Continued fraction expansions of algebraic numbers

TL;DR

This paper investigates the properties of continued fraction expansions of algebraic numbers, establishing conditions for their independence and symmetry, and applying these results to approximation theorems similar to Roth's theorem.

Contribution

It introduces new criteria for independence and symmetry in continued fractions of algebraic numbers and applies these to prove a Roth-like approximation theorem.

Findings

Same long sub-words imply identical tails in continued fractions.

Mirror symmetry in expansions indicates quadratic algebraic numbers.

Results lead to a Roth-type approximation theorem for algebraic numbers.

Abstract

In this paper we establish properties of independence for the continued fraction expansions of two algebraic numbers. Roughly speaking, if the continued fraction expansions of two irrational algebraic numbers have the same long sub-word, then the two continued fraction expansions have the same tails. If the two expansions have mirror symmetry long sub-words, then both the two algebraic numbers are quadratic. Applying the above results, we prove a theorem analogous to the Roth's theorem about approximation by algebraic numbers.

Equations

$$(|A_n|+|A_n^{\prime }|)/|B_n|\le L,$$

$$\alpha =[[\alpha ];a_1,a_2,\cdots ]$$

$$p(\text{a},n):=Card\{a_{i+1}\cdots a_{i+n}|i\ge 0\}.$$

$$\underset{n\to +\infty }{\lim }\frac{p(\text{a},n)}{n}=+\infty ,$$

$$(|A_n|+|A_n^{\prime }|)/|B_n|\le L,$$

$$\alpha =[\alpha ]+0.a_1{a}_2\cdots ,$$

$${\alpha }^{\prime }=[\alpha ]+0.a_1^{\prime }{a}_2^{\prime }\cdots$$

$$\alpha =[[\alpha ];a_1,a_2,\cdots ],$$

$${\alpha }^{\prime }=[[{\alpha }^{\prime }];a_1^{\prime },a_2^{\prime },\cdots ]$$

$$\underset{n\to \infty }{\mathrm{limsup}}\mid |A_n|-|A_n^{\prime }|\mid =+\infty ,$$

$$A=a_1{a}_2\cdots a_n$$

$$B=b_1{b}_2\cdots b_n$$

$$b_n{b}_{n-1}\cdots b_1=a_i\cdots a_n{a}_1\cdots a_{i-1}.$$

$$(q_{k_n}{q}_{l_n}^{\prime }{)}^{1+\delta }<L{q}_{k_n+m_n}{q}_{l_n+m_n}^{\prime }$$

$$\begin{array}{c}A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in PSL(2,\mathbb{Z})\end{array}$$

$$|\xi -\frac{a\infty +b}{c\infty +d}|<\Vert A{\Vert }^{-2-ϵ},$$

$$\begin{array}{c}A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in PSL(2,\mathbb{Z}),\end{array}$$

$$\begin{array}{c}A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\alpha =\frac{a\alpha +b}{c\alpha +d}.\end{array}$$

$$\Vert \beta \Vert =\Vert A\Vert ,$$

$$c_1\Vert \cdot {\Vert }_{{\alpha }^{\prime }}<\Vert \cdot {\Vert }_{\alpha }<c_2\Vert \cdot {\Vert }_{{\alpha }^{\prime }}.$$

$$|\xi -\beta |<\Vert \beta {\Vert }^{-2-ϵ},$$

$$\Vert \beta \Vert =|\frac{1}{\beta -{\beta }^{\sigma }}|=|\frac{(c\alpha +d)(c{\alpha }^{\sigma }+d)}{\alpha -{\alpha }^{\sigma }}|.$$

$$|\xi -\beta |<\Vert \beta {\Vert }^{-1-ϵ},$$

$$[a_0;a_1,a_2,\cdots ,a_n]$$

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

$$[a_0;a_1,a_2,\cdots ,a_n,\cdots ]$$

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

$$[a_0;a_1,a_2,\cdots ,a_{k-1},\overline{a_k,\cdots ,a_{k+m-1}}],$$

$$a=[a_0;a_1,a_2,\cdots ,a_n,\cdots ]$$

Full text

*College of Mathematics and Computer Science, Fujian Normal University,

*  *Fuzhou, 350108, China;

*  Email: linxianzu@126.com

Keywords: continued fraction, subspace theorem, irrational algebraic number.

Mathematics Subject Classification 2010: 11J70, 11J81.

1. Introduction

It is a well-known fact that every quadratic irrational real number can be represented by an eventually periodic continued fraction. By contrast, no analogous results are known for algebraic numbers of higher degree. In fact we can not write down explicitly the continued fraction expansion of a single real algebraic number of degree higher than 2, and we do not know whether the partial quotients of such expansions are bounded or unbounded.

In the past 10 years, some breakthroughs have been obtained in this direction by Adamczewski, Bugeaud and other people [1, 4, 5, 6, 7], highlighted in [7]. Before introducing the main result in [7], we need some preparations.

