On quadratic approximation for hyperquadratic continued fractions

TL;DR

This paper investigates quadratic approximations of hyperquadratic continued fractions over finite fields, addressing Diophantine exponents and degree calculations for specific families of algebraic Laurent series.

Contribution

It provides new insights into quadratic approximations of hyperquadratic continued fractions and answers a previously open question about Diophantine exponents.

Findings

Resolved a question on Diophantine exponents for algebraic Laurent series

Determined degrees of specific hyperquadratic continued fraction families

Enhanced understanding of approximation properties in finite field Laurent series

Abstract

We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning Diophantine exponents for algebraic Laurent series. As the second application, we determine the degrees of these families in particular case.

Equations

$$0<\left|P(\xi )\right|\le H(P{)}^{-w}(\text{resp. }0<\left|\xi -\alpha \right|\le H(\alpha {)}^{-w^{\ast }-1})$$

$$w_n(\alpha )=w_n^{\ast }(\alpha )=\min \{n,d-1\},$$

$$w_1(\alpha )=w_1^{\ast }(\alpha )=\dots =w_n(\alpha )=w_n^{\ast }(\alpha )=w.$$

$$w_n(\alpha )=w_n^{\ast }(\alpha )$$

$$r\ge \frac{3n+2+\sqrt{9n^2+4n+4}}{2}.$$

$$w_n(\alpha )\ne w_n^{\ast }(\alpha )$$

$$A{\alpha }^{r+1}+B{\alpha }^r+C\alpha +D=0.$$

$$0<\left|P(\xi )\right|\le H(P{)}^{-w}(\text{resp. }0<\left|\xi -\alpha \right|\le H(\alpha {)}^{-w^{\ast }-1})$$

$$\xi =a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\cdots }}},$$

$$\{\begin{array}{l}{p}_{-1}=1,p_0=a_0,p_n=a_n{p}_{n-1}+p_{n-2},n\ge 1,\\ q_{-1}=0,q_0=1,q_n=a_n{q}_{n-1}+q_{n-2},n\ge 1.\end{array}$$

$${\Theta }_k^t(\lambda )=\{\begin{array}{ll}[0,T^{[k]},{⨁}_{i\ge 1}(T,(\lambda T,\mu T{)}^{[(r^i-1)/2]}{)}^{[k+1]}]& \text{if}p\ne 2,\\ [0,T^{[k]}{⨁}_{i\ge 1}(T,\lambda T^{[r^i-1]}{)}^{[k+1]}]& \text{if}p=2.\end{array}$$

$$\{\begin{array}{l}{q}_k{X}^{r+1}-p_k{X}^r+(\lambda \mu {)}^{(r-1)/2}{q}_{k+r}X-(\lambda \mu {)}^{(r-1)/2}{p}_{k+r}=0\text{if}p\ne 2,\\ q_k{X}^{r+1}-p_k{X}^r+q_{k+r}X-p_{k+r}=0\text{if}p=2,\end{array}$$

$$w_2^{\ast }({\Theta }_k^t(\lambda ))\ge \max \left\{\frac{r-1}{k+1},r-1-\frac{r(r-1)(r-4)}{2r(k+1)+(r-1)(r-2)}\right\},$$

$$w_2({\Theta }_k^t(\lambda ))\ge \max \left\{\frac{r-1}{k+1}+1,r-\frac{r(r-1)(r-3)}{2r(k+1)+(r-1)(r-2)}\right\}.$$

$$k\le \frac{2(r-1)}{d+\sqrt{d^2+4(2r-1)d+4}}-1,$$

$$w_n^{\ast }({\Theta }_k^t(\lambda ))=\frac{r-1}{k+1},w_n({\Theta }_k^t(\lambda ))=\frac{r-1}{k+1}+1$$

$$2rd\le \left(r-2-\frac{r(r-1)(r-4)}{2r(k+1)+(r-1)(r-2)}\right)\left(r-d-\frac{r(r-1)(r-4)}{2r(k+1)+(r-1)(r-2)}\right),$$

$$w_n^{\ast }({\Theta }_k^t(\lambda ))=r-1-\frac{r(r-1)(r-4)}{2r(k+1)+(r-1)(r-2)},$$

$$w_n({\Theta }_k^t(\lambda ))=r-\frac{r(r-1)(r-3)}{2r(k+1)+(r-1)(r-2)}$$

$$w_1(\xi )=w_1^{\ast }(\xi )=\underset{n⟶\infty }{\mathrm{limsup}}\frac{\mathrm{deg}{q}_{n+1}}{\mathrm{deg}{q}_n},$$

