_{\textrm{red}}\in\operatorname{\overline{\textup{{FCF}}}}(\mathbf{Z})$ .

3.2. Mapping finite continued fractions to matrices

Let

\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})=\{g\in\operatorname{GL_{2}}(\mathcal{O})\mid\det(g)=\pm 1\}.

The kernel of the determinant map, $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ , is normal of index 2 in $\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ .

Proposition 3.12.

The map $M:\operatorname{\textup{{FCF}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ given by

M([c_{1},\ldots,c_{n}])=D(c_{1})\cdots D(c_{n})\quad\text{so that}\quad M(\mathbf{D}(x))=D(x)

(7)

induces a well-defined homomorphism of groups $\overline{M}:\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ . We therefore have a homomorphism of groups $\mathbf{M}\colonequals\overline{M}\circ\overline{W}_{\mathsf{F}}^{\,-1}:\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}\rightarrow\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ .

Proof.

We have specified the map $M$ on $\operatorname{\textup{{FCF}}}(\mathcal{O})$ . We must show that $M$ respects equivalence under (6). By direct computation,

M(\mathbf{D}(x)\mathbf{D}(0)\mathbf{D}(y))=D(x)D(0)D(y)=D(x+y)=M(\mathbf{D}(x+y)).\qed

Proposition 3.13.

Suppose $F_{1},F_{2}\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ and $F_{1}\sim F_{2}$ .

(a)

The values of $F_{1}$ and $F_{2}$ are equal: $\hat{F}_{1}=\hat{F}_{2}$ .

(b)

We have $\operatorname{length}(F_{1})\equiv\operatorname{length}(F_{2})\bmod 2$ , so there is a well-defined function

\operatorname{\overline{\operatorname{length}}}:\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\rightarrow\mathbf{Z}/2\mathbf{Z}\quad\text{given by}\quad\operatorname{\overline{\operatorname{length}}}(\overline{F})=\operatorname{length}(F)\pmod{2}.

Proof.

(a): Since $F_{1}\sim F_{2}$ , $M(F_{1})=M(F_{2})\in\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ by Proposition 3.12 We have that $\hat{F}_{1}=\hat{F}_{2}$ , because $\hat{F}_{1}=M(F_{1})_{11}/M(F_{1})_{21}$ and likewise for $\hat{F}_{2}$ by (7).
(b): Reformation of a word by relation (6) maintains the parity of the word length. ∎

Proposition 3.14.

Let $\chi:\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\rightarrow\langle\pm 1\rangle$ be $\det\circ\,\overline{M}$ . Let $\tilde{\chi}:\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}\rightarrow\langle\pm 1\rangle$ be the group homomorpism to $\langle\pm 1\rangle$ that maps $j$ to $-1$ and is trivial on the second factor. For $w\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ , we have $\chi(\overline{w})=(-1)^{\operatorname{\overline{\operatorname{length}}}(\overline{w})}$ and $\tilde{\chi}\circ\overline{W}_{\mathsf{F}}=\chi$ .

Proof.

The maps $\chi$ and $\tilde{\chi}\circ\overline{W}_{\mathsf{F}}$ are determined by their action on the generators $\overline{\mathbf{D}}(x)$ of $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ , namely $\chi(\overline{\mathbf{D}}(x))=-1$ and $\tilde{\chi}\circ\overline{W}_{\mathsf{F}}(\overline{\mathbf{D}}(x))=\tilde{\chi}(xj)=-1$ . Hence $\chi=\tilde{\chi}\circ W_{\mathsf{F}}$ and for $w\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ , $\chi(\overline{w})=(-1)^{\operatorname{length}(w)}=(-1)^{\operatorname{\overline{\operatorname{length}}}(\overline{w})}$ . ∎

Definition 3.15.

Set $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O})=\ker(\operatorname{\overline{\operatorname{length}}})\triangleleft\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ .

Proposition 3.16.

Let $\tilde{\chi}:\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}\rightarrow\langle\pm 1\rangle$ be as in Proposition 3.14. There is an isomorphism $\psi:\mathcal{O}\ast\mathcal{O}\rightarrow\ker{\tilde{\chi}}\subseteq\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ given by sending $x\in\mathcal{O}$ in the first factor to $x\in\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ and $y\in\mathcal{O}$ in the second factor to $jyj\in\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ . Hence $\overline{W}_{\mathsf{F}}$ maps $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O})$ isomorphically onto $\mathcal{O}\ast\mathcal{O}\simeq\ker(\tilde{\chi})$ .

Proof.

The definition of the free product $\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ implies that $\psi$ is injective. A word in $\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ with an even number of $j$ ’s is of the form $x_{1}jy_{1}jx_{2}jy_{2}j\cdots x_{n}jy_{n}j$ with possibly $x_{1}$ , $y_{n}$ , or both 0, which is in the image of $\psi$ . ∎

Definition 3.17.

Set

W_{\mathsf{F}^{+}}\colonequals\psi^{-1}\circ W_{\mathsf{F}}|_{\operatorname{\textup{{FCF}}}^{+}(\mathcal{O})}:\operatorname{\textup{{FCF}}}^{+}(\mathcal{O})\longrightarrow\mathcal{O}\ast\mathcal{O}.

Proposition 3.18.

(a)

The map $W_{\mathsf{F}^{+}}$ of Definition 3.17 induces an isomorphism

\overline{W}_{\mathsf{F}^{+}}:\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathcal{O}\ast\mathcal{O}.

(b)

Let $\mathbf{J},\mathbf{U}(x),\mathbf{L}(x)\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ for $x\in\mathcal{O}$ be as in Definition 3.5. We have $\overline{W}_{\mathsf{F}^{+}}(\overline{\mathbf{U}}(x))=x$ in the first factor of $\mathcal{O}\ast\mathcal{O}$ and $\overline{W}_{\mathsf{F}^{+}}(\overline{\mathbf{L}}(y))=y$ in the second factor. In particular $\{\overline{\mathbf{U}}(x),\overline{\mathbf{L}}(y)\mid x,y\in\mathcal{O}\}$ generates $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O})$ .

Proof.

The fact that the equivalence relation does not change the parity of the number of $\mathbf{J}$ ’s implies (a). (b) follows from Proposition 3.16 after unwinding the definitions. ∎

Remark 3.19.

The map $\overline{M}$ of Proposition 3.12 on $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ takes $\overline{\mathbf{J}}$ to $J\colonequals D(0)=\left[\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right].$ For $x\in\mathcal{O}$ ,

\overline{M}(\overline{\mathbf{D}}(x))=D(x)\colonequals\begin{bmatrix}x&1\\ 1&0\end{bmatrix},\,\,\overline{M}(\overline{\mathbf{U}}(x))=U(x)\colonequals\begin{bmatrix}1&x\\ 0&1\end{bmatrix},\,\,\overline{M}(\overline{\mathbf{L}}(x))=L(x)\colonequals\begin{bmatrix}1&0\\ x&1\end{bmatrix}.

Definition 3.20.

(a)

The elementary matrices in $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ are $\{L(c),U(c)\mid c\in\mathcal{O}\}$ .
(b)

The elementary matrices in $\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ are $\{L(c),U(c),J\mid c\in\mathcal{O}\}$ .
(c)

The group ${\mathbb{E}}_{2}(\mathcal{O})$ is the subgroup of $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ generated by elementary matrices in $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ .
(d)

The group ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ is the subgroup of $\operatorname{\operatorname{SL}_{2}}^{\pm}(\mathcal{O})$ generated by elementary matrices in $\operatorname{\operatorname{SL}_{2}}^{\pm}(\mathcal{O})$ .
(e)

Set $\overline{M}^{+}=\overline{M}|_{\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O})}:\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O})\longrightarrow\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ .

Proposition 3.21.

The image of $\overline{M}:\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ is ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ and $\overline{M}(\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O}))={\mathbb{E}}_{2}(\mathcal{O})$ .

Proof.

By Remark 3.19, $\overline{M}$ maps generators of $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ to generators of ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ and generators of $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O})$ to generators of ${\mathbb{E}}_{2}(\mathcal{O})$ . ∎

Corollary 3.22.

We have ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})=\langle D(x):x\in\mathcal{O}\rangle$ and ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})\cap\operatorname{\operatorname{SL}_{2}}(\mathcal{O})={\mathbb{E}}_{2}(\mathcal{O}).$

Proof.

Since $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ is generated by $\mathbf{D}(x)$ , the equality ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})=\langle D(x):x\in\mathcal{O}\rangle$ is an immeditate consequence of Proposition 3.21. The kernel of the determinant map applied to ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ is ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})\cap\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ . The group $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathcal{O})$ is the kernel of determinant composed with $\overline{M}$ by Proposition 3.14 and Definition 3.15. Thus, the equality ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})\cap\operatorname{\operatorname{SL}_{2}}(\mathcal{O})={\mathbb{E}}_{2}(\mathcal{O})$ follows. ∎

Remark 3.23.

Note that ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})=\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ if and only if ${\mathbb{E}}_{2}(\mathcal{O})=\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ . For a number field $K$ with integers $\mathcal{O}_{K}$ , it is known that ${\mathbb{E}}_{2}(\mathcal{O}_{K})=\operatorname{\operatorname{SL}_{2}}(\mathcal{O}_{K})$ unless $K=\mathbf{Q}(\sqrt{-D})$ where $D\neq 1,2,3,7,11$ and is squarefree. Vaseršteĭn [Vas72] proved that ${\mathbb{E}}_{2}(\mathcal{O}_{K})=\operatorname{\operatorname{SL}_{2}}(\mathcal{O}_{K})$ if $K$ is not imaginary quadratic (see also [Lie81]), and Cohn [Coh66, Thm. 6.1] had previously settled the imaginary quadratic case.

Nica [Nic11] provides a short constructive proof that for $\mathcal{O}_{K}$ the ring of integers of an imaginary quadratic field $K$ , either ${\mathbb{E}}_{2}(\mathcal{O}_{K})=\operatorname{\operatorname{SL}_{2}}(\mathcal{O}_{K})$ or ${\mathbb{E}}_{2}(\mathcal{O}_{K})$ is an infinite index non-normal subgroup of $\operatorname{\operatorname{SL}_{2}}(\mathcal{O}_{K})$ .

3.3. The kernel of \excepttoc $\overline{M}$ \fortoc $\overline{M}$ and presentations of \excepttoc ${\mathbb{E}}_{2}(\mathcal{O})$ \fortoc ${\mathbb{E}}_{2}(\mathcal{O})$

The map $\overline{M}:\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ of Proposition 3.12 has image ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ by Proposition 3.21. It is a difficult problem to find the kernel of $\overline{M}$ . We begin with the observation that $\overline{M}$ is not injective.

Remark 3.24.

Let

x,y,z\in\mathcal{O}\quad\text{with}\quad w=-x-z-xyz.

Suppose $x,y,z\neq 0$ , and there exists $a,b\in\mathcal{O}$ such that $xy=aw$ , $yz=bw$ . Note $\mathcal{O}$ is an integral domain since $\mathcal{O}\subset\mathbf{C}$ , so $x,y\neq 0$ implies $w\neq 0$ . We also have $y+a+b+awb=0$ because $w(y+a+b+awb)=wy+xy+yz+xy^{2}z=y(w+x+z+xyz)=0$ . This is sufficient to compute that

	$\displaystyle\overline{M}(\|\![x,y,z,a,w,b]\!\|)$	$\displaystyle=\left(D(x)D(y)D(z)\right)\left(D(a)D(w)D(b)\right)$
		$\displaystyle=\begin{bmatrix}-w&aw+1\\ bw+1&y\end{bmatrix}\begin{bmatrix}-y&aw+1\\ bw+1&w\end{bmatrix}=1\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O}).$

Hence, for example we have the following nontrivial elements in the kernel of $\overline{M}$ :

$\displaystyle\|\![x,-x^{-1},x$	$\displaystyle,x^{-1},-x,x^{-1}]\!\|,$
$\displaystyle\|\![x,-4x^{-1},x$	$\displaystyle,-2x^{-1},2x,-2x^{-1}]\!\|,$	(8)
$\displaystyle\|\![x,-3x^{-1},x$	$\displaystyle,-3x^{-1},x,-3x^{-1}]\!\|,$	(9)
$\displaystyle\|\![x,\alpha,-\alpha^{-1}$	$\displaystyle,x\alpha^{2},\alpha^{-1},-\alpha]\!\|,$	(10)

provided $1/x,2/x,3/x,1/\alpha\in\mathcal{O}$ , respectively.

A ring $\mathcal{O}\subset\mathbf{C}$ has a norm $|\cdot|$ given by the usual absolute value.

Definition 3.25.

A ring $\mathcal{O}\subset\mathbf{C}$ is discretely normed [Coh66, p. 16] when every unit $x\in\mathcal{O}^{\times}$ has norm 1, and every nonzero, nonunit $x\in\mathcal{O}$ has norm at least $2$ .

An imaginary quadratic number ring $\mathcal{O}$ is discretely normed unless it has discriminant

-D\in\{-3,-4,-7,-8,-11,-12\}.

The exceptions arise from elements $x$ such that $|x|=\sqrt{2}$ when $-D=-4,-7,-8$ or $|x|=\sqrt{3}$ when $-D=-3,-8,-11,-12$ , namely

x=1+i,\,\frac{1+\sqrt{-7}}{2},\,\sqrt{-2},\,\quad\text{and}\quad x=\frac{3+\sqrt{-3}}{2},\,1+\sqrt{-2},\,\frac{1+\sqrt{-11}}{2},\,\sqrt{-3}

up to signs. These all give nontrivial kernel elements when substituted into expressions (8) and (9), respectively, such that all partial quotients are nonzero nonunits.

If $\mathcal{O}$ is any discretely normed ring and $|\![c_{1},c_{2},\ldots,c_{n}]\!|\in\ker\overline{M}$ , then some $c_{i}$ is a unit or zero as shown by a modification of [Coh66, Lemma 5.1].