We say that an infinite word $\textbf{a}=a_{1}a_{2}\cdots$ of elements from an alphabet $\Omega$ has $long\ repetition$ if it satisfies (i), (ii) and (iii) of Condition 1.1, where the length of a finite word $A$ is denoted by $|A|$, and the mirror image $a_{n}a_{n-1}\cdots a_{1}$ of a finite word $B=a_{1}a_{2}\cdots a_{n}$ is denoted by $\overline{B}$. We say $\textbf{a}=a_{1}a_{2}\cdots$ has $long\ mirror\ repetition$ if it satisfies (i’), (ii) and (iii) of Condition 1.1

Condition 1.1**.**

There exist three sequences of finite nonempty words $\{A_{n}\}_{n\geq 1}$, $\{A^{\prime}_{n}\}_{n\geq 1}$, $\{B_{n}\}_{n\geq 1}$ such that:

1. (i)

for any $n\geq 1$, $A_{n}B_{n}A^{\prime}_{n}B_{n}$ is a prefix of a; 2. (i’)

for any $n\geq 1$, $A_{n}B_{n}A^{\prime}_{n}\overline{B_{n}}$ is a prefix of a; 3. (ii)

the sequence $\{|B_{n}|\}_{n\geq 1}$ is strictly increasing; 4. (iii)

there exists a positive constant $L$ such that

[TABLE]

for every $n\geq 1$.

The main results in [7] are the following two theorems:

Theorem 1.2**.**

Let

[TABLE]

be the continued fraction expansions of a real algebraic number of degree higher than 2, and let $\{\frac{p_{n}}{q_{n}}\}_{n\geq 0}$ be the sequence of convergents. Assume that the sequence $\{(q_{n})^{1/n}\}_{n\geq 0}$ is bounded. Then, the infinite word $a_{1}a_{2}\cdots$ has no long repetitions.

Theorem 1.3**.**

Let assumptions be as above. Then, the infinite word $a_{1}a_{2}\cdots$ has no long mirror repetitions.

For an infinite word $\textbf{a}=a_{1}a_{2}\cdots$ of elements from an alphabet $\Omega$ and a positive integer $n$, set

[TABLE]

Then $p(\textbf{a},n)$ is the number of distinct blocks of $n$ consecutive letters occurring in a. Theorem 1.2 implies that

[TABLE]

where a is the continued fraction expansion of an algebraic number of degree higher than 2. This result combining with a fundamental property about automatic sequences (cf.[9]) immediately implies that the continued fraction expansion of an algebraic number of degree higher than 2 can not be generated by a finite automaton.

Theorem 1.2 and its corollaries are very similar to the corresponding results about expansions of algebraic numbers to integer bases [2]. In [3], Adamczewski and Bugeaud further explored the independence of $b$-ary expansions of two irrational real algebraic numbers $\alpha$ and $\beta$.

Let $\textbf{a}=a_{1}a_{2}\cdots$ and $\textbf{a}^{\prime}=a^{\prime}_{1}a^{\prime}_{2}\cdots$ be two infinite words of elements from an alphabet $\Omega$. The following is a condition about the pair $(\textbf{a},\textbf{a}^{\prime})$:

Condition 1.4**.**

There exist three sequences of finite nonempty words $\{A_{n}\}_{n\geq 1}$, $\{A^{\prime}_{n}\}_{n\geq 1}$, $\{B_{n}\}_{n\geq 1}$ such that:

1. (i)

for any $n\geq 1$, the word $A_{n}B_{n}$ is a prefix of the word a and the word $A^{\prime}_{n}B_{n}$ is a prefix of the word $\textbf{a}^{\prime}$; 2. (i’)

for any $n\geq 1$, the word $A_{n}B_{n}$ is a prefix of the word a and the word $A^{\prime}_{n}\overline{B_{n}}$ is a prefix of the word $\textbf{a}^{\prime}$; 3. (ii)

the sequence $\{|B_{n}|\}_{n\geq 1}$ tends to infinity; 4. (iii)

there exists a positive constant $L$ such that

[TABLE]

for each $n\geq 1$.

The main result in [3] is:

Theorem 1.5**.**

Let $b\geq 2$ be a fixed integer. Let $\alpha$ and $\alpha^{\prime}$ be two irrational real algebraic numbers. If their $b$-ary expansions

[TABLE]

and

[TABLE]

satisfy (i), (ii)and (iii) of Condition 1.4, then the two infinite words $\textrm{a}=a_{1}a_{2}\cdots$ and $\textrm{a}^{\prime}=a^{\prime}_{1}a^{\prime}_{2}\cdots$ have the same tail.