$$r>\frac{3d+2+\sqrt{9d^2+4d+4}}{2},$$

$$k=0\text{or}k>\frac{(r^2-4r+2)(r-1)}{2r}-1,$$

$$F_0=1,F_1=T,F_{n+1}=T{F}_n+F_{n-1}\text{for}n\ge 1.$$

$$q_{\ell -1}{X}^{r+1}+p_{\ell -1}{X}^r+({\epsilon }_1{q}_{\ell -2}{F}_{r-1}+{\epsilon }_2{q}_{\ell -1}{F}_{r-2})X+{\epsilon }_1{p}_{\ell -2}{F}_{r-1}+{\epsilon }_2{p}_{\ell -1}{F}_{r-2}=0.$$

$${\Phi }_{\ell }^t(\mathbf{\lambda };\mathbf{\epsilon })=[{\lambda }_{1}T,{\lambda }_{2}T,\dots ],$$

$${\lambda }_{\ell +rm+1}=({\epsilon }_2/{\epsilon }_1){\epsilon }_2^{(-1{)}^{m+1}}{\lambda }_{m+1}^r,$$

$${\lambda }_{\ell +rm+i}=({\epsilon }_1/{\epsilon }_2{)}^{(-1{)}^i}\text{for}m\ge 0\text{and}2\le i\le r.$$

$$w_2^{\ast }({\Phi }_{\ell }^t(\mathbf{\lambda };\mathbf{\epsilon }))\ge \frac{m}{\ell }(r-1),w_2({\Phi }_{\ell }^t(\mathbf{\lambda };\mathbf{\epsilon }))\ge 1+\frac{m}{\ell }(r-1).$$

$$\frac{\ell }{m}\le \frac{2(r-1)}{d+\sqrt{d^2+4(2r-1)d+4}},$$

$$w_n^{\ast }({\Phi }_{\ell }^t(\mathbf{\lambda };\mathbf{\epsilon }))=\frac{m}{\ell }(r-1),w_n({\Phi }_{\ell }^t(\mathbf{\lambda };\mathbf{\epsilon }))=1+\frac{m}{\ell }(r-1)$$

$$w_2^{\ast }({\Phi }_{\ell }^t(\mathbf{\lambda };\mathbf{\epsilon }))\ge r-1-\frac{r(r-1)(r-4)}{2\ell r+(r-1)(r-2)},$$

$$w_2({\Phi }_{\ell }^t(\mathbf{\lambda };\mathbf{\epsilon }))\ge r-\frac{r(r-1)(r-3)}{2\ell r+(r-1)(r-2)}.$$

Full text

1. Introduction

Let $n\geq 1$ be an integer and $\xi\in\mathbb{R}$. We denote by $w_{n}(\xi)$ (resp. $w_{n}^{*}(\xi)$) the supremum of the real numbers $w$ (resp. $w^{*}$) which satisfy

[TABLE]

for infinitely many integer polynomials $P(X)$ of degree at most $n$ (resp. algebraic numbers $\alpha\in\mathbb{C}$ of degree at most $n$). Here, $H(P)$ is defined to be the maximum of the absolute values of the coefficients of $P(X)$ and $H(\alpha)$ is equal to $H(Q)$, where $Q(X)$ is the minimal polynomial of $\alpha$ over $\mathbb{Z}$. The functions $w_{n}$ and $w_{n}^{*}$ are called Diophatine exponents. Liouville proved that $w_{1}(\alpha)\leq\deg\alpha-1$ for any algebraic real number $\alpha$. Roth [13] improved Liouville Theorem, that is, he showed that $w_{1}(\alpha)=1$ for all algebraic irrational real numbers $\alpha$. For higher Diophantine exponents, from the Schmidt Subspace Theorem, it is known that

[TABLE]

where $n\geq 1$ is an integer and $\alpha$ is an algebraic real number of degree $d$ (see [2, Section 3]).