In parallel with [Coh66, p. 27], we define the following subgroup of $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O}).$

Definition 3.26.

(a)

For $a\in\mathcal{O}^{\times}$ , let $\mathbf{c}(a)=|\![a,-a^{-1},a]\!|\in\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O}).$

(b)

Let $\mathbf{K}(\mathcal{O})$ be the normal closure in $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ of the group generated by

\mathbf{c}(1)^{2},

(11)

\mathbf{c}(a)\mathbf{c}(b)\mathbf{c}(a^{-1}b^{-1})\mathbf{c}(1),\ a,b\in\mathcal{O}^{\times},

(12)

\overline{\mathbf{D}}(x)\mathbf{c}(a)\overline{\mathbf{D}}(0)\overline{\mathbf{D}}(a^{2}x)\overline{\mathbf{D}}(0)\mathbf{c}(-a),\ a\in\mathcal{O}^{\times},x\in\mathcal{O}.

(13)

Calculating that $\overline{M}(\mathbf{c}(a))=\delta(a,-a^{-1})$ in the notation of Definition 2.2, it is routinely verified that (11) – (13) map to $I\in\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ under $\overline{M}$ and hence $\mathbf{K}(\mathcal{O})$ is a normal subgroup of $\ker\overline{M}$ . Relations (11) and (12) give $\langle\mathbf{c}(a):a\in\mathcal{O}^{\times}\rangle\,\,\text{mod}\,\mathbf{K}(\mathcal{O})\,\subset\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})/\mathbf{K}(\mathcal{O})$ the same commutative group structure as $\langle\overline{M}(\mathbf{c}(a))\rangle\subset\operatorname{\operatorname{SL}_{2}}^{\pm}(\mathcal{O})$ . In particular,

\mathbf{c}(1)\mathbf{c}(-1)\,\,\text{mod}\,\mathbf{K}(\mathcal{O})=\mathbf{c}(-1)\mathbf{c}(1)\,\,\text{mod}\,\mathbf{K}(\mathcal{O})

has order $2$ and maps by $\overline{M}$ to $-I$ . Relation (13) implies

\mathbf{c}(-a)\overline{\mathbf{D}}(x)\mathbf{c}(a)=\overline{\mathbf{D}}(-a^{2}x)\bmod\mathbf{K}(\mathcal{O}),

from which we see that $\mathbf{c}(1)\mathbf{c}(-1)\bmod\mathbf{K}(\mathcal{O})$ is central in $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})/\mathbf{K}(\mathcal{O})$ .

Definition 3.27.

Borrowing the terminology of [Coh66, p. 8], we say a ring $\mathcal{O}$ is universal for ${\mathbb{E}}_{2}^{\pm}$ when $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})/\mathbf{K}(\mathcal{O})\cong{\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ , in other words, if $\ker\overline{M}=\mathbf{K}(\mathcal{O})$ .

Remark 3.28.

Note that taking $a=-1,b=1$ in (12) gives $\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\in\mathbf{K}(\mathcal{O})$ for any $\mathcal{O}$ , implying the same for its conjugate $\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)$ . Set

\mathbf{K^{\prime}}(\mathcal{O})\colonequals\operatorname{ncl}\langle\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1),|\![x,1,-1,x,1,-1]\!|,|\![x,-1,1,x,-1,1]\!|,x\in\mathcal{O}\rangle,

(14)

the normal closure being taken in $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ . Note that $\mathbf{c}(1)^{2},\,\mathbf{c}(-1)^{2}\in\mathbf{K}^{\prime}(\mathcal{O})$ by taking $x=1$ , $x=-1$ in (14). From (12), (13) we see that $\mathbf{K}^{\prime}(\mathcal{O})\lhd\,\mathbf{K}(\mathcal{O})$ . When $\mathcal{O}^{\times}=\langle\pm 1\rangle$ , we have $\mathbf{K}^{\prime}(\mathcal{O})=\mathbf{K}(\mathcal{O})$ .

Proposition 3.29.

Let

\mathbf{K}^{\prime\prime}(\mathbf{Z})=\operatorname{ncl}\langle\mathbf{c}(1)^{2},\mathbf{c}(-1)^{2},\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)\rangle,

(15)

the normal closure being taken in $\operatorname{\overline{\textup{{FCF}}}}(\mathbf{Z})$ . Then $\mathbf{K}(\mathbf{Z})=\mathbf{K}^{\prime}(\mathbf{Z})=\mathbf{K}^{\prime\prime}(\mathbf{Z})$ and $\mathbf{K}(\mathbf{Z})$ is a normal subgroup of $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ , equal to the normal closure of $\langle\mathbf{c}(1)^{2},\mathbf{c}(-1)^{2},\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)\rangle$ in $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ .

Proof.

To show that $\mathbf{K}^{\prime\prime}(\mathbf{Z})=\mathbf{K}^{\prime}(\mathbf{Z})=\mathbf{K}(\mathbf{Z})$ , it suffices to show that

\langle|\![x,1,-1,x,1,-1]\!|,|\![x,-1,1,x,-1,1]\!|,x\in\mathcal{O}\rangle\subseteq\mathbf{K}^{\prime\prime}(\mathbf{Z})

(16)

in light of Remark 3.28. Verify that

\mathbf{c}(1)^{2}|\![x]\!|^{-1}|\![x-1,1,-1,x-1,1,-1]\!||\![x]\!|=|\![x]\!|^{-1}|\![x,1,-1,x,1,-1]\!||\![x]\!|

for $x\in\mathcal{O}$ , and hence

|\![x-1,1,-1,x-1,1,-1]\!|\in\mathbf{K}^{\prime\prime}(\mathbf{Z})\Longleftrightarrow|\![x,1,-1,x,1,-1]\!|\in\mathbf{K}^{\prime\prime}(\mathbf{Z}).

Since $\mathbf{c}(-1)^{2}=|\![-1,1,-1,-1,1,-1]\!|\in\mathbf{K}^{\prime\prime}(\mathbf{Z})$ , it follows by induction that

|\![x,1,-1,x,1,-1]\!|\in\mathbf{K}^{\prime\prime}(\mathbf{Z})\text{ for all $x\in\mathbf{Z}$.}

Likewise the identity

\mathbf{c}(-1)^{2}|\![x]\!|^{-1}|\![x+1,-1,1,x+1,-1,1]\!||\![x]\!|=|\![x]\!|^{-1}|\![x,-1,1,x,-1,1]\!||\![x]\!|

implies that

|\![x+1,-1,1,x+1,-1,1]\!|\in\mathbf{K}^{\prime\prime}(\mathbf{Z})\Longleftrightarrow|\![x,-1,1,x,-1,1]\!|\in\mathbf{K}^{\prime\prime}(\mathbf{Z}).

Since $|\![1,-1,1,1,-1,1]\!|=\mathbf{c}(1)^{2}\in\mathbf{K}^{\prime\prime}(\mathbf{Z})$ , induction shows that $|\![x,-1,1,x,-1,1]\!|\in\mathbf{K}^{\prime\prime}(\mathbf{Z})$ for all $x\in\mathbf{Z}$ , establishing (16).

We have $\mathbf{K}(\mathbf{Z})\leq\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ since each of its generators in (15) has even length, satisfying Definition 3.15. Since $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ is a normal subgroup of index $2$ with the nontrivial coset of $\operatorname{\overline{\textup{{FCF}}}}(\mathbf{Z})/\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ containing $|\![0]\!|=|\![0]\!|^{-1}$ , to show that the normal closure of $\langle\mathbf{c}(1)^{2},\mathbf{c}(-1)^{2},\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)\rangle$ in $\operatorname{\overline{\textup{{FCF}}}}(\mathbf{Z})$ is equal to its normal closure in $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ , it suffices to observe that

	$\displaystyle\|\![0]\!\|\mathbf{c}(-1)^{2}\|\![0]\!\|$	$\displaystyle=\mathbf{c}(1)^{-2}$
	$\displaystyle\|\![0]\!\|\mathbf{c}(1)^{2}\|\![0]\!\|$	$\displaystyle=\mathbf{c}(-1)^{-2}$
	$\displaystyle\|\![0]\!\|\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)\|\![0]\!\|$	$\displaystyle=[\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)]^{-1}.\qed$

Theorem 3.30 (Cohn).

The map $\overline{M}:\operatorname{\overline{\textup{{FCF}}}}(\mathbf{Z})\rightarrow{\mathbb{E}}_{2}^{\pm}(\mathbf{Z})=\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathbf{Z})$ is surjective with kernel $\mathbf{K}(\mathbf{Z})$ . Likewise $\overline{M}^{\,+}:\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})\rightarrow{\mathbb{E}}_{2}(\mathbf{Z})$ is surjective with kernel $\mathbf{K}(\mathbf{Z})$ .

Proof.

Cohn [Coh66, Thm. 5.2] shows that discretely normed rings are universal for ${\mathbb{E}}_{2}^{\pm}$ as in Definition 3.27, and $\mathbf{Z}$ is discretely normed. The statement for $\overline{M}$ implies that for $\overline{M}^{\,+}$ . Of course, it follows from the Euclidean algorithm that ${\mathbb{E}}_{2}^{\pm}(\mathbf{Z})=\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathbf{Z})$ ; see, for example,[Art91, Chap. 12, Sect. 4]. ∎

Knowing the kernel of $\overline{M}^{\,+}:\operatorname{\overline{\textup{{FCF}}}}^{+}(\mathcal{O})\cong\mathcal{O}\ast\mathcal{O}\rightarrow{\mathbb{E}}_{2}(\mathcal{O})\leq\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ gives a presentation of ${\mathbb{E}}_{2}(\mathcal{O})$ , and a presentation of $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ if $\overline{M}^{\,+}$ is surjective. More generally, information on $\ker(\overline{M}^{\,+})$ gives information on the generation of $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ if ${\mathbb{E}}_{2}(\mathcal{O})=\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ . It is an interesting question whether information on $\ker(\overline{M}^{\,+})$ , such as knowing $\mathbf{K}(\mathcal{O})\leq\ker(\overline{M}^{\,+})$ , gives results on bounded generation in case $\mathcal{O}^{\times}$ is infinite as in [MRS18] and [JZ19].

We show that the presentation of $\operatorname{\operatorname{SL}_{2}}(\mathbf{Z})$ given by Theorem 3.30 gives the familiar amalgamated product presentation.

Corollary 3.31.

We have $\operatorname{\operatorname{SL}_{2}}(\mathbf{Z})\cong\mathbf{Z}/6\mathbf{Z}\ast_{\mathbf{Z}/2\mathbf{Z}}\mathbf{Z}/4\mathbf{Z}$ .

Proof.

Note that by Remark 3.7, $\mathbf{c}(-1)^{-1}=|\![0,1,-1,1,0]\!|$ . Let $\alpha=|\![1,-1]\!|\in\operatorname{\overline{\textup{{FCF}}}}^{+}(\mathbf{Z})$ and $\beta=|\![1,-1,1,0]\!|=\mathbf{c}(1)|\![0]\!|\in\operatorname{\overline{\textup{{FCF}}}}^{+}(\mathbf{Z})$ , so that

\alpha^{3}=\mathbf{c}(1)\mathbf{c}(-1),\,\beta^{2}=\mathbf{c}(1)\mathbf{c}(-1)^{-1},\,\overline{M}(\alpha)=\begin{bmatrix}0&1\\ -1&1\end{bmatrix},\,\text{ and }\overline{M}(\beta)=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}.

(17)

We now consider the group $\mathbf{Z}/6\mathbf{Z}\ast_{\mathbf{Z}/2\mathbf{Z}}\mathbf{Z}/4\mathbf{Z}$ . A presentation of this group is the free group on two generators $\bm{\alpha}$ and $\bm{\beta}$ modulo the obvious relations:

\mathbf{Z}/6\mathbf{Z}\ast_{\mathbf{Z}/2\mathbf{Z}}\mathbf{Z}/4\mathbf{Z}\cong\frac{\langle\bm{\alpha}\rangle\ast\langle\bm{\beta}\rangle}{\operatorname{ncl}\langle\bm{\alpha}^{6},\bm{\alpha}^{3}\bm{\beta}^{2},\bm{\beta}^{4}\rangle},

(18)

where $\operatorname{ncl}\langle\bm{\alpha}^{6},\bm{\alpha}^{3}\bm{\beta}^{2},\bm{\beta}^{4}\rangle$ is the normal closure of $\langle\bm{\alpha}^{6},\bm{\alpha}^{3}\bm{\beta}^{2},\bm{\beta}^{4}\rangle$ in $\langle\bm{\alpha}\rangle\ast\langle\bm{\beta}\rangle$ .