Let $p$ be a prime, $q=p^{s}$, where $s\geq 1$ is an integer and $\mathbb{F}_{q}$ be the finite field containing $q$ elements. We denote by $\mathbb{F}_{q}[T],\mathbb{F}_{q}(T),$ and $\mathbb{F}_{q}((T^{-1}))$ respectively the ring of polynomials, the field of rational functions, the field of Laurent series in $T^{-1}$ over $\mathbb{F}_{q}$. We consider analogues of Diophantine exponents $w_{n}$ and $w_{n}^{*}$ for Laurent series (see Section 2 for the precisely definition). As an analogue of Liouville Theorem, Mahler [7] showed $w_{1}(\alpha)\leq\deg\alpha-1$ for any algebraic Laurent series $\alpha\in\mathbb{F}_{q}((T^{-1}))$. However, there exist counter examples of analogue of Roth Theorem for Laurent series over a finite field. Let $r_{1}=p^{t_{1}}$, where $t_{1}\geq 1$ is an integer. He proved that the Laurent series $\alpha_{1}:=\sum_{n=0}^{\infty}T^{-r_{1}^{n}}$ is algebraic of degree $r_{1}$ with $w_{1}(\alpha_{1})=r_{1}-1$. After that many authors investigated rational approximations (e.g. [3, 9, 14, 15]) and algebraic approximations (e.g. [12, 16, 17]) for algebraic Laurent series. In particular, the second author ([12]) proved that, for any rational number $w>2n-1$, there exists an algebraic Laurent series $\alpha\in\mathbb{F}_{q}((T^{-1}))$ such that

[TABLE]

The first purpose of this paper is to consider Problem 2.2 in [12].

Problem 1.1**.**

Is it true that

[TABLE]

for an integer $n\geq 1$ and an algebraic Laurent series $\alpha\in\mathbb{F}_{q}((T^{-1}))$?

Note that it is well-known and easy to see that $w_{1}(\xi)=w_{1}^{*}(\xi)$ for all $\xi\in\mathbb{F}_{q}((T^{-1}))$. We give a negative answer of Problem 1.1 for any $n\geq 2$ and $q\geq 4$.

Theorem 1.2**.**

Let $n\geq 2$ be an integer, $q=p^{s}$, and $r=q^{t}$ with $s,t\geq 1$ be integers and

[TABLE]

If $q\geq 4$, then there exist algebraic Laurent series $\alpha\in\mathbb{F}_{q}((T^{-1}))$ with $\deg\alpha=r+1$ such that

[TABLE]

Let $\alpha$ be in $\mathbb{F}_{q}((T^{-1}))$ and $r=p^{t}$, where $t\geq 0$ is an integer. We call $\alpha$ hyperquadratic if $\alpha$ is irrational and there exists $(A,B,C,D)\in\mathbb{F}_{q}[T]^{4}\setminus\{{\bf 0}\}$ such that

[TABLE]

For example, quadratic Laurent series are hyperquadratic, $\alpha_{1}$ is hyperquadratic and satisfies $T\alpha_{1}^{r}-T\alpha_{1}+1=0$. Baum and Sweet [1] started a study of hyperquadratic continued fractions in characteristic two. In 1986, Mills and Robbins [10] showed the existence of hyperquadratic continued fractions with all partial quotient of degree one in odd characteristic with a prime field. We refer the reader to [6] for a survey of recent works of hyperquadratic continued fractions. We observe that the degree of hyperquadratic Laurent series which satisfies (1) is less than or equal to $r+1$. However, the exact degree of these is open. In this paper, we discuss the following problem.

Problem 1.3**.**

Given a hyperquadratic Laurent series $\alpha\in\mathbb{F}_{q}((T^{-1}))$, determine the exact degree of $\alpha$.

In order to determine the exact degree, we approach to quadratic approximations. Since $w_{2}(\alpha)\leq\deg\alpha-1$ for $\alpha$ as in (1) (see Lemma 4.1), if $w_{2}(\alpha)>r-1$, then $\deg\alpha=r+1$. In this paper, we deal with two families for hyperquadratic continued fractions introduced in [4, 5]. We determine the exact degree of these families in particular cases.

This paper is organized as follows. In Section 2, we recall notations which are used in this paper. In Section 3, we state main results of quadratic approximation for hyperquadratic continued fractions and applications of the results. In Section 4, we give lemmas for the proof of main results. In Section 5, we prove the main results.

2. Notations

A nonzero Laurent series $\xi\in\mathbb{F}_{q}((T^{-1}))$ is represented by $\xi=\sum_{n=N}^{\infty}a_{n}T^{-n}$, where $N\in\mathbb{Z},a_{n}\in\mathbb{F}_{q},$ and $a_{N}\neq 0$. The field $\mathbb{F}_{q}((T^{-1}))$ has a non-Archimedean absolute value $\left\lvert\xi\right\rvert=q^{-N}$ and $\left\lvert 0\right\rvert=0$. The absolute value can be uniquely extended to the algebraic closure of $\mathbb{F}_{q}((T^{-1}))$ and we also write $\left\lvert\cdot\right\rvert$ for the extended absolute value.

The height of $P(X)\in(\mathbb{F}_{q}[T])[X]$, denoted by $H(P)$, is defined to be the maximum of the absolute values of the coefficients of $P(X)$. For $\alpha\in\overline{\mathbb{F}_{q}(T)}$, there exists a unique non-constant, irreducible, primitive polynomial $P(X)\in(\mathbb{F}_{q}[T])[X]$ whose leading coefficients are monic polynomials in $T$ such that $P(\alpha)=0$. The polynomial $P(X)$ is called the minimal polynomial of $\alpha$. The height (resp. the degree, the inseparable degree) of $\alpha$, denoted by $H(\alpha)$ (resp. $\deg\alpha$, $\operatorname{insep}\alpha$), is defined to be the height of $P(X)$ (resp. the degree of $P(X)$, the inseparable degree of $P(X)$). Let $n\geq 1$ be an integer and $\xi$ be in $\mathbb{F}_{q}((T^{-1}))$. We denote by $w_{n}(\xi)$ (resp. $w_{n}^{*}(\xi)$) the supremum of the real numbers $w$ (resp. $w^{*}$) which satisfy

[TABLE]

for infinitely many $P(X)\in(\mathbb{F}_{q}[T])[X]$ of degree at most $n$ (resp. $\alpha\in\overline{\mathbb{F}_{q}(T)}$ of degree at most $n$).

Let $\alpha\in\overline{\mathbb{F}_{q}(T)}$ be a quadratic number. If $\operatorname{insep}\alpha=1$, let $\alpha^{\prime}\neq\alpha$ be the Galois conjugate of $\alpha$. If $\operatorname{insep}\alpha=2$, let $\alpha^{\prime}=\alpha$.

A Laurent series $\xi\in\mathbb{F}_{q}((T^{-1}))$ can be expressed as a continued fraction:

[TABLE]

where $a_{n}\in\mathbb{F}_{q}[T]$ for all $n\geq 0$ and $\deg_{T}a_{m}\geq 1$ for all $m\geq 1$. We write the expansion $\xi=[a_{0},a_{1},a_{2},\ldots]$. The continued fraction expansion of $\xi$ is finite if and only if $\xi$ is in $\mathbb{F}_{q}(T)$. It is known that the continued fraction expansion of $\xi$ is ultimately periodic if and only if $\xi$ is quadratic (see Théorème 4 in [8, CHAPITRE IV]). We define sequences $(p_{n})_{n\geq-1}$ and $(q_{n})_{n\geq-1}$ by

[TABLE]

We call $(p_{n}/q_{n})_{n\geq 0}$ the convergent sequence of $\xi$ and $p_{n}/q_{n}$ the $n$-th convergent of $\xi$.

Let $n\geq 1$ be an integer and $W,V$ be finite words. We denote by $W\bigoplus V$ the word of concatenation of $W$ and $V$. We put $W^{[n]}:=W\bigoplus\cdots\bigoplus W$ ($n$ times) and $\overline{W}:=W\bigoplus W\bigoplus\cdots\bigoplus W\bigoplus\cdots$ (infinitely many times). We write $W^{[0]}$ the empty word.

Let $A,B$ be real numbers with $B\neq 0$. We write $A\ll B$ (resp. $A\ll_{a}B$) if $\left\lvert A\right\rvert\leq C\left\lvert B\right\rvert$ for some constant (resp. some constant depending at most on $a$) $C>0$. We write $A\asymp B$ (resp. $A\asymp_{a}B$) if $A\ll B$ and $B\ll A$ (resp. $A\ll_{a}B$ and $B\ll_{a}A$) hold.

3. Main results

We recall a family of hyperquadratic continued fractions. Let $s,t\geq 1$ be integers and we put $q:=p^{s}$ and $r:=q^{t}$. Let $k\geq 0$ be an integer and $\lambda$ be in $\mathbb{F}_{q}^{*}$. If $p\neq 2$, then we assume that $\lambda\neq 2$ and put $\mu:=2-\lambda$. We define $\Theta_{k}^{t}(\lambda)\in\mathbb{F}_{q}((T^{-1}))$ by

[TABLE]

Lasjaunias and Ruch [4] proved that $\Theta_{k}^{t}(\lambda)$ is hyperquadratic and satisfies the algebraic equation

[TABLE]

where $(p_{n}/q_{n})_{n\geq 0}$ is the convergent sequence of $\Theta_{k}^{t}(\lambda)$. Note that if $\lambda=1$, then $\Theta_{k}^{t}(\lambda)$ is ultimately periodic continued fraction, that is, $\Theta_{k}^{t}(\lambda)$ is quadratic.