We now state and prove a series of claims:
Claim 1. Set $\mathbf{K}^{\prime\prime}(\mathbf{Z})\colonequals\operatorname{ncl}\langle\alpha^{6},\alpha^{3}\beta^{2},\beta^{4}\rangle$ , the normal closure of $\langle\alpha^{6},\alpha^{3}\beta^{2},\beta^{4}\rangle$ in $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ . Then $\mathbf{K}^{\prime\prime}(\mathbf{Z})$ is the normal closure of $\langle\mathbf{c}(1)^{2},\mathbf{c}(-1)^{2},\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)\rangle$ in $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ . So by Proposition 3.29, $\mathbf{K}^{\prime\prime}(\mathbf{Z})=\mathbf{K}(\mathbf{Z})=\mathbf{K}^{\prime}(\mathbf{Z})$ .
Proof. $\mathbf{K}^{\prime\prime}(\mathbf{Z})\subseteq\mathbf{K}(\mathbf{Z})$ : Using (17), verify that $\overline{M}(\alpha^{6})=\overline{M}(\alpha^{3}\beta^{2})=\overline{M}(\beta^{4})=1\in\operatorname{\operatorname{SL}_{2}}(\mathbf{Z})$ . Hence $\mathbf{K}^{\prime\prime}(\mathbf{Z})\subseteq\ker(\overline{M})=\ker(\overline{M}^{\,+})=\mathbf{K}(\mathbf{Z})$ by Theorem 3.30.
$\mathbf{K}(\mathbf{Z})\subseteq\mathbf{K}^{\prime\prime}(\mathbf{Z})$ : It suffices to note the following:

	$\displaystyle\mathbf{c}(1)\mathbf{c}(-1)\mathbf{c}(1)\mathbf{c}(-1)$	$\displaystyle=\alpha^{6},$
	$\displaystyle\mathbf{c}(-1)^{2}$	$\displaystyle=\beta^{-2}\alpha^{3}=(\alpha^{3}\beta^{2})^{-1}\alpha^{6},$
	$\displaystyle\mathbf{c}(1)^{2}$	$\displaystyle=\beta\alpha^{-3}\beta=\beta^{-1}[\beta^{4}(\alpha^{3}\beta^{2})^{-1}]\beta\in\operatorname{ncl}\langle\alpha^{6},\alpha^{3}\beta^{2},\beta^{4}\rangle\equalscolon\mathbf{K}^{\prime\prime}(\mathbf{Z}),$

proving Claim 1.
Claim 2. The elements $\alpha$ and $\beta$ generate $\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})$ .
Proof. By Proposition 3.18, $\operatorname{\overline{\textup{{PCF}}}}^{\,+}(\mathbf{Z})$ is a free group on the two generators $\overline{\mathbf{U}}(-1)=\overline{\mathbf{U}}(1)^{-1}$ and $\overline{\mathbf{L}}(-1)=\overline{\mathbf{L}}(1)^{-1}$ . But

	$\displaystyle\overline{\mathbf{U}}(-1)$	$\displaystyle=\|\![-1,0]\!\|=\|\![0,0,-1,1,-1,0]\!\|\|\![1,-1]\!\|=\beta^{-1}\alpha$
	$\displaystyle\overline{\mathbf{L}}(-1)$	$\displaystyle=\|\![0,-1]\!\|=\|\![0,0,-1,1,-1,0]\!\|\|\![1,-1]\!\|\|\![1,-1]\!\|=\beta^{-1}\alpha^{2},$

proving Claim 2.
We can now conclude the proof of Corollary 3.31. By Theorem 3.30 we have

	$\displaystyle\operatorname{\operatorname{SL}_{2}}(\mathbf{Z})$	$\displaystyle\cong\frac{\operatorname{\overline{\textup{{FCF}}}}^{\,+}(\mathbf{Z})}{\mathbf{K}(\mathbf{Z})}\cong\frac{\langle\alpha\rangle\ast\langle\beta\rangle}{\operatorname{ncl}\langle\alpha^{6},\alpha^{3}\beta^{2},\beta^{4}\rangle}\quad\text{by Claims 1 and 2}$
		$\displaystyle\cong\mathbf{Z}/6\mathbf{Z}\ast_{\mathbf{Z}/2\mathbf{Z}}\mathbf{Z}/4\mathbf{Z}\quad\text{by \eqref{warm}}.\qed$

When $\mathcal{O}$ is a discretely normed quadratic imaginary number ring, Cohn shows $\mathbf{K}(\mathcal{O})=\ker\overline{M}$ and, in [Coh66, Thm. 6.1], ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})\subset\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ is a proper containment.

4. Periodic continued fractions

An infinite continued fraction $C$ is a formal expression $[c_{1},c_{2},\ldots]$ where the sequence of partial quotients does not terminate. The infinite continued fraction $C=[c_{1},c_{2},\ldots]$ may also be expressed as a nonterminating version of (5). As in the finite case, the convergent $\mathcal{C}_{k}(C)$ is the evaluation of $[c_{1},c_{2},\ldots,c_{k}].$ For $M$ as in (7), set $M_{k}(C)\colonequals M([c_{1},c_{2},\ldots,c_{k}])$ with $M_{0}(C)=I$ so that $\mathcal{C}_{k}(C)=M_{k}(C)_{11}/M_{k}(C)_{21}$ . The value (or limit) of $C$ is

\hat{C}=\lim_{k\to\infty}\mathcal{C}_{k}(C)

when the limit exists in $\mathbb{P}^{1}(\mathbf{C})$ , in which case $C$ converges.

A periodic continued fraction (PCF) $P$ is an infinite continued fraction $[c_{1},c_{2},\ldots]$ together with a type $(N,k)$ with $N\geq 0$ , $k\geq 1$ such that $c_{n+k}=c_{n}$ for $n>N$ . We denote the sequence of partial quotients of the PCF $P$ by

P=[b_{1},\ldots,b_{N},\overline{a_{1},a_{2},\ldots,a_{k}}\,].

(19)

The natural number $k$ is the period of $P$ ; the initial part of $P$ in (19) is the FCF $\operatorname{In}(P)=[b_{1},\ldots,b_{N}]$ ; and the repeated part of $P$ is the FCF $\operatorname{Rep}(P)=[a_{1},\ldots,a_{k}]$ . The PCF $P$ as in (19) has Galois dual

P^{\ast}\colonequals[b_{1},\ldots,b_{N},0,\overline{-a_{k},\ldots,-a_{1}}\,];

(20)

see, e.g., [BEJ21, p. 384]. For a ring $\mathcal{O}\subseteq\mathbf{C}$ containing 1, say $P=[b_{1},\ldots,b_{N},\overline{a_{1},a_{2},\ldots,a_{k}}\,]$ is an $\mathcal{O}$ - $\operatorname{\textup{{PCF}}}$ if $b_{i},a_{j}\in\mathcal{O}$ for $1\leq i\leq N$ , $1\leq j\leq k$ . Let $\operatorname{\textup{{PCF}}}(\mathcal{O})$ denote the set of all $\mathcal{O}$ -PCFs. If $P\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ , then $P^{\ast}\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ .

A PCF with type $(0,k)$ is purely periodic. For brevity, we abbreviate purely periodic continued fraction as RCF, after the phrase repeating continued fraction. The set of RCFs which are $\mathcal{O}$ -PCFs is denoted $\operatorname{\textup{{RCF}}}(\mathcal{O})$ . An untyped periodic continued fraction (UCF) $U$ is an infinite continued fraction which is periodic of some type $(N,k)$ . Notice that the type of a UCF is not uniquely determined: A UCF $U$ with type $(N,k)$ also has type $(N^{\prime},mk)$ for every $N^{\prime}\geq N$ and every multiple $mk$ of $k$ .

Definition 4.1.

There are three equality relationships between a PCF $P$ of type $(N,k)$ and a PCF $P^{\prime}$ of type $(N^{\prime},k^{\prime})$ .

(a)

$P$ and $P^{\prime}$ are CF-equal (written $=_{\textrm{CF}}$ ) when $P$ and $P^{\prime}$ are the same as UCFs.
(b)

$P$ and $P^{\prime}$ are k-equal (written $=_{\textrm{k}}$ ) when $P$ and $P^{\prime}$ are the same as UCFs and $k=k^{\prime}$ .
(c)

$P$ and $P^{\prime}$ are equal (written $=$ ) when $P$ and $P^{\prime}$ are the same as PCFs, that is, $P=_{\textrm{CF}}P^{\prime}$ and $(N,k)=(N^{\prime},k^{\prime})$ .

For example, $[1,\overline{2}]=_{\textrm{CF}}[1,\overline{2,2}]$ , but $[1,\overline{2}\,]\neq_{\textrm{k}}[1,\overline{2,2}\,]$ . And $[1,\overline{2,1}\,]=_{\textrm{k}}[1,2,\overline{1,2}\,]$ , but $[1,\overline{2,1}\,]\neq[1,2,\overline{1,2}\,]$ .

4.1. An equivalence relation and group law on PCFs

We give an equivalence relation on $\operatorname{\textup{{PCF}}}(\mathcal{O})$ and a group law on the equivalence classes by means of the group law on $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ .

Definition 4.2.

The map $W_{\mathsf{P}}:\operatorname{\textup{{PCF}}}(\mathcal{O})\rightarrow\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ maps the PCF $P$ with initial part $\operatorname{In}(P)$ and repeating part $\operatorname{Rep}(P)$ to

W_{\mathsf{P}}(P)=W_{\mathsf{F}}(\operatorname{In}(P))W_{\mathsf{F}}(\operatorname{Rep}(P))W_{\mathsf{F}}(\operatorname{In}(P))^{-1}=W_{\mathsf{F}}(\operatorname{In}(P))W_{\mathsf{F}}(\operatorname{Rep}(P))W_{\mathsf{F}}(\operatorname{In}(P)^{\ast}),

(21)

where $W_{\mathsf{F}}$ is as in Definition 3.5 and $\operatorname{In}(P)^{\ast}$ is as in Remark 3.7.

The type $(N,k)$ is essential data attached to $P$ for computing the map $W_{\mathsf{P}}$ . We also have maps $\operatorname{\mathscr{F}}:\operatorname{\textup{{PCF}}}(\mathcal{O})\rightarrow\operatorname{\textup{{FCF}}}(\mathcal{O})$ and $\operatorname{\mathscr{R}}:\operatorname{\textup{{FCF}}}(\mathcal{O})\rightarrow\operatorname{\textup{{RCF}}}(\mathcal{O})\subseteq\operatorname{\textup{{PCF}}}(\mathcal{O})$ defined by

	$\displaystyle\operatorname{\mathscr{F}}([b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k}}\,])$	$\displaystyle=\begin{cases}[b_{1},\ldots,b_{N},a_{1},\ldots,a_{k},0,-b_{N},\ldots,-b_{1},0],\ N>0\\ [a_{1},\ldots,a_{k}],\ N=0.\end{cases}$		(22)
	$\displaystyle\operatorname{\mathscr{R}}([c_{1},\ldots,c_{n}])$	$\displaystyle=[\,\overline{c_{1},\ldots,c_{n}}\,].$		(23)

Note that $\operatorname{\mathscr{F}}\circ\operatorname{\mathscr{R}}$ is the identity on $\operatorname{\textup{{FCF}}}(\mathcal{O})$ and the map $W_{\mathsf{P}}$ factors as

W_{\mathsf{P}}=W_{\mathsf{F}}\circ\operatorname{\mathscr{F}}.

(24)

The maps $\operatorname{\mathscr{R}}$ and $\operatorname{\mathscr{F}}|_{\operatorname{\textup{{RCF}}}(\mathcal{O})}$ give inverse bijections:

\operatorname{\mathscr{F}}|_{\operatorname{\textup{{RCF}}}(\mathcal{O})}:\operatorname{\textup{{RCF}}}(\mathcal{O})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{\textup{{FCF}}}(\mathcal{O})\quad\text{and}\quad\operatorname{\mathscr{R}}:\operatorname{\textup{{FCF}}}(\mathcal{O})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{\textup{{RCF}}}(\mathcal{O}).

(25)

The equivalence relation $\sim$ on $\operatorname{\textup{{FCF}}}(\mathcal{O})$ defined in Section 3.1 induces an equivalence relation, also denoted $\sim$ . on $\operatorname{\textup{{PCF}}}(\mathcal{O})$ .

Definition 4.3.

For $P,P^{\prime}\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ , say

	$\displaystyle P\sim P^{\prime}$	$\displaystyle\Longleftrightarrow\operatorname{\mathscr{F}}(P)\sim\operatorname{\mathscr{F}}(P^{\prime})\Longleftrightarrow W_{\mathsf{F}}(\operatorname{\mathscr{F}}(P))=W_{\mathsf{F}}(\operatorname{\mathscr{F}}(P^{\prime}))$
		$\displaystyle\Longleftrightarrow W_{\mathsf{P}}(P)=W_{\mathsf{P}}(P^{\prime})\in\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}.$

Remark 4.4.

For $F,F^{\prime}\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ we have

F\sim F^{\prime}\Longleftrightarrow\operatorname{\mathscr{R}}(F)\sim\operatorname{\mathscr{R}}(F^{\prime})\text{ since }\operatorname{\mathscr{F}}\left(\operatorname{\mathscr{R}}(F)\right)=F.

(26)

Proposition 4.5.

If $P,P^{\prime}\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ and $P=_{\textrm{k}}P^{\prime}$ as in Definition 4.1(b), then $P\sim P^{\prime}$ .

Proof.

First consider the case that $P$ is of type $(N,k)$ and $P^{\prime}$ is of type $(N+1,k)$ with $P=_{\textrm{k}}P^{\prime}$ . Then we must have

P=[b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k}}]\quad\text{and}\quad P^{\prime}=[b_{1},\ldots,b_{N},a_{1},\overline{a_{2},\ldots,a_{k},a_{1}}].

We have

	$\displaystyle\operatorname{\mathscr{F}}(P)$	$\displaystyle=[b_{1},\ldots,b_{N},a_{1},\ldots,a_{k},0,-b_{N},\ldots,-b_{1},0]\text{ and}$
	$\displaystyle\operatorname{\mathscr{F}}(P^{\prime})$	$\displaystyle=[b_{1},\dots,b_{N},a_{1},a_{2},\ldots,a_{k},a_{1},0,-a_{1},-b_{N},\ldots,-b_{1},0]$

with $\operatorname{\mathscr{F}}$ as in (22). But then $\operatorname{\mathscr{F}}(P)\sim\operatorname{\mathscr{F}}(P^{\prime})$ by applying (6).

The proposition then follows from this by iteration. ∎

Definition 4.6.

An RCF $R=[\,\overline{a_{1},\ldots a_{n}}\,]\in\operatorname{\textup{{RCF}}}(\mathcal{O})$ is reduced when $\operatorname{\mathscr{F}}(R)\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ is reduced as in Definition 3.8, i.e., $a_{i}\neq 0$ for $2\leq i\leq n-1$ .

Remark 4.7.

Note that $F\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ is reduced if and only if $\operatorname{\mathscr{R}}(F)\in\operatorname{\textup{{RCF}}}(\mathcal{O})$ is reduced since $\operatorname{\mathscr{F}}\circ\operatorname{\mathscr{R}}(F)=F$ .

Proposition 4.8.

For $P\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ , let $\operatorname{\mathscr{F}}(P)_{\mathrm{red}}\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ be as in Proposition 3.9. Then the element $P_{\mathrm{red}}=\operatorname{\mathscr{R}}(\operatorname{\mathscr{F}}(P)_{\mathrm{red}})$ is the unique reduced element of $\operatorname{\textup{{RCF}}}(\mathcal{O})$ such that $P_{\mathrm{red}}\sim P.$

Proof.

For $P\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ we have that

P_{\mathrm{red}}=\operatorname{\mathscr{R}}\left(\operatorname{\mathscr{F}}(P)_{\mathrm{red}}\right)\in\operatorname{\textup{{RCF}}}(\mathcal{O})

is reduced, as stated in Remark 4.7. We claim that $P\sim P_{\mathrm{red}}$ . Note that $\operatorname{\mathscr{F}}(P)\sim\operatorname{\mathscr{F}}(P)_{\mathrm{red}}$ implies that $\operatorname{\mathscr{R}}\left(\operatorname{\mathscr{F}}(P)\right)\sim\operatorname{\mathscr{R}}\left(\operatorname{\mathscr{F}}(P)_{\mathrm{red}}\right)$ as in Remark 4.4. By Definition 4.3, $\operatorname{\mathscr{F}}(P)=\operatorname{\mathscr{F}}(\operatorname{\mathscr{R}}(\operatorname{\mathscr{F}}(P)))$ implies $P\sim\operatorname{\mathscr{R}}(\operatorname{\mathscr{F}}(P))$ . Thus, $P\sim\operatorname{\mathscr{R}}\left(\operatorname{\mathscr{F}}(P)_{\mathrm{red}}\right)$ . Suppose $R,R^{\prime}\in\operatorname{\textup{{RCF}}}(\mathcal{O})$ are both reduced and $R\sim R^{\prime}$ . In this case, $\operatorname{\mathscr{F}}(R),\operatorname{\mathscr{F}}(R^{\prime})\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ are both reduced and $\operatorname{\mathscr{F}}(R)\sim\operatorname{\mathscr{F}}(R^{\prime})$ by Definition 4.3. Proposition 3.9 shows that $\operatorname{\mathscr{F}}(R)=\operatorname{\mathscr{F}}(R^{\prime})\in\operatorname{\textup{{FCF}}}(\mathcal{O})$ and hence $R=R^{\prime}$ by (25). ∎

Write $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\colonequals\operatorname{\textup{{PCF}}}(\mathcal{O})/\sim$ . For $P=[b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k}}\,]\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ , let $\overline{P}=|\![b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k}}\,]\!|\in\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ be the equivalence class containing $P$ . By Definition 4.3 and Remark 4.4, the maps $\operatorname{\mathscr{F}}$ and $\operatorname{\mathscr{R}}$ induce inverse bijections

\overline{\operatorname{\mathscr{F}}}:\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\quad\text{and}\quad\overline{\operatorname{\mathscr{R}}}:\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O}).

Via the bijections $\overline{\operatorname{\mathscr{F}}}$ , $\overline{\operatorname{\mathscr{R}}}$ the binary operation $\star$ of Proposition 3.4 on $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})$ induces a binary operation, also denoted $\star$ , on $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ : for $P,P^{\prime}\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ we have

\overline{\operatorname{\mathscr{F}}}(\overline{P}\star\overline{P}^{\prime})=\overline{\operatorname{\mathscr{F}}}(\overline{P})\star\overline{\operatorname{\mathscr{F}}}(\overline{P}^{\prime})\quad\text{and}\quad\overline{P}\star\overline{P}^{\prime}=\overline{\operatorname{\mathscr{R}}}\left(\overline{\operatorname{\mathscr{F}}}(\overline{P})\star\overline{\operatorname{\mathscr{F}}}(\overline{P}^{\prime})\right).

(27)

Theorem 4.9.

The semigroup $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ with its binary operation $\star$ is a group isomorphic to $\operatorname{\overline{\textup{{FCF}}}}(\mathcal{O})\cong\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ . The map $W_{\mathsf{P}}:\operatorname{\textup{{PCF}}}(\mathcal{O})\rightarrow\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ of Definition 4.2 induces an isomorphism of groups $\overline{W}_{\mathsf{P}}:\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ .

For the convenience of the reader we give in Figure 1 the maps defined thus far and the relations between them.

Figure 1. The commutative diagrams formed by the maps in Sections 3 and 4.1. The maps

W_{\mathsf{P}}=W_{\mathsf{F}}\circ\operatorname{\mathscr{F}}:\operatorname{\textup{{PCF}}}(\mathcal{O})\rightarrow\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}

and

\overline{W}_{\mathsf{P}}=\overline{W}_{\mathsf{F}}\circ\overline{\operatorname{\mathscr{F}}}:\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\rightarrow\mathbf{Z}/2\ast\mathcal{O}

can be added to the diagrams.

Remark 4.10.

If $\overline{P}\in\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ with $P\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ , then Proposition 4.8 shows that the equivalence class $\overline{P}$ contains a unique reduced RCF $P_{\textrm{red}}=\overline{P}_{\textrm{red}}$ . We call the $P_{\textrm{red}}=\overline{P}_{\textrm{red}}$ the normal form of $\overline{P}$ . Two elements $\overline{P},\overline{P}^{\prime}\in\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ are equal if and only if they have the same normal forms:

\overline{P}=\overline{P}^{\prime}\Longleftrightarrow\overline{P}_{\textrm{red}}=\overline{P}^{\prime}_{\textrm{red}}\in\operatorname{\textup{{RCF}}}(\mathcal{O}).

For $\mathcal{O}\neq\mathbf{C}$ , there are PCFs $P$ with not all partial quotients in $\mathcal{O}$ with normal form $P_{\mathrm{red}}\in\operatorname{\textup{{RCF}}}(\mathcal{O})$ ; most simply, $[\,\overline{c,0,-c}\,]\sim[\,\overline{0}\,]$ for any $c\in\mathbf{C}$ . We test the equivalence of $P,P^{\prime}\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ via their normal forms $P_{\mathrm{red}},P^{\prime}_{\mathrm{red}}\in\operatorname{\textup{{RCF}}}(\mathcal{O})$ . It is never necessary to make an excursion outside $\mathcal{O}$ to determine equivalence of PCFs with partial quotients in $\mathcal{O}$ .

We give an example of how to compute $P_{\mathrm{red}}$ using Proposition 4.8 and Remark 3.11. Suppose $P=[3,0,-3,4,\overline{0,1,0,-5}\,]\in\operatorname{\textup{{PCF}}}(\mathbf{Z})$ . Then

\operatorname{\mathscr{F}}(P)=[3,0,-3,4,0,1,0,-5,0,-4,3,0,-3,0]\in\operatorname{\textup{{FCF}}}(\mathbf{Z}).

We have

\operatorname{\mathscr{F}}(P)\sim[0,4,0,1,0,-5,0,-4,3,0,-3,0]\sim[0,5,0,-9,0,0]\sim[0,-4,0,0]\sim[0,-4].

Hence $\operatorname{\mathscr{F}}(P)_{\mathrm{red}}=[0,-4]\in\operatorname{\textup{{FCF}}}(\mathbf{Z})$ and $P_{\mathrm{red}}=\operatorname{\mathscr{R}}([0,-4])=[\,\overline{0,-4}\,]\in\operatorname{\textup{{RCF}}}(\mathbf{Z})$ with

|\![3,0,-3,4,\overline{0,1,0,-5}\,]\!|=\overline{P}=\overline{P}_{\mathrm{red}}=|\![\,\overline{0,-4}\,]\!|\in\operatorname{\overline{\textup{{PCF}}}}(\mathbf{Z}).

In practice, $\overline{P}\star\overline{P}^{\prime}\in\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ is computed using the normal forms $\overline{P}_{\textrm{red}}$ , $\overline{P}^{\prime}_{\textrm{red}}$ : if $\overline{P}_{\textrm{red}}=[\,\overline{c_{1},\ldots,c_{n}}\,]\in\operatorname{\textup{{RCF}}}(\mathcal{O})$ and $\overline{P}^{\prime}_{\textrm{red}}=[\,\overline{c^{\prime}_{1},\ldots,c^{\prime}_{n^{\prime}}}\,]\in\operatorname{\textup{{RCF}}}(\mathcal{O})$ , then

\overline{P}\star\overline{P}^{\prime}=|\![\,\overline{c_{1},\ldots,c_{n},c^{\prime}_{1},\ldots,c^{\prime}_{n^{\prime}}}\,]\!|\in\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O}).

Remark 4.11.

Suppose

P=[b_{1},\overline{a_{1},\ldots,a_{k}}\,],\,P^{\prime}=[b_{1}^{\prime},\overline{a_{1}^{\prime},\ldots,a_{k^{\prime}}^{\prime}}\,]\in\operatorname{\textup{{PCF}}}(\mathcal{O})

are of types $(1,k)$ and $(1,k^{\prime})$ respectively. Then the product takes a particularly simple form with a representative of type $(1,k+k^{\prime})$ :

$\displaystyle\overline{P}\star\overline{P^{\prime}}$	$\displaystyle=\|\![\,\overline{b_{1},a_{1},\ldots,a_{k},0,-b_{1},0,b_{1}^{\prime},a_{1}^{\prime},\ldots a_{k^{\prime}}^{\prime},0,-b_{1}^{\prime},0}\,]\!\|\text{ by \eqref{status}}$
	$\displaystyle=\|\![\,\overline{b_{1},a_{1},\ldots,a_{k}+b_{1}^{\prime}-b_{1},a_{1}^{\prime},\ldots,a_{k^{\prime}}^{\prime}-b_{1}^{\prime},0}\,]\!\|\text{ using \eqref{muster}}$
	$\displaystyle=\|\![b_{1},\overline{a_{1},\ldots,a_{k-1},a_{k}+b^{\prime}_{1}-b_{1},a^{\prime}_{1},\ldots,a^{\prime}_{k^{\prime}-1},a^{\prime}_{k^{\prime}}-b_{1}^{\prime},0,b_{1}}\,]\!\|$
	$\displaystyle=\|\![b_{1},\overline{a_{1},\ldots,a_{k-1},a_{k}+b_{1}^{\prime}-b_{1},a_{1}^{\prime},\ldots,a^{\prime}_{k^{\prime}-1},a^{\prime}_{k^{\prime}}-b^{\prime}_{1}+b_{1}}\,]\!\|.$	(28)

A similar exercise shows that the product is simple for $P,P^{\prime}\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ with $\operatorname{In}(P)=\operatorname{In}(P^{\prime})$ . Suppose

P=[b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k}}\,]\text{ and }P^{\prime}=[b_{1},\ldots,b_{N},\overline{a_{1}^{\prime},\ldots,a_{k^{\prime}}^{\prime}}\,].

Then

\overline{P}\star\overline{P^{\prime}}=|\![b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k},a_{1}^{\prime},\ldots,a_{k^{\prime}}^{\prime}}\,]\!|.

(29)

Remark 4.12.

For every PCF of the form $P=[b_{1},\ldots,b_{N},\overline{0}\,]$ , $W_{\mathsf{P}}(P)\in\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ is an involution, but, in general, $P$ is not equivalent to $[\,\overline{0}\,]$ .

Every PCF of the form $[b_{1},\ldots,b_{N},\overline{0,0}\,]$ is in the class of the identity in $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ , that is, $P\sim[\,\overline{0,0}\,]$ .

5. The subgroup $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)\leq\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ and its representations

For $P\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ , [BEJ21, Defn. 2.4] defines a matrix $E(P)$ which plays a large role in the study of $P$ . We review the definition, using the notation of this paper.

Definition 5.1.

Let $E:\operatorname{\textup{{PCF}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ be the map $E=M\circ\operatorname{\mathscr{F}}$ with $M$ as in (7) and $\operatorname{\mathscr{F}}$ as in (22).

We denote the quadratic polynomial $\operatorname{Quad}(E(P))$ as in (3) by $\operatorname{Quad}(P)\in\mathcal{O}[x]$ and the multiset of its roots $\operatorname{Roots}(E(P))$ by $\operatorname{Roots}(P)$ .

Proposition 5.2.

The map $E$ induces a homomorphism $\overline{E}:\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ with image ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})\leq\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ as in Section 3.

Proof.

From Definition 4.3 and Proposition 3.12 we see that $E$ is well-defined on equivalence classes in $\operatorname{\textup{{PCF}}}(\mathcal{O})$ . By Proposition 3.21, the image of $E$ is ${\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ . ∎

Definition 5.3.

Let $\beta\in\mathbb{P}^{1}(\mathbf{C}).$ Set

\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta)=\{\overline{P}\in\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\mid\overline{E}(\overline{P})\beta=\beta\}.

Recall from Proposition 2.5 that $T(\beta)$ is the subgroup $\{M\in\operatorname{GL_{2}}(\mathbf{C})\mid M\beta=\beta\}$ and $T(\beta)$ has a multiplicative character $\lambda_{\beta}$ mapping $M$ to its $v(\beta)$ -eigenvalue. The preimage of $T(\beta)\cap{\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ under $\overline{E}$ is $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta)$ , so $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta)$ is a subgroup of $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ . The multiplicative character $\lambda_{\beta}\circ\overline{E}$ maps an element $\overline{P}\in\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta)$ to the $v(\beta)$ -eigenvalue of $\overline{E}(\overline{P})$ .

Definition 5.4.

(a)

For $Q\in\mathcal{O}[X]$ with $\deg Q\leq 2,$ set

\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(Q)=\{\overline{P}\in\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\mid\overline{E}(\overline{P})\in G(Q)\},

where $G(Q)$ is the subgroup of $\operatorname{GL_{2}}(\mathbf{C})$ defined in Proposition 2.6.