Our first main result is the following.

Theorem 3.1**.**

Let $d\geq 2$ be an integer and $\Theta_{k}^{t}(\lambda)\in\mathbb{F}_{q}((T^{-1}))$ be as in (2). Assume that $\lambda\neq 1$. Then we have

[TABLE]

If

[TABLE]

then we have

[TABLE]

for all integers $2\leq n\leq d$. If

[TABLE]

then we have

[TABLE]

for all integers $2\leq n\leq d$.

Remark**.**

It is well-known and easy to see that for any $\xi\in\mathbb{F}_{q}((T^{-1}))$,

[TABLE]

where $(p_{n}/q_{n})_{n\geq 0}$ is the convergent sequence of $\xi$. Since $\Theta_{k}^{t}(\lambda)$ has a bounded partial quotient, we have $w_{1}(\Theta_{k}^{t}(\lambda))=w_{1}^{*}(\Theta_{k}^{t}(\lambda))=1$. If

[TABLE]

there exists an effectively computable positive constant $C_{1}(r,d)$, depending only on $r$ and $d$, such that we have (9) and (10) for all $k\geq C_{1}(r,d)$.

The key point of the proof of Theorem 3.1 is to find two sequences of very good quadratic approximation, that is, ultimately periodic continued fractions.

As an application of Theorem 3.1, we give a sufficient condition to determine the exact degree of $\Theta_{k}^{t}(\lambda)$.

Corollary 3.2**.**

Let $\Theta_{k}^{t}(\lambda)\in\mathbb{F}_{q}((T^{-1}))$ be as in (2). Assume that $\lambda\neq 1$. If

[TABLE]

then we have $\deg\Theta_{k}^{t}(\lambda)=r+1$.

We recall another family of hyperquadratic continued fractions. We define a sequence $(F_{n})_{n\geq 0}$ of polynomials in $\mathbb{F}_{q}[T]$ by

[TABLE]

Note that the sequence can be regarded as the analogue of the Fibonacci sequence. Let $s,t\geq 1$ be integers and we put $q:=2^{s},r:=2^{t}$. Let $\ell\geq 1$ be an integer, ${\bm{\lambda}}=(\lambda_{1},\ldots,\lambda_{\ell})\in(\mathbb{F}_{q}^{*})^{\ell},{\bm{\varepsilon}}=(\varepsilon_{1},\varepsilon_{2})\in(\mathbb{F}_{q}^{*})^{2}$, and $(p_{n}/q_{n})_{0\leq n\leq\ell-1}$ be the convergent sequence of the continued fraction $[\lambda_{1}T,\ldots,\lambda_{\ell}T]$. We consider the algebraic equation

[TABLE]

Lasjaunias [5] proved that (14) has a unique root $\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})$ in $\mathbb{F}_{q}((T^{-1}))$ with $\left\lvert\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})\right\rvert\geq q$, $\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})$ is hyperquadratic and its continued fraction expansion is the following:

[TABLE]

where a sequence $(\lambda_{n})_{n\geq 1}$ satisfy

[TABLE]

Note that if $\varepsilon_{1}=\varepsilon_{2}=1$ and $\lambda_{j}=1$ for all $1\leq j\leq\ell$, then $\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})$ is ultimately periodic continued fraction, that is, $\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})$ is quadratic.

In this paper, we consider the continued fractions $\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})$ in particular cases.

Theorem 3.3**.**

Let $\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})$ be as in (15), $m\geq 1,d\geq 2$ be integers with $m\leq\ell$, and $\lambda$ be in $\mathbb{F}_{q}^{*}$ with $\lambda\neq 1$.

1. (1)

Assume that $\lambda_{1}=\lambda,\lambda_{i}=1$ for all $2\leq i\leq m,\varepsilon_{1}=\varepsilon_{2}=1$ and if $m<\ell$, then $\lambda_{m+1}\neq 1$. Then we have

[TABLE]

If

[TABLE]

then we have

[TABLE]

for all integers $2\leq n\leq d$. 2. (2)

Assume that $r$ is a power of $q,\varepsilon_{1}=\varepsilon_{2}=1$ and $\lambda_{i}=\lambda$ for all $1\leq i\leq\ell$. Then we have

[TABLE]

If

[TABLE]

then we have

[TABLE]

for all integers $2\leq n\leq d$.