(b)

For $A\in\operatorname{GL_{2}}(\mathcal{O})$ , set

$\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)=\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\operatorname{Quad}(A)).$

(c)

Recall from Proposition 5.2 that $E:\operatorname{\textup{{PCF}}}(\mathcal{O})\rightarrow\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O})$ factors through $\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ :

E:\operatorname{\textup{{PCF}}}(\mathcal{O})\longrightarrow\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})\stackrel{{\scriptstyle\overline{E}}}{{\longrightarrow}}\operatorname{\operatorname{\operatorname{SL}_{2}}^{\raisebox{0.72229pt}{\!\!$\scriptstyle\pm$}}}(\mathcal{O}).

For $P\in\operatorname{\textup{{PCF}}}(\mathcal{O})$ , set

\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(P)=\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\overline{P})=\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(E(P))=\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\overline{E}(\overline{P})).

For $Q\in\mathcal{O}[X]$ with $\deg Q\leq 2$ , the preimage of $G(Q)\cap{\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ under $\overline{E}$ is $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(Q)$ , so $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(Q)$ is a group. For $\beta\in\operatorname{Roots}(Q)$ , the group $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(Q)$ has a multiplicative character $\lambda_{\beta}\circ\overline{E}$ inherited from the containment $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(Q)\subset\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta)$ .

Theorem 5.5.

Let $A\in\operatorname{GL_{2}}(\mathcal{O})$ .

(a)

If $\operatorname{Roots}(A)=\{\beta,\beta^{*}\}$ , $\beta\neq\beta^{*}$ , and $A$ has corresponding eigenvectors $v(\beta),v(\beta^{*})$ , then

\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)=\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta)\cap\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta^{*}),

and the values of the characters $\lambda_{\beta}\circ\overline{E}$ and $\lambda_{\beta^{*}}\circ\,\overline{E}$ on $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)$ are quadratic integral over $\mathcal{O}$ and in $\mathcal{O}[\beta]^{\times}$ . The product of these characters is $\det\circ\,\overline{E}$ .

(b)

If $\operatorname{Roots}(A)$ has one element $\beta$ of multiplicity $2$ , $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)$ has a character $\lambda_{\beta}\circ\,\overline{E}$ whose square is $\det\circ\,\overline{E}$ .
(c)

Otherwise, $A$ is a scalar multiple of the identity and $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)$ has a character $\lambda\circ\,\overline{E}$ equal to $\lambda_{\beta}\circ\overline{E}$ for every $\beta.$ Its square is $\det\circ\,\overline{E}$ .

Proof.

Let $\overline{P}\in\operatorname{\overline{\textup{{PCF}}}}(A)$ and $E=\overline{E}(\overline{P})\in{\mathbb{E}}_{2}^{\pm}(\mathcal{O})$ .

(a): If $\operatorname{Quad}(A)$ is not zero and not a square, then $\operatorname{Roots}(A)$ has distinct elements $\beta,\beta^{*}$ and $E$ has corresponding eigenvalues $\lambda_{\beta}(E),\lambda_{\beta^{*}}(E)$ that solve the monic quadratic equation $\det(E-\lambda I)$ . It follows from Propostion 2.6 that

\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)=\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta)\cap\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(\beta^{*}).

Relation (4) shows the eigenvalue $\lambda_{\beta}(E)\in\mathcal{O}[\beta]$ . The sum $\lambda_{\beta}(E)+\lambda_{\beta^{*}}(E)$ is in $\mathcal{O}$ , so $\lambda_{\beta^{*}}(E)\in\mathcal{O}[\beta]$ . The product $\lambda_{\beta}(E)\lambda_{\beta^{*}}(E)$ is $\det E=\pm 1$ , so both $\lambda_{\beta}(E)$ and $\lambda_{\beta^{*}}(E)$ are units.

(b): When $\operatorname{Quad}(A)$ is a nonzero square, $A$ has a single eigenvector $v(\beta)$ . For $E$ as above, the eigenvalue $\lambda_{\beta}(E)$ solves a monic linear equation over $\mathcal{O}$ and $\lambda_{\beta}(E)^{2}=\pm 1$ .

(c): When $\operatorname{Quad}(A)$ is zero, every $E\in\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)$ is $\mu_{E}I$ , a scalar multiple of $I$ . In this case, $\lambda_{\beta}(E)=\mu_{E}$ for every $\beta$ . ∎

Remark 5.6.

Let $P=[b_{1},\overline{a_{1},\ldots,a_{k}}]$ be a PCF of type $(1,k)$ . The convergent $\mathcal{C}_{k}(P)$ relates $\operatorname{Roots}(P)$ and the eigenvalues of $E=M(\operatorname{\mathscr{F}}(P))$ . When

P\sim[\overline{b_{1},a_{1},\ldots,a_{k-1},a_{k}-b_{1},0}],

(30)

we have $E(P)=M([b_{1},a_{1},\ldots,a_{k-1},a_{k}-b_{1},0])$ . For $M=M([b_{1},a_{1},\ldots,a_{k-1}])=\left[\begin{smallmatrix}m_{11}&m_{12}\\ m_{21}&m_{22}\end{smallmatrix}\right]$ , we have

E=\begin{bmatrix}E_{11}&E_{12}\\ E_{21}&E_{22}\end{bmatrix}=\begin{bmatrix}m_{11}&m_{12}\\ m_{21}&m_{22}\end{bmatrix}\begin{bmatrix}1&a_{k}-b_{1}\\ 0&1\end{bmatrix}=\begin{bmatrix}m_{11}&m_{11}(a_{k}-b_{1})+m_{12}\\ m_{21}&m_{21}(a_{k}-b_{1})+m_{22}\end{bmatrix}.

Thus, $E_{11}/E_{21}$ is the $k$ th convergent $\mathcal{C}_{k}(P)$ . If $\operatorname{Roots}(P)$ has two distinct elements $\beta,\beta^{*}$ and $E$ has associated eigenvalues $\lambda,\lambda^{*}$ ,

\lambda=E_{11}-E_{21}\beta^{\ast}.

If $\operatorname{Roots}(P)$ has one element $\beta$ of multiplicity two, $E_{11}-E_{21}\beta$ is the eigenvalue of $E$ . If $E$ is a scalar multiple of $I$ , its eigenvalue is $E_{11}$ .

Remark 5.7.

The maps defined in this section form a commutative diagram as given in Figure 2.

Figure 2. The commutative diagram formed by the maps in Section 5 for

Q\in\mathcal{O}[X]

with

\deg(Q)\leq 2

and

\beta\in\operatorname{Roots}(Q)

6. Examples of Theorem 5.5 arising from number rings

6.1. \excepttoc $P_{1}=[1,\overline{2}],\hat{P}_{1}=\sqrt{2}$ \fortoc $P_{1}=[1,\overline{2}],\hat{P}_{1}=\sqrt{2}$

The $\mathbf{Z}$ -PCF $P_{1}$ is of type $(N,k)=(1,1)$ and

E(P_{1})=D(1)D(2)D(1)^{-1}=\begin{bmatrix}1&2\\ 1&1\end{bmatrix},

so $\operatorname{Quad}(P_{1})=x^{2}-2$ , $\operatorname{Roots}(P_{1})=\{\sqrt{2},-\sqrt{2}\}$ , and the first convergent $\mathcal{C}_{1}(P_{1})=1/1$ . We have $\overline{P}_{1}\in\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{1})$ , with $\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{1})$ as in Definition 5.4(c). The characters

\lambda\colonequals\lambda_{\sqrt{2}}\circ\overline{E},\,\,\lambda^{*}\colonequals\lambda_{-\sqrt{2}}\circ\overline{E}\,\,\text{ of $\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{1})$}

(31)

map $\overline{P}_{1}$ to the eigenvalues of $E(P_{1})$ :

\lambda(\overline{P}_{1})=1+\sqrt{2}\text{ and }\lambda^{\ast}(\overline{P}_{1})=1-\sqrt{2},

(32)

which are units in $\mathbf{Z}[\sqrt{2}]$ , and $\lambda(\overline{P}_{1})\lambda^{\ast}(\overline{P}_{1})=\det(E({P}_{1}))=(-1)^{k}=-1$ .

6.2. \excepttoc $P_{2}=[1,\overline{2,2,2}],\hat{P}_{2}=\sqrt{2}$ \fortoc $P_{2}=[1,\overline{2,2,2}],\hat{P}_{2}=\sqrt{2}$

The $\mathbf{Z}$ -PCF $P_{2}$ has type $(N,k)=(1,3)$ , and

\displaystyle E(P_{2})

\displaystyle=D(1)D(2)^{3}D(1)^{-1}=\begin{bmatrix}7&10\\ 5&7\end{bmatrix}=E(P_{1})^{3}=\begin{bmatrix}1&2\\ 1&1\end{bmatrix}^{3},

so $\operatorname{Quad}(P_{2})=5x^{2}-10=5\operatorname{Quad}(P_{1})$ and $\operatorname{Roots}(P_{2})=\operatorname{Roots}(P_{1})=\{\sqrt{2},-\sqrt{2}\}$ . By (29), the equivalence class $\overline{P}_{2}$ satisfies

\overline{P}_{2}=\overline{P}_{1}\star\overline{P}_{1}\star\overline{P}_{1}\in\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{2})=\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{1}).

The convergent $\mathcal{C}_{3}(P_{2})$ is 7/5. The characters $\lambda$ , $\lambda^{\ast}$ of $\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{2})=\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{1})$ in (31) map $\overline{P}_{2}$ to the eigenvalues of $E(P_{2})$ :

\lambda(\overline{P}_{2})=7+5\sqrt{2}\text{ and }\lambda^{\ast}(\overline{P}_{2})=7-5\sqrt{2},

which are units in $\mathbf{Z}[\sqrt{2}]$ , and $\lambda(\overline{P}_{2})\lambda^{\ast}(\overline{P}_{2})=\det(\overline{E}(\overline{P}_{2}))=(-1)^{3}=-1$ . Consequent to $\lambda$ being a homomorphism,

\lambda(\overline{P}_{2})=\lambda(\overline{P}_{1}\star\overline{P}_{1}\star\overline{P}_{1})=5\sqrt{2}+7=(1+\sqrt{2})^{3}=\lambda(\overline{P}_{1})^{3}

using (32).

6.3. \excepttoc $P_{3}=[2,\overline{-2,4}],\hat{P}_{3}=\sqrt{2}$ \fortoc $P_{3}=[2,\overline{-2,4}],\hat{P}_{3}=\sqrt{2}$

The $\mathbf{Z}$ -PCF $P_{3}$ is of type $(N,k)=(1,2)$ and

\displaystyle E(P_{3})

\displaystyle=D(2)D(-2)D(4)D(2)^{-1}=\begin{bmatrix}-3&-4\\ -2&-3\end{bmatrix},

so $\operatorname{Quad}(P_{3})=-2x^{2}+4=-2\operatorname{Quad}(P_{1})$ and $\operatorname{Roots}(P_{3})=\operatorname{Roots}(P_{1})$ . The convergent $\mathcal{C}_{2}(P_{3})$ is $3/2$ . The class $\overline{P}_{3}$ is in $\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{1})=\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{3})$ . For the characters $\lambda$ , $\lambda^{\ast}$ as in (31), we have

\lambda(\overline{P}_{3})=-3-2\sqrt{2}\text{ and }\lambda^{\ast}(\overline{P}_{3})=-3+2\sqrt{2},

(33)

with which are units in $\mathbf{Z}[\sqrt{2}]$ , and $\lambda(\overline{P}_{3})\lambda^{\ast}(\overline{P}_{3})=\det(\overline{E}(\overline{P}_{3}))=(-1)^{2}=1$ .

6.4. \excepttoc $P_{4}=[2,\overline{3,-2,3}],\hat{P}_{4}=\sqrt{2}$ \fortoc $P_{4}=[2,\overline{3,-2,3}],\hat{P}_{4}=\sqrt{2}$

The $\mathbf{Z}$ -PCF $P_{4}$ is of type $(N,k)=(1,3)$ and

\displaystyle E(P_{4})

\displaystyle=D(1)D(3)D(-2)D(3)D(1)^{-1}=\begin{bmatrix}-7&-10\\ -5&-7\end{bmatrix},

so $\operatorname{Quad}(P_{4})=-5x^{2}+10=-5\operatorname{Quad}(P_{1})$ and $\operatorname{Roots}(P_{4})=\operatorname{Roots}(P_{1})$ . The convergent $\mathcal{C}_{3}(P_{4})$ is $7/5$ . We have $\overline{P}_{4}\in\operatorname{\overline{\textup{{PCF}}}}_{Z}(P_{4})=\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}}(P_{1})$ and, by (4.11),

\overline{P}_{4}=|\![1,\overline{2-1+2,-2,4-2+1}]\!|=|\![1,\overline{2}]\!||\![2,\overline{-2,4}]\!|=\overline{P}_{1}\star\overline{P}_{3}.

The values of the characters $\lambda,\lambda^{\ast}$ in (31) on $\overline{P}_{4}$ are

\lambda(\overline{P}_{4})=-7-5\sqrt{2}\text{ and }\lambda^{\ast}(\overline{P}_{4})=-7+5\sqrt{2}.

We have $\lambda(\overline{P}_{4})\lambda^{\ast}(\overline{P}_{4})=\det(\overline{E}(\overline{P}_{4}))=(-1)^{3}=-1.$ From Example 6.1 $\lambda(\overline{P}_{1})=1+\sqrt{2}$ and from Example 6.3 $\lambda(\overline{P}_{3})=-2\sqrt{2}-3$ . We verify using (32) and (33)

\lambda(\overline{P}_{4})=\lambda(\overline{P}_{1}\star\overline{P}_{3})=-5\sqrt{2}-7=(1+\sqrt{2})(-2\sqrt{2}-3)=\lambda(\overline{P}_{1})\lambda(\overline{P}_{3}).