Remark**.**

By (11), we have $w_{1}(\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}}))=w_{1}^{*}(\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}}))=1$. If

[TABLE]

there exists an effectively computable positive constant $C_{2}(r,d)$, depending only on $r$ and $d$, such that we have (22) and (23) for all $\ell\geq C_{2}(r,d)$.

Corollary 3.4**.**

Let $\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})$ be as in (15), $m\geq 1,d\geq 2$ be integers with $m\leq\ell$, and $\lambda$ be in $\mathbb{F}_{q}^{*}$ with $\lambda\neq 1$.

1. (1)

Assume that $\lambda_{1}=\lambda,\lambda_{i}=1$ for all $2\leq i\leq m,\varepsilon_{1}=\varepsilon_{2}=1$ and if $m<\ell$, then $\lambda_{m+1}\neq 1$. If

[TABLE]

then we have $\deg\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})=r+1$. 2. (2)

Assume that $r$ is a power of $q,\varepsilon_{1}=\varepsilon_{2}=1$ and $\lambda_{i}=\lambda$ for all $1\leq i\leq\ell$. If

[TABLE]

then we have $\deg\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})=r+1$.

In the last part of this section, we mention a problem associated to Problem 1.1, Corollary 3.2 and 3.4.

Problem 3.5**.**

Let $n\geq 2$ be an integer. Determine the set of all values taken by $w_{n}-w_{n}^{*}$ over the set of algebraic Laurent series.

4. Preliminaries

In this section, we gather lemmas for the proof of main results.

Lemma 4.1**.**

([11, Theorem 5.2]).* Let $n\geq 1$ be an integer and $\alpha\in\mathbb{F}_{q}((T^{-1}))$ be an algebraic Laurent series. Then we have*

[TABLE]

Lemma 4.2 and 4.3 are immediately seen.

Lemma 4.2**.**

Let $\xi=[0,a_{1},a_{2},\ldots],\zeta=[0,b_{1},b_{2},\ldots]$ be in $\mathbb{F}_{q}((T^{-1}))$. Assume that there exists an integer $k\geq 1$ such that $a_{n}=b_{n}$ for all $1\leq n\leq k$ and $a_{k+1}\neq b_{k+1}$. Then we have

[TABLE]

where $(p_{n}/q_{n})_{n\geq 0}$ is the convergent sequence of $\xi$.

Lemma 4.3**.**

Let $\xi=[a_{0},a_{1},\ldots]$ be in $\mathbb{F}_{q}((T^{-1}))$ and $(p_{n}/q_{n})_{n\geq 0}$ be the convergent sequence of $\xi$. Then, for any $n\geq 1$, we have

[TABLE]

Lemma 4.4**.**

([11, Lemma 4.6]).* Let $r,s\geq 1$ be integers and $\alpha=[0,a_{1},\ldots,a_{r},\overline{a_{r+1},\ldots,a_{r+s}}]\in\mathbb{F}_{q}((T^{-1}))$ be an ultimately periodic continued fraction with $a_{r}\neq a_{r+s}$. Let $(p_{n}/q_{n})_{n\geq 0}$ be the convergent sequence of $\alpha$. Then we have*

[TABLE]

The following lemma is well-known and easy to see.

Lemma 4.5**.**

Let $r\geq 0,s\geq 1$ be integers and $\alpha=[0,a_{1},\ldots,a_{r},\overline{a_{r+1},\ldots,a_{r+s}}]\in\mathbb{F}_{q}((T^{-1}))$ be an ultimately periodic continued fraction. Let $(p_{n}/q_{n})_{n\geq 0}$ be the convergent sequence of $\xi$. Then we have $H(\alpha)\leq|q_{r}q_{r+s}|$.

Lemma 4.6**.**

Let $b,c,d\in\mathbb{F}_{q}[T]$ be distinct polynomials with $\deg_{T}b,\deg_{T}c,\deg_{T}d\geq 1$. Let $n,m,\ell\geq 1$ be integers and $a_{1},\ldots,a_{n-1}\in\mathbb{F}_{q}[T]$ be polynomials with $\deg_{T}a_{i}\geq 1$ for all $1\leq i\leq n-1$. We put

[TABLE]

Let $(p_{i,k}/q_{i,k})_{k\geq 0}$ be the convergent sequence of $\xi_{i}$ for $i=1,2,3,4$. Then we have

[TABLE]

Proof.