6.5. \excepttoc $P_{5}=[442+312\sqrt{2},\overline{-298532+211094\sqrt{2},884+624\sqrt{2}}],\hat{P}_{5}=\sqrt{2+\sqrt{2}}$ \fortoc $P_{5}=[442+312\sqrt{2},\overline{-298532+211094\sqrt{2},884+624\sqrt{2}}],\hat{P}_{5}=\sqrt{2+\sqrt{2}}$

The $\mathbf{Z}[\sqrt{2}]$ -PCF $P_{5}$ from [BEJ21, Cor. 8.19] is of type $(N,k)=(1,2)$ . We have

	$\displaystyle E=E(P_{5})$	$\displaystyle=D(442+312\sqrt{2})D(-298532+211094\sqrt{2})D(884+624\sqrt{2})D(442+312\sqrt{2})^{-1}$
		$\displaystyle=\begin{bmatrix}-228487+161564\sqrt{2}&-174876+123656\sqrt{2}\\ -298532+211094\sqrt{2}&-228487+161564\sqrt{2}\end{bmatrix},\text{ so}$
	$\displaystyle\operatorname{Quad}(P_{5})$	$\displaystyle=(-298532+211094\sqrt{2})x^{2}+174876-123656\sqrt{2}$
		$\displaystyle=(-298532+211094\sqrt{2})(x^{2}-2-\sqrt{2}),$

$\operatorname{Roots}(P_{5})=\{\beta\colonequals\sqrt{2+\sqrt{2}},-\beta\}$ , and the convergent $\mathcal{C}_{2}(P_{5})=(11502+8105\sqrt{2})/52\approx 441.6$ , which is a horrible approximation to $\beta$ in keeping with [BEJ21, p. 415], which remarks on and quantifies the extremely slow convergence of $P_{5}$ . The values of the characters

\mu\colonequals\lambda_{\beta}\circ\overline{E},,\mu^{\ast}\colonequals\lambda_{-\beta}\circ\overline{E}:\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}[\sqrt{2}]}(P_{5})\rightarrow\mathbf{Z}[\sqrt{2}][\beta]^{\times}

on the class $\overline{P}_{5}\in\operatorname{\overline{\textup{{PCF}}}}_{\mathbf{Z}[\sqrt{2}]}(P_{5})$ are

	$\displaystyle\mu(\overline{P}_{5})$	$\displaystyle=(-228487+161564\sqrt{2})+(-298532+211094\sqrt{2})\beta\text{ and}$
	$\displaystyle\mu^{\ast}(\overline{P}_{5})$	$\displaystyle=(-228487+161564\sqrt{2})-(-298532+211094\sqrt{2})\beta,$

which are both units in the ring $\mathbf{Z}[\sqrt{2}][\beta]=\mathbf{Z}[\beta]$ . In fact

\operatorname{Nm}_{\mathbf{Q}(\beta)/\mathbf{Q}(\sqrt{2})}\mu(\overline{P}_{5})=\operatorname{Nm}_{\mathbf{Q}(\beta)/\mathbf{Q}(\sqrt{2})}\mu^{\ast}(\overline{P}_{5})=\mu(\overline{P}_{5})\mu^{\ast}(\overline{P}_{5})=\det(\overline{E}(\overline{P}_{5}))=1.

7. Refined notions of convergence for periodic continued fractions

For a PCF $P$ , the multiset $\operatorname{Roots}(P)$ contains the limit $\hat{P}$ of $P$ when that limit exists; see, e.g., [BEJ21, Appendix].

Theorem 7.1.

Let $P=[b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k}}]$ and $P^{\prime}=[b^{\prime}_{1},\ldots,b^{\prime}_{N^{\prime}},\overline{a^{\prime}_{1},\ldots,a^{\prime}_{k^{\prime}}}]$ be PCFs of types $(N,k)$ and $(N^{\prime},k^{\prime})$ that are $\operatorname{CF}$ -equal.

(a)

If $k^{\prime}=k$ so that $P$ and $P^{\prime}$ are $\operatorname{k}$ -equal, then $E(P)=E(P^{\prime})$ , $\operatorname{Quad}(P)=\operatorname{Quad}(P^{\prime})$ , and $\operatorname{Roots}(P)=\operatorname{Roots}(P^{\prime})$ .
(b)

If $k^{\prime}\neq k$ , then $E(P)^{k^{\prime}}=E(P^{\prime})^{k}$ , and $\operatorname{Quad}(P)$ and $\operatorname{Quad}(P^{\prime})$ are linearly dependent. If $\operatorname{Quad}(P)$ and $\operatorname{Quad}(P^{\prime})$ are both nonzero, then $\operatorname{Roots}(P)=\operatorname{Roots}(P^{\prime})$ .
(c)

The PCF $P$ converges if and only if $P^{\prime}$ converges and in this case their limits are equal: $\hat{P}=\hat{P}^{\prime}\in\operatorname{Roots}(P)=\operatorname{Roots}(P^{\prime})$ .

Proof.

(a): Suppose $k=k^{\prime}$ . If $N=N^{\prime}$ , then $P$ equals $P^{\prime}$ . In the alternative $N\neq N^{\prime}$ , we may assume $N^{\prime}>N$ with $N^{\prime}=N+j$ . In this case,

P^{\prime}=[b_{1},\ldots,b_{N},b_{N+1},\ldots,b_{N+j},\overline{a_{j+1},\ldots,a_{j+k}}],

where we take the subscripts of the $a$ ’s $\bmod\ k$ to put them in the range 1 to $k$ and $b_{i}=a_{i-N}$ for $i>N$ . So

	$\displaystyle E(P^{\prime})$	$\displaystyle=M([b_{1},\ldots,b_{N},b_{N+1},\ldots,b_{N+j},a_{j+1},\ldots,a_{j+k},0,-b_{N+j},\ldots,-b_{1},0])$
		$\displaystyle=M([b_{1},\ldots,b_{N},a_{1},a_{2},\ldots,a_{j+k},0,-a_{j},\ldots,-a_{1},-b_{N},\ldots,-b_{1},0])$
		$\displaystyle=M([b_{1},\ldots,b_{N},a_{1},a_{2},\ldots,a_{j+k-1},0,-a_{j-1},\ldots,-a_{1},-b_{N},\ldots,-b_{1},0])$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\vdots$
		$\displaystyle=M([b_{1},\ldots,b_{N},a_{1},\ldots,a_{k},0,-b_{N},\ldots,-b_{1},0])=E(P)$

using the fact that $M([a,0,-a])=M([0])$ and $a_{i}=a_{i+k}$ $j$ times. The rest of (a) follows trivially.

(b): Since the choice of $N$ does not affect $E$ , we may assume $N=N^{\prime}$ . The equality $E(P)^{k^{\prime}}=E(P^{\prime})^{k}$ follows from the observation that the concatenation of $k^{\prime}$ copies of $[a_{1},\ldots,a_{k}]$ equals the concatenation of $k$ copies of $[a^{\prime}_{1},\ldots,a^{\prime}_{k^{\prime}}]$ . The Cayley–Hamilton theorem implies that any power of a $2\times 2$ matrix $M$ is a linear combination of $M$ and $I$ . This implies $E(P)$ , $E(P^{\prime})$ , and $I$ are linearly dependent, which implies $\operatorname{Quad}(P)$ and $\operatorname{Quad}(P^{\prime})$ are linearly dependent, which implies they have the same roots if both are nonzero.

(c): This is trivial, since $P$ and $P^{\prime}$ are $\operatorname{CF}$ -equal. ∎

For the PCF $P=[b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k}}]$ of type $(N,k)$ , we introduce notation for the matrix $M_{N+j}(P)^{-1}M_{N+j+k}(P)$ and its entries:

R_{j}(P)=M_{N+j}(P)^{-1}M_{N+j+k}(P)=M([a_{j+1},\ldots,a_{k},a_{1},\ldots,a_{j}])=\begin{bmatrix}r_{j}(P)&s_{j}(P)\\ t_{j}(P)&u_{j}(P)\end{bmatrix},

and for the limits, if they exist,

\hat{P}_{j}=\lim_{n\to\infty}\mathcal{C}_{N+j+nk}(P).

(34)

Definition 7.2.

The matrix $R_{j}(P)$ is heavy when

t_{j}(P)=0\ \textrm{and}\ |u_{j}(P)|>1.

Rephrased with this terminology, the condition [BEJ21, Thm. 4.3(b), INEQ 4.1] says that $R_{j}(P)$ is heavy for some $0\leq j\leq k-1$ .

The matrix $R_{0}(P)$ is $M(\operatorname{Rep}(P))$ . The matrices $R_{j}(P)$ are all conjugate to $E(P)$ . From the definition, we see $\hat{P}_{j}=\hat{P}_{j+k}$ .

Proposition 7.3.

Let $P$ be a PCF. Suppose $R_{j}(P)$ is heavy.

(a)

The limit $\hat{P}_{j}$ exists and $v(\hat{P}_{j})$ is an $r_{j}(P)$ -eigenvector of $E(P)$ .
(b)

$R_{j+1}(P)$ is not heavy.

Proof.

(a): The heavy condition implies $R_{j}(P)$ has an eigenvector $\binom{1}{0}$ with eigenvector $r_{j}(P)$ . Therefore, $E=M_{N+j}R_{j}M_{N+j}(P)^{-1}$ has eigenvector $M_{N+j}(P)\binom{1}{0}$ . We compute the limit

\hat{P}_{j}=\lim_{n\to\infty}M_{N+j+nk}(P)\infty=\lim_{n\to\infty}M_{N+j}(P)R_{j}^{n}\infty=M_{N+j}\infty.

We are now done because $v(\hat{P}_{j})$ and $M_{N+j}(P)\binom{1}{0}$ are both elements of the equivalence class $M_{N+j}(P)\infty$ .

(b): The calculation, with argument $P$ suppressed and $a=a_{j+1-nk}$ , for the value of $nk$ such that $1\leq j+1-nk\leq k$ ,

\begin{bmatrix}r_{j+1}&s_{j+1}\\ t_{j+1}&u_{j+1}\end{bmatrix}=\begin{bmatrix}0&1\\ 1&-a\end{bmatrix}\begin{bmatrix}r_{j}&s_{j}\\ 0&u_{j}\end{bmatrix}\begin{bmatrix}a&1\\ 1&0\end{bmatrix}=\begin{bmatrix}u_{j}&0\\ (r_{j}-u_{j})a+s_{j}&r_{j}\end{bmatrix}

shows that $R_{j}(P)$ and $R_{j+1}(P)$ cannot both be heavy. If they are, $u_{j+1}=r_{j}$ , yielding $|u_{j+1}|<1$ , a contradiction. ∎

We recall a theorem proved in the appendix of [BEJ21], with the cases aligned to match Proposition 2.3.

Theorem 7.4.

([BEJ21, Thm. 4.3]) Let $P$ be a PCF, let $\lambda_{\pm}$ be the eigenvalues of $E=E(P)$ chosen so that $|\lambda_{+}|\geq 1\geq|\lambda_{-}|$ , and let $v(\beta_{\pm})$ be the corresponding eigenvectors. If $P$ converges, its limit $\hat{P}=\beta_{+}$ . Exactly one of the following holds.

(a)

$\operatorname{Quad}(P)$ has one root $\beta_{+}=\beta_{-}$ of multiplicity $2$ , and $\hat{P}=\beta_{+}=\beta_{-}.$
(b)

$\operatorname{Quad}(P)=0$ , $E=\lambda_{+}I=\lambda_{-}I$ , and $P$ diverges.
(c)

$\operatorname{Quad}(P)$ has roots $\beta_{+},\beta_{-}$ with $|\lambda_{+}|>|\lambda_{-}|$ .
1. (i)
  
  For some $j\geq 0$ , $R_{j}(P)$ is heavy, $\hat{P}_{j}=\beta_{-}$ , $\hat{P}_{j+1}=\beta_{+}$ , and $P$ diverges.
2. (ii)
  
  For all $j\geq 0$ , $R_{j}(P)$ is not heavy, and $P$ converges to $\hat{P}=\beta_{+}$ .
(d)

$\operatorname{Quad}(P)$ has distinct roots $\beta_{+},\beta_{-}$ with $|\lambda_{+}|=|\lambda_{-}|$ , and $P$ diverges. In fact, $\hat{P}_{j}$ does not exist for some $j$ .

Proof.

In case (a), by Proposition 2.3(a), for all $j$ , $\hat{P}_{j}=\beta_{+}$ , so $P$ converges to $\beta_{+}$ .

In case (b), $R_{j}(P)=E$ for all $j,$ so $\hat{P}_{j}=\mathcal{C}_{N+j}(P)$ . No two consecutive ones are equal: $\hat{P}_{j}\neq\hat{P}_{j+1}$ by the more general rule that no two consecutive convergents can be equal, so $P$ diverges.

Now assume that cases (a) and (b) do not hold. $\operatorname{Quad}(P)$ has two distinct roots, $\beta_{-}$ , $\beta_{+}$ . The matrix $E$ has corresponding distinct eigenvalues $\lambda_{-}$ , $\lambda_{+}$ .

In case (c), $|\lambda_{+}|>|\lambda_{-}|,$ and Proposition 2.3(c) shows all $\beta_{j}$ exist. Suppose, as in the first subcase, there exists $j$ such that $R_{j}(P)$ is heavy, Proposition 7.3 shows $\hat{P}_{j}=\beta_{-}$ and $R_{j+1}$ is not heavy. Proposition 2.3(c) shows $\hat{P}_{j+1}=\beta_{+},$ so $P$ diverges.

In the alternative subcase of (c), where there is no $j$ such that $R_{j}$ is heavy, we have $\hat{P}_{j}=\beta_{+}$ for all $j$ , and $P$ converges to $\beta_{+}$ .

In case (d) of divergence, we have $|\lambda_{+}|=|\lambda_{-}|$ and $\lambda_{+}\neq\lambda_{-}$ . By Proposition 2.3(d), the limit $\hat{P}_{j}$ exists if and only if $t_{j}=t_{j}(P)=0$ . We claim there exists $j$ such that $t_{j}(P)\neq 0$ .