See Lemma 4.7 in [11] for the proof of (25).

It follows from Lemma 4.5 that $H(\alpha_{3})\leq|q_{3,n}q_{3,n+m+\ell+1}|$. We put

[TABLE]

Let $P_{3}(X)$ and $Q_{3}(X)$ be the minimal polynomial of $\alpha_{3}$ and $\beta_{3}$, respectively. Since $P_{3}$ and $Q_{3}$ do not have a common root, we have

[TABLE]

By Lemma 4.2, 4.3 and 4.4, we obtain

[TABLE]

Therefore, by Lemma 4.3 and 4.5, we have $H(\alpha_{3})\gg_{b,c}|q_{3,n}q_{3,n+m+\ell+1}|$.

In the same way to the above proof, we obtain $H(\alpha_{4})\asymp_{b,c,d}\left\lvert q_{4,n}q_{4,n+2m+2\ell+1}\right\rvert$. ∎

Lemma 4.7**.**

([11, Proposition 5.6]).* Let $n\geq 1$ be an integer and $\xi\in\mathbb{F}_{q}((T^{-1}))$. Then we have*

[TABLE]

Lemma 4.8**.**

Let $n,m\geq 1$ be integers with $n\geq m$ and $\xi\in\mathbb{F}_{q}((T^{-1}))$ be not algebraic of degree at most $m$. Then we have $w_{n}(\xi)\geq m$.

Proof.

Since the sequence $(w_{k}(\xi))_{k\geq 1}$ is increasing, we have $w_{n}(\xi)\geq m$ (see e.g. [2, page 200]). ∎

The following lemma is well-known and immediately seen.

Lemma 4.9**.**

Let $P(X)$ be in $(\mathbb{F}_{q}[T])[X]$. Assume that $P(X)$ can be factorized as

[TABLE]

where $A\in\mathbb{F}_{q}[T]$ and $\alpha_{i}\in\overline{\mathbb{F}_{q}(T)}$ for $1\leq i\leq n$. Then we have

[TABLE]

The following lemma is immediately seen by the definition of discriminant.

Lemma 4.10**.**

Let $\alpha\in\overline{\mathbb{F}_{q}(T)}$ be a quadratic number. If $\alpha\neq\alpha^{\prime}$, then we have

[TABLE]

The following two lemmas are key lemmas for proving Theorem 3.1 and 3.3.

Lemma 4.11**.**

([12, Lemma 3.19]).* Let $d\geq 2$ be an integer. Let $\xi$ be in $\mathbb{F}_{q}((T^{-1}))$, $\theta,\delta$ be positive numbers, and $\varepsilon$ be a non-negative number. Assume that there exist a sequence $(\alpha_{j})_{j\geq 1}$ and positive number $c$ such that for any $j\geq 1$, $\alpha_{j}\in\overline{\mathbb{F}_{q}(T)}$ is quadratic with $0<|\alpha_{j}-\alpha_{j}^{\prime}|\leq c$ and $\xi\neq\alpha_{j}$, $(H(\alpha_{j}))_{j\geq 1}$ is a divergent increasing sequence, and*

[TABLE]

If $2d\theta\leq(d-2+\delta)\delta$, then we have for all $2\leq n\leq d$,

[TABLE]

Lemma 4.12**.**

Let $\xi$ be in $\mathbb{F}_{q}((T^{-1}))\setminus\mathbb{F}_{q}(T)$. Assume that there exists a sequence $(\alpha_{j})_{j\geq 1}$ such that for any $j\geq 1$, $\alpha_{j}\in\overline{\mathbb{F}_{q}(T)}$ is quadratic with $\alpha_{j}\neq\alpha_{j}^{\prime}$ and $\xi\neq\alpha_{j}$, and $(H(\alpha_{j}))_{j\geq 1}$ is a divergent increasing sequence. If there exist limits of the sequences

[TABLE]

then we have

[TABLE]

Proof.

The inequality (28) is immediately seen. In what follows, we show (29). We put

[TABLE]

We may assume that $\gamma_{1}>1$ and $\gamma_{2}>0$. In fact, if $\gamma_{2}\leq 0$, then we have (29) by Lemma 4.7. By Lemma 4.10, we have $\gamma_{2}\leq 1$, which implies $\gamma_{2}<\gamma_{1}$. If $\gamma_{1}\leq 1$, then we obtain (29) by Lemma 4.8.

Then we obtain

[TABLE]

for all sufficiently large $j$. Therefore, we have

[TABLE]

for all sufficiently large $j$. It follows from Lemma 4.9 that

[TABLE]

for all sufficiently large $j$, where $P_{j}(X)=A_{j}(X-\alpha_{j})(X-\alpha_{j}^{\prime})$ is the minimal polynomial of $\alpha_{j}$. Hence, we have

[TABLE]

for all sufficiently large $j$. Thus, we obtain

[TABLE]

which implies (29). ∎

5. Proof of Main results

In this section, we prove the main results, that is, Theorem 1.2, 3.1, 3.3, Corollary 3.2 and 3.4.

Proof of Theorem 3.1.

For an integer $n\geq 3$, we put

[TABLE]

Then we have

[TABLE]

if $p\neq 2$, and

[TABLE]

if $p=2$. It follows from Lemma 4.3 and 4.6 that

[TABLE]

Since $\Theta_{k}^{t}(\lambda)$ and $\alpha_{n}$ have the same first $((k+1)\sum_{i=0}^{n-1}r^{i}+r^{n}-1)$-th partial quotients, while the next partial quotient are different, we have

[TABLE]

by Lemma 4.2 and 4.3. Similary, since $\Theta_{k}^{t}(\lambda)$ and $\beta_{n}$ have the same first $((k+1)\sum_{i=0}^{n}r^{i}+r^{n}-1)$-th partial quotients, while the next partial quotient are different, we get

[TABLE]

By Lemma 4.3 and 4.4, we obtain

[TABLE]

Therefore, we deduce that

[TABLE]

Hence, we obtain (4) and (5) by Lemma 4.12.

If (6) holds, then we heve

[TABLE]

Therefore, by Lemma 4.11, we obtain (7).

Similaly, if (8) holds, then we heve (9) and (10) by Lemma 4.11. ∎

Proof of Corollary 3.2.

It follows from (1) that $\deg\Theta_{k}^{t}(\lambda)\leq r+1$. If (12) holds, then we have $w_{2}(\Theta_{k}^{t}(\lambda))>r-1$ by Theorem 3.1. Therefore, by Lemma 4.1, we obtain $\deg\Theta_{k}^{t}(\lambda)=r+1$. ∎

Proof of Theorem 1.2.

It follows from $q\geq 4$ that we can take $\lambda\in\mathbb{F}_{q}^{*}$ with $\lambda\neq 1$ and $\lambda\neq 2$. Let $n\geq 2$ be integers. Since $r\geq(3n+2+\sqrt{9n^{2}+4n+4})/2$, we take $r=p^{t}$, where $t\geq 0$ is an integer with

[TABLE]

Then, by Theorem 3.1 and Corollary 3.2, we have $w_{n}(\Theta_{0}^{t}(\lambda))\neq w_{n}^{*}(\Theta_{0}^{t}(\lambda))$ and $\deg\Theta_{0}^{t}(\lambda)=r+1$. ∎

Proof of Theorem 3.3.

Since $w_{n}(\xi)=w_{n}(\xi^{-1})$ and $w_{n}^{*}(\xi)=w_{n}^{*}(\xi^{-1})$ for all $0\neq\xi\in\mathbb{F}_{q}((T^{-1}))$ and $n\geq 1$, we consider $\Phi:=\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})^{-1}=[0,\lambda_{1}T,\lambda_{2}T,\ldots]$ instead of $\Phi_{\ell}^{t}({\bm{\lambda}};{\bm{\varepsilon}})$. By the assumption, we obtain

[TABLE]

First, we prove (1). For an integer $n\geq 1$, we put

[TABLE]

where $i(n)=1+\ell\sum_{j=0}^{n-1}r^{j}$. Then we have $\lambda_{i(n)}=\lambda^{r^{n}}$. In a similar way to the proof of Theorem 3.1, we obtain

[TABLE]

Therefore, we have (16) by Lemma 4.12. If (17) holds, then we obtain (18) by Lemma 4.11.

Next, we prove (2). For an integer $n\geq 1$, we put

[TABLE]

Then we have

[TABLE]

and $\lambda_{i(n)-r^{n-1}}=\lambda$. In a similar way to the proof of Theorem 3.1, we obtain

[TABLE]

Therefore, we have (19) and (20) by Lemma 4.12. If (21) holds, then we obtain (22) and (23) by Lemma 4.11. ∎

Proof of Corollary 3.4.

In the same way to the proof Corollary 3.2, we prove Corollary 3.4. ∎

Acknowledgements

The author would like to thank the referee for helpful comments.