Assume, to the contrary, that $t_{j}=0$ for all $j$ . To deduce a contradiction, we show $t_{j}=t_{j+1}=t_{j+2}=0$ implies $a_{j+2}=0$ . From the equality

R_{j+1}\begin{bmatrix}a_{j+2}&1\\ 1&0\end{bmatrix}=\begin{bmatrix}a_{j+2}&1\\ 1&0\end{bmatrix}R_{j+2},

(35)

we see that $t_{j+1}=t_{j+2}=0$ implies $s_{j+2}=0$ . Similarly, $t_{j}=t_{j+1}=0$ implies $s_{j+1}=0$ , so $R_{j+1}$ and $R_{j+2}$ are diagonal with distinct diagonal entries $\lambda_{\pm}$ . Equation (35) implies $a_{j+2}=0$ .

Since $t_{j}=0$ for all $j$ , we have that $a_{j}=0$ for all $j$ . If $k$ is odd then $t_{j}=1$ for every j, a contradiction. If $k$ is even then we are in case (b), also a contradiction, thus proving the claim. Hence, for some $j$ , $\hat{P}_{j}$ does not exist, and $P$ diverges. ∎

We name the convergence behaviors of a PCF $P$ identified in Theorem 7.4.

Definition 7.5.

In cases 7.4(a) and 7.4(c), $P$ is quasiconvergent. In cases 7.4(b) and 7.4(d), $P$ is strictly divergent. In the divergent subcase of 7.4(ci), $P$ is strictly quasiconvergent.

Divergent means strictly divergent or strictly quasiconvergent. Quasiconvergent means convergent or strictly quasiconvergent.

Theorem 7.6.

Let PCFs $P$ and $Q$ be equivalent. $P$ is quasiconvergent if and only if $Q$ is quasiconvergent. If $P$ and $Q$ both converge, then $\hat{P}=\hat{Q}$ .

Proof.

PCFs $P$ and $Q$ are quasiconvergent together, because $E(P)=E(Q)$ . For the same reason, if $P$ and $Q$ both converge, they converge to the same value. ∎

The following lemma and theorem characterize strict quasiconvergence.

Lemma 7.7.

Let $P=[b_{1},\ldots,b_{N},\overline{a_{1},\ldots,a_{k}}]$ be a PCF. If $R_{j}(P)$ and $R_{j+2}(P)$ are both heavy, then $a_{j+2-n^{\prime}k}=0$ for the value of $n^{\prime}$ such that $1\leq j+2-n^{\prime}k\leq k$ .

Proof.

Inspect the relation, with argument $P$ suppressed, $a=a_{j+1-nk}$ , and $a^{\prime}=a_{j+2-n^{\prime}k}$ :

\begin{bmatrix}r_{j+2}&s_{j+2}\\ t_{j+2}&u_{j+2}\end{bmatrix}=\begin{bmatrix}1&-a\\ -a^{\prime}&aa^{\prime}+1\end{bmatrix}\begin{bmatrix}r_{j}&s_{j}\\ t_{j}&u_{j}\end{bmatrix}\begin{bmatrix}aa^{\prime}+1&a\\ a^{\prime}&1\end{bmatrix}.

Observe that $t_{j}=0$ implies

	$\displaystyle r_{j+2}=r_{j}+aa^{\prime}r_{j}-aa^{\prime}u_{j}+a^{\prime}s_{j},$
	$\displaystyle t_{j+2}=a^{\prime}(u_{j}-r_{j+2}).$

If $t_{j+2}=0$ and $a^{\prime}\neq 0$ , then $r_{j+2}=u_{j}$ , contradicting $|u_{j}|>1>|r_{j+2}|$ . Hence, $t_{j+2}=0$ implies $a^{\prime}=a_{j+2-n^{\prime}k}=0.$ ∎

Theorem 7.8.

The PCF $P$ is strictly quasiconvergent if and only if for each $j$ , $1\leq j\leq k$ , $\hat{P}_{j}$ exists, a strict majority, but not all, of which are the same.

Proof.

Let the eigenvalues of $E=E(P)$ be $\lambda_{+}$ and $\lambda_{-}$ , with $|\lambda_{+}|\geq 1$ . If both eigenvalues have magnitude $1$ , arbitrarily assign one to be $\lambda_{+}$ . Define $\beta_{+}$ and $\beta_{-}$ such that $v(\beta_{+})$ , $v(\beta_{-})$ are eigenvectors of $E$ with corresponding eigenvalues $\lambda_{+}$ and $\lambda_{-}$ .

If $P$ is strictly quasiconvergent, then there exists $j$ such that $R_{j}(P)$ is heavy and $\lambda_{+}=u_{j}.$

Proposition 7.3 shows that for every $j$ such that $R_{j}(P)$ is heavy, $\hat{P}_{j}=\beta_{-}$ and $R_{j+1}(P)$ is not heavy. For those $j$ such that $R_{j}(P)$ is not heavy, $\hat{P}_{j}=\beta_{+}.$

If $R_{j}(P)$ is heavy for every even $j$ or every odd $j$ , then either $k$ is even and, by Lemma 7.7, every second partial quotient of $\operatorname{Rep}(P)$ is zero, or $k$ is odd, and all partial quotients of $\operatorname{Rep}(P)$ are zero. In the even case, $E$ is conjugate to an elementary matrix and $P$ satisfies 7.4(a) or 7.4(b), a contradiction. In the odd case, $E$ is conjugate to the matrix $J$ of Remark 3.19 and $P$ satisfies 7.4(d). Hence, for only a minority of $j$ , $1\leq j\leq k$ , can $R_{j}(P)$ be heavy and $\hat{P}_{j}=\beta_{-}$ .

Conversely, suppose for each $j$ , $1\leq j\leq k$ , $\hat{P}_{j}$ exists, a strict majority, but not all, of which are the same.

If $P$ satisfies Theorem 7.4(b), then $\hat{P}_{j}\neq\hat{P}_{j+1}$ . So a majority of the $\hat{P}_{j}$ , $1\leq j\leq k$ , are not equal to a particular value.

The PCF $P$ satisfies Theorem 7.4(d) if and only if $E$ has eigenvalues $\lambda_{+}\neq\lambda_{-}$ with $|\lambda_{+}|=|\lambda_{-}|=1$ . We claim that for $1\leq j\leq k$ , either $\hat{P}_{j}$ does not exist, or $\hat{P}_{j}$ exists and $\mathcal{C}_{N+j+nk}(P)$ is a constant independent of $n$ . We shall prove this for $j=k$ . The result follows for all $j$ by rotating the periodic part of $P$ . Write

v(\hat{P}_{k})=cv(\beta_{+})+dv(\beta_{-}),

and observe

E^{n}v(\hat{P}_{k})=c\lambda_{+}^{n}v(\beta_{+})+d\lambda_{-}^{n}v(\beta_{-}).

If $\hat{P}_{k}$ exists, then $v(\hat{P}_{k})$ is an eigenvector of $E$ , so either $c=0$ or $d=0$ . This proves the claim.

By assumption that the limits $\hat{P}_{j}$ exist and the fact that consecutive convergents cannot be equal, we cannot have a strict majority of $\hat{P}_{j}$ agree and $P$ satisfy Theorem 7.4(d)

The only possibility left is that $P$ satisfies Theorem 7.4(ci), so $P$ is strictly quasiconvergent. ∎

8. Examples for quasiconvergence

8.1. Strict majority in Theorem 7.8 is necessary

The strict majority hypothesis in the sufficiency part of the proof of Theorem 7.8 is essential. For example, $P=[a,b,\overline{0,0}]$ has limits $\hat{P}_{1}=a$ and $\hat{P}_{2}=a+1/b$ . The split is equal and $P$ is not quasiconvergent. It is strictly divergent because it satisfies Theorem 7.4(b).

8.2. The fraction of \excepttoc $\hat{P}_{j}=\beta_{+}$ \fortoc $\hat{P}_{j}=\beta_{+}$ can be made arbitrarily close to \excepttoc $1/2$ \fortoc $1/2$

We can make the fraction of $\hat{P}_{j}=\beta_{+}$ arbitrarily close to $1/2$ . Let $c\neq 0$ and

P=[a,b,\overline{c,-1/c,c,0,\ldots,0}\,],

have odd length period $k\geq 3$ . We have $\hat{P}_{2j}=a$ and $\hat{P}_{2j+1}=a+1/b$ , $1\leq j<k/2$ . If $c^{2}=-1$ we are in case (b) of Theorem 7.4, and $\hat{P}_{1}=a+1/(b-c)$ . If $|c|=1$ , $c^{2}\neq-1$ , we are in case (d), and $\hat{P}_{1}$ does not exist. If $|c|\neq 1$ we are in case (ci) and

\hat{P}_{1}=\beta_{+}=\begin{cases}a+1/b&\text{if }|c|>1,\\ a&\text{if }|c|<1,\end{cases}

so $\beta_{+}$ has a slim $(k+1)/2$ majority. A similar construction can be made for even periods by taking

P^{\prime}=[a,b,\overline{c,-1/c,c,0,\ldots,0,d,-1/d,d,0,\ldots,0}\,]

where the length of the strings of 0’s have the same parity.

8.3. The fraction of \excepttoc $\hat{P}_{j}=\beta_{+}$ \fortoc $\hat{P}_{j}=\beta_{+}$ can be made arbitrarily close to \excepttoc $1$ \fortoc $1$

Likewise we can make the fraction of $\hat{P}_{j}=\beta_{+}$ arbitrarily close to $1$ . Let $\{F_{n}\}_{n=1}^{\infty}=1,1,2,3,5,\ldots$ be the Fibonacci sequence. Let $b\neq 0$ and

P\colonequals Q_{k}=[a,b,\overline{1,\ldots,1,-F_{k-2}/F_{k-1}}\,],

where there are $(k-1)$ $1$ ’s. The continued fraction $P$ for $k\geq 4$ provides an example where exactly $k-1$ of the $k$ $\hat{P}_{j}$ ’s are $\beta_{+}$ . (If $k=3$ , $\hat{P}_{2}$ doesn’t exist, and if $k=2$ , $\beta_{+}=\beta_{-}=\hat{P}$ .)

Proposition 8.1.

If $k\geq 4$ and $b\neq 0$ , then $\hat{P}_{j}=\beta_{+}(Q_{k})=a+\frac{F_{k-1}}{bF_{k-1}+F_{k-2}}$ for $1\leq j\leq k-1$ , but $\hat{P}_{k}=\beta_{-}(Q_{k})=a+1/b$ .

Proof.

We compute $M=M(k)\colonequals M([1,1,\dots,1,-F_{k-2}/F_{k-1}])$ :

$\displaystyle M$	$\displaystyle=\begin{bmatrix}1&1\\ 1&0\end{bmatrix}^{k-1}\begin{bmatrix}-F_{k-2}/F_{k-1}&1\\ 1&0\end{bmatrix}$
	$\displaystyle=\begin{bmatrix}F_{k}&F_{k-1}\\ F_{k-1}&F_{k-2}\end{bmatrix}\begin{bmatrix}-F_{k-2}/F_{k-1}&1\\ 1&0\end{bmatrix}$
	$\displaystyle=\begin{bmatrix}(-1)^{k}/F_{k-1}&F_{k}\\ 0&F_{k-1}\end{bmatrix},$	(36)

using Cassini’s identity $F_{k-1}^{2}-F_{k}F_{k-2}=(-1)^{k}$ ([Knu97, p. 81]).

The next step is to compute the $\hat{P}_{j}$ , $1\leq j\leq k$ , as in (34). We begin with $\hat{P}_{k}$ . Set

A=D(a)D(b)=\begin{bmatrix}a&1\\ 1&0\end{bmatrix}\begin{bmatrix}b&1\\ 1&0\end{bmatrix}=\begin{bmatrix}ab+1&a\\ b&1\end{bmatrix}.

(37)

Using (8.3), we have

	$\displaystyle M(n,k)\colonequals M_{2+nk}(Q_{k})$	$\displaystyle=AM^{n}=\begin{bmatrix}ab+1&a\\ b&1\end{bmatrix}\begin{bmatrix}(-1)^{k}/F_{k-1}&F_{k}\\ 0&F_{k-1}\end{bmatrix}^{n}$
		$\displaystyle=\begin{bmatrix}ab+1&a\\ b&1\end{bmatrix}\begin{bmatrix}(-1)^{kn}/F_{k-1}^{n}&\ast\\ 0&F_{k-1}^{n}\end{bmatrix}$
		$\displaystyle=\begin{bmatrix}(ab+1)(-1)^{kn}/F_{k-1}^{n}&\ast\\ b(-1)^{kn}/F_{k-1}^{n}&\ast\end{bmatrix}.$

Hence

\mathcal{C}_{N+nk}=M(n,k)_{11}/M(n,k)_{21}=\frac{ab+1}{b}=a+1/b

for all $n\geq 1$ and

\hat{P}_{k}=\lim_{n\to\infty}\mathcal{C}_{N+nk}=a+1/b.

For the other $\hat{P}_{j}$ , we need to calculate $M_{j}\colonequals D(1)^{-j}MD(1)^{j}$ for $1\leq j\leq k-1$ , so that we can apply Proposition 2.3. Firstly note that

D(1)^{j}=\begin{bmatrix}F_{j+1}&F_{j}\\ F_{j}&F_{j-1}\end{bmatrix},\quad\text{and so}\quad D(1)^{-j}=(-1)^{j}\begin{bmatrix}F_{j-1}&-F_{j}\\ -F_{j}&F_{j+1}\end{bmatrix}.

(38)

Hence from (38) and (8.3) we get

	$\displaystyle M_{j}$	$\displaystyle=(-1)^{j}\begin{bmatrix}F_{j-1}&-F_{j}\\ -F_{j}&F_{j+1}\end{bmatrix}\begin{bmatrix}(-1)^{k}/F_{k-1}&F_{k}\\ 0&F_{k-1}\end{bmatrix}\begin{bmatrix}F_{j+1}&F_{j}\\ F_{j}&F_{j-1}\end{bmatrix}$
		$\displaystyle=(-1)^{j}\begin{bmatrix}\ast&\ast\\ F_{j}((-1)^{k+1}F_{j+1}/F_{k-1}+F_{j+1}F_{k-1}-F_{j}F_{k})&\ast\end{bmatrix}.$		(39)

If $(M_{j})_{21}=0$ , then we would have $F_{k-1}|F_{j+1}$ from (8.3). But this is not possible for $j\leq k-1$ unless $j=k-2$ : $F_{k-1}\not|F_{k}$ takes care of $j=k-1\geq 3$ and if $j\leq k-3$ , then $F_{k-1}>F_{j+1}$ since $k\geq 4$ . Hence $(M_{j})_{21}\neq 0$ if $1\leq j\leq k-1$ , $j\neq k-2$ , so in this case $\hat{P}_{j}=\beta_{+}(Q_{k})$ by Proposition 2.3(c), with the “ $M$ ” there equal to $AMA^{-1}$ and the “ $\beta$ ” there equal to $AD(1)^{j}\infty$ .

For the exceptional case $j=k-2$ , explicit computation of $M_{k-2}$ using the Cassini/Vajda identities for Fibonacci numbers gives

M_{k-2}=\begin{bmatrix}F_{k-1}&F_{k}F_{k-3}/F_{k-1}\\ 0&(-1)^{k}/F_{k-1}\end{bmatrix}.

(40)

Hence by (40) $(M_{k-2})_{21}=0$ and $|(M_{k-2})_{11}|>|(M_{k-2})_{22}|$ since $k\geq 4$ . Hence $\hat{P}_{k-2}=\beta_{+}(Q_{k})$ again by Proposition 2.3(c), with the “ $M$ ” there equal to $AMA^{-1}$ and the “ $\beta$ ” there equal to $AD(1)^{k-2}\infty$ .

Lastly we show the computation giving $\beta_{-}(Q_{k})$ and $\beta_{+}(Q_{k})$ :

\beta_{-}(Q_{k})=a+1/b\quad\text{and}\quad\beta_{+}(Q_{k})=a+\frac{F_{k-1}}{bF_{k-1}+F_{k-2}}.

(41)

Calculating $\beta_{-}(Q_{k})$ and $\beta_{+}(Q_{k})$ entails finding the eigenvectors of $E(Q_{k})=AMA^{-1}$ with $A$ as in (37) and $M$ as in (8.3). Let

v_{-}\colonequals\begin{pmatrix}1\\ 0\end{pmatrix},\,\,\lambda_{-}\colonequals\frac{(-1)^{k}}{F_{k-1}},\quad\text{and}\quad v_{+}\colonequals\begin{pmatrix}F_{k-1}\\ F_{k-2}\end{pmatrix},\,\,\lambda_{+}\colonequals F_{k-1}.

Observe that $|\lambda_{+}|>|\lambda_{-}|$ since $k\geq 4$ . Using the Cassini identity yet again, verify that

Mv_{-}=\lambda_{-}v_{-}\quad\text{and}\quad Mv_{+}=\lambda_{+}v_{+}.

So $E(Q_{k})=AMA^{-1}$ will have eigenvectors $Av_{-}$ , $Av_{+}$ with eigenvalues $\lambda_{-}$ , $\lambda_{+}$ , respectively. We have

	$\displaystyle Av_{-}$	$\displaystyle=\begin{pmatrix}ab+1\\ b\end{pmatrix}=b\begin{pmatrix}a+1/b\\ 1\end{pmatrix}=bv(\beta_{-})\text{ and}$
	$\displaystyle Av_{+}$	$\displaystyle=\begin{pmatrix}(ab+1)F_{k-1}+aF_{k-2}\\ bF_{k-1}+F_{k-2}\end{pmatrix}=(bF_{k-1}+F_{k-2})\begin{pmatrix}a+\frac{F_{k-1}}{bF_{k-1}+F_{k-2}}\\ 1\end{pmatrix}$
		$\displaystyle=(bF_{k-1}+F_{k-2})v(\beta_{+}),$

establishing the formulas (41) for $\beta_{-}(Q_{k})$ and $\beta_{+}(Q_{k})$ . ∎

8.4. The equivalence \excepttoc $\sim$ \fortoc $\sim$ on \excepttoc $\operatorname{\textup{{PCF}}}(\mathcal{O})$ \fortoc $\operatorname{\textup{{PCF}}}(\mathcal{O})$ does not respect convergence

Theorem 7.6 shows quasiconvergence is a property of PCF equivalence class. It is possible for $P\sim Q$ , both quasiconvergent, with $P$ strictly quasiconvergent (and hence divergent) and $Q$ convergent. For example, take $a\neq 0$ and let $P=[\overline{a,0,-1/a}]$ , $Q=[\overline{a-1/a}]$ . Let the convergents of $P$ be $\mathcal{C}_{i}(P)$ , $i\geq 1$ . If $|a|<1$ , then $P$ satisfies Theorem 7.4(ci), and hence is strictly quasiconvergent. The limits $\hat{P}_{j}$ for $1\leq j\leq 3$ are

\begin{array}[]{ccccc}\hat{P}_{1}&=&\lim_{i\to\infty}\mathcal{C}_{1+3i}(P)&=&a,\\ \hat{P}_{2}&=&\lim_{i\to\infty}\mathcal{C}_{2+3i}(P)&=&-1/a,\\ \hat{P}_{3}&=&\lim_{i\to\infty}\mathcal{C}_{3+3i}(P)&=&-1/a.\end{array}

The PCF $Q$ converges to $-1/a$ , since the convergents are just $\mathcal{C}_{i}(Q)=\mathcal{C}_{3i}(P)$ .

References

[Art91] Michael Artin. Algebra. Prentice Hall, Inc., Englewood Cliffs, NJ, 1991.
[BEJ21] Bradley W. Brock, Noam D. Elkies, and Bruce W. Jordan. Periodic continued fractions over $S$ -integers in number fields and Skolem’s $p$ -adic method. Acta Arith., 197(4):379–420, 2021.
[Coh66] P. M. Cohn. On the structure of the $\operatorname{GL}_{2}$ of a ring. Inst. Hautes Études Sci. Publ. Math., (30):5–53, 1966.
[JZ19] Bruce W. Jordan and Yevgeny Zaytman. On the bounded generation of arithmetic $\operatorname{SL}_{2}$ . Proc. Natl. Acad. Sci. USA, 116(38):18880–18882, 2019.
[Knu97] Donald E. Knuth. The art of computer programming. Vol. 1. Addison-Wesley, Reading, MA, 1997. Fundamental algorithms, Third edition [of MR0286317].
[Lie81] Bernhard Liehl. On the group $\operatorname{SL}_{2}$ over orders of arithmetic type. J. Reine Angew. Math., 323:153–171, 1981.
[MRS18] Aleksander V. Morgan, Andrei S. Rapinchuk, and Balasubramanian Sury. Bounded generation of $\operatorname{SL}_{2}$ over rings of $S$ -integers with infinitely many units. Algebra Number Theory, 12(8):1949–1974, 2018.
[Nic11] Bogdan Nica. The unreasonable slightness of $\textrm{E}_{2}$ over imaginary quadratic rings. Amer. Math. Monthly, 118(5):455–462, 2011.
[Vas72] L. N. Vaseršteĭn. The group $\operatorname{SL}_{2}$ over Dedekind rings of arithmetic type. Mat. Sb. (N.S.), 89(131):313–322, 351, 1972.

$\displaystyle\|\![x,-x^{-1},x$	$\displaystyle,x^{-1},-x,x^{-1}]\!\|,$
$\displaystyle\|\![x,-4x^{-1},x$	$\displaystyle,-2x^{-1},2x,-2x^{-1}]\!\|,$	(8)
$\displaystyle\|\![x,-3x^{-1},x$	$\displaystyle,-3x^{-1},x,-3x^{-1}]\!\|,$	(9)
$\displaystyle\|\![x,\alpha,-\alpha^{-1}$	$\displaystyle,x\alpha^{2},\alpha^{-1},-\alpha]\!\|,$	(10)

$\displaystyle\overline{P}\star\overline{P^{\prime}}$	$\displaystyle=\|\![\,\overline{b_{1},a_{1},\ldots,a_{k},0,-b_{1},0,b_{1}^{\prime},a_{1}^{\prime},\ldots a_{k^{\prime}}^{\prime},0,-b_{1}^{\prime},0}\,]\!\|\text{ by \eqref{status}}$
	$\displaystyle=\|\![\,\overline{b_{1},a_{1},\ldots,a_{k}+b_{1}^{\prime}-b_{1},a_{1}^{\prime},\ldots,a_{k^{\prime}}^{\prime}-b_{1}^{\prime},0}\,]\!\|\text{ using \eqref{muster}}$
	$\displaystyle=\|\![b_{1},\overline{a_{1},\ldots,a_{k-1},a_{k}+b^{\prime}_{1}-b_{1},a^{\prime}_{1},\ldots,a^{\prime}_{k^{\prime}-1},a^{\prime}_{k^{\prime}}-b_{1}^{\prime},0,b_{1}}\,]\!\|$
	$\displaystyle=\|\![b_{1},\overline{a_{1},\ldots,a_{k-1},a_{k}+b_{1}^{\prime}-b_{1},a_{1}^{\prime},\ldots,a^{\prime}_{k^{\prime}-1},a^{\prime}_{k^{\prime}}-b^{\prime}_{1}+b_{1}}\,]\!\|.$	(28)

A composition law and refined notions of convergence for periodic continued fractions

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

2. Actions of 2×22\times 2 matrices

2.1. Linear fractional transformations

Proposition 2.1.

Proof.

Definition 2.2.

Proposition 2.3.

Proof.

2.2. Subgroups of \excepttoc𝐆𝐋𝟐⁡(𝐂)\operatorname{GL_{2}}(\mathbf{C})\fortocGL2⁡(𝐂)\operatorname{GL_{2}}(\mathbf{C}) determined by eigenspaces

Proposition 2.4.

Proof.

Proposition 2.5.

Proof.

Proposition 2.6.

Proof.

3. Finite continued fractions

Definition 3.1.

3.1. An equivalence relation on finite continued fractions

Definition 3.2.

Definition 3.3.

Proposition 3.4.

Proof.

Definition 3.5.

Proposition 3.6.

Proof.

Remark 3.7.

Definition 3.8.

Proposition 3.9.

Proof.

Definition 3.10.

Remark 3.11.

3.2. Mapping finite continued fractions to matrices

Proposition 3.12.

Proof.

Proposition 3.13.

Proof.

Proposition 3.14.

Proof.

Definition 3.15.

Proposition 3.16.

Proof.

Definition 3.17.

Proposition 3.18.

Proof.

Remark 3.19.

Definition 3.20.

Proposition 3.21.

Proof.

Corollary 3.22.

Proof.

Remark 3.23.

3.3. The kernel of \excepttoc𝑴¯\overline{M}\fortocM¯\overline{M} and presentations of \excepttoc𝔼𝟐​(𝓞){\mathbb{E}}_{2}(\mathcal{O})\fortoc 𝔼2​(𝒪){\mathbb{E}}_{2}(\mathcal{O})

Remark 3.24.

Definition 3.25.

Definition 3.26.

Definition 3.27.

Remark 3.28.

Proposition 3.29.

Proof.

Theorem 3.30 (Cohn).

Proof.

Corollary 3.31.

Proof.

4. Periodic continued fractions

Definition 4.1.

4.1. An equivalence relation and group law on PCFs

Definition 4.2.

Definition 4.3.

Remark 4.4.

Proposition 4.5.

Proof.

Definition 4.6.

Remark 4.7.

Proposition 4.8.

Proof.

Theorem 4.9.

2. Actions of $2\times 2$ matrices

2.2. Subgroups of \excepttoc $\operatorname{GL_{2}}(\mathbf{C})$ \fortoc $\operatorname{GL_{2}}(\mathbf{C})$ determined by eigenspaces

3.3. The kernel of \excepttoc $\overline{M}$ \fortoc $\overline{M}$ and presentations of \excepttoc ${\mathbb{E}}_{2}(\mathcal{O})$ \fortoc ${\mathbb{E}}_{2}(\mathcal{O})$

5. The subgroup $\operatorname{\overline{\textup{{PCF}}}}_{\mathcal{O}}(A)\leq\operatorname{\overline{\textup{{PCF}}}}(\mathcal{O})$ and its representations

6.1. \excepttoc $P_{1}=[1,\overline{2}],\hat{P}_{1}=\sqrt{2}$ \fortoc $P_{1}=[1,\overline{2}],\hat{P}_{1}=\sqrt{2}$

6.2. \excepttoc $P_{2}=[1,\overline{2,2,2}],\hat{P}_{2}=\sqrt{2}$ \fortoc $P_{2}=[1,\overline{2,2,2}],\hat{P}_{2}=\sqrt{2}$

6.3. \excepttoc $P_{3}=[2,\overline{-2,4}],\hat{P}_{3}=\sqrt{2}$ \fortoc $P_{3}=[2,\overline{-2,4}],\hat{P}_{3}=\sqrt{2}$

6.4. \excepttoc $P_{4}=[2,\overline{3,-2,3}],\hat{P}_{4}=\sqrt{2}$ \fortoc $P_{4}=[2,\overline{3,-2,3}],\hat{P}_{4}=\sqrt{2}$

8.2. The fraction of \excepttoc $\hat{P}_{j}=\beta_{+}$ \fortoc $\hat{P}_{j}=\beta_{+}$ can be made arbitrarily close to \excepttoc $1/2$ \fortoc $1/2$

8.3. The fraction of \excepttoc $\hat{P}_{j}=\beta_{+}$ \fortoc $\hat{P}_{j}=\beta_{+}$ can be made arbitrarily close to \excepttoc $1$ \fortoc $1$

8.4. The equivalence \excepttoc $\sim$ \fortoc $\sim$ on \excepttoc $\operatorname{\textup{{PCF}}}(\mathcal{O})$ \fortoc $\operatorname{\textup{{PCF}}}(\mathcal{O})$ does not respect convergence