In this paper, we show that similar results of independence hold for the continued fraction expansions of two algebraic numbers. Our main results are:

Theorem 1.6**.**

Let

[TABLE]

and

[TABLE]

be the continued fraction expansions of two irrational real algebraic numbers, and let $\{\frac{p_{n}}{q_{n}}\}_{n\geq 0}$ and $\{\frac{p^{\prime}_{n}}{q^{\prime}_{n}}\}_{n\geq 0}$ be respectively the sequence of convergents of $\alpha$ and $\alpha^{\prime}$. Assume that the sequence $\{(q_{n}q^{\prime}_{n})^{1/n}\}_{n\geq 0}$ is bounded. If the two infinite words $\textbf{a}=a_{1}a_{2}\cdots$ and $\textbf{a}^{\prime}=a^{\prime}_{1}a^{\prime}_{2}\cdots$ satisfy $(i),(ii)$ and $(iii)$ of Condition 1.4, then they have the same tail. Moreover, if

[TABLE]

then both $\alpha$ and $\alpha^{\prime}$ are quadratic irrationals

We say that two finite words

[TABLE]

and

[TABLE]

are $cycle\ mirror\ symmetry$ if there exists an positive integer $i\leq n$ such that

[TABLE]

Theorem 1.7**.**

Let assumptions be as above. If the two infinite words $\textbf{a}=a_{1}a_{2}\cdots$ and $\textbf{a}^{\prime}=a^{\prime}_{1}a^{\prime}_{2}\cdots$ satisfy (i’), (ii) and (iii) of Condition 1.4, then both $\alpha$ and $\alpha^{\prime}$ are quadratic irrationals. Moreover the shortest periods of the a and $\textbf{a}^{\prime}$ are cycle mirror symmetry.

Remark 1.8**.**

The proofs below show that Theorems 1.6 and 1.7 are still valid if we replace (iii) of Condition 1.4 and the boundness of $\{(q_{n}q^{\prime}_{n})^{1/n}\}_{n\geq 0}$ with the following condition

Condition 1.9**.**

There exist positive numbers $\delta$ and $L$ such that

[TABLE]

for each $n\geq 1$, where $k_{n}=|A_{n}|$, $l_{n}=|A^{\prime}_{n}|$, and $m_{n}=|B_{n}|$.

Applying Theorems 1.6 and 1.7 to the case $\alpha=\alpha^{\prime}$, we recover Theorems 1.2 and 1.3 immediately.

Another main result of this paper is Diophantine approximation by algebraic numbers. The classical theory of Diophantine approximation of reals by rationals has the geometric interpretation of approximating elements of the boundary of the hyperbolic plane by the orbit of infinity under the modular group. For example, the celebrated Roth’s Theorem [13] can be stated as:

Theorem 1.10** (Roth).**

Let $\epsilon$ be a positive number and let $\xi$ be an irrational real number. If there exist infinitely many

[TABLE]

such that

[TABLE]

then $\xi$ is transcendental.

In this paper we will show that an analogy to Roth’s theorem holds when the point of infinity is replaced by an irrational real algebraic number. Let $PSL(2,\mathbb{Z})$ be the projective linear group of $2\times 2$ matrices with integer coefficients and unit determinant. For any

[TABLE]

Set $\|A\|=\max(|c|,|d|)$. For an irrational real number $\alpha$, an element

[TABLE]

acts on $\alpha$ by

[TABLE]

Let $\Theta_{\alpha}=PSL(2,\mathbb{Z})\alpha$ be the orbit of $\alpha$ for the action of $PSL(2,\mathbb{Z})$. When $\alpha$ is of degree higher than 2, for any $\beta\in\Theta_{\alpha}$, set

[TABLE]

where $A$ is the unique element of $PSL(2,\mathbb{Z})$ such that $\beta=A\alpha$.

Remark 1.11**.**

The definition of the norm $\|\cdot\|$ above depends upon $\alpha$. But it is easy to see that when $\alpha\neq\alpha^{\prime}\in\Theta_{\alpha}$, there exist two positive constants $c_{1}$ and $c_{2}$ such that the corresponding norms $\|\cdot\|_{\alpha}$ and $\|\cdot\|_{\alpha^{\prime}}$ satisfy

[TABLE]

Theorem 1.12**.**

Let $\alpha$ be an irrational real algebraic number of degree $d>2$. Let $\epsilon$ be a positive number and let $\xi$ be an irrational real number not in $\Theta_{\alpha}$. If there exist infinitely many $\beta\in\Theta_{\alpha}$ such

[TABLE]

then $\xi$ is transcendental.

Similar result holds for quadratic real number. Let $\alpha$ be a fixed quadratic irrational real number, and let $x^{\sigma}$ be the conjugate of any quadratic number $x$. For any $\beta=\frac{a\alpha+b}{c\alpha+d}\in\Theta_{\alpha}$, set

[TABLE]

The proof of Theorem 1.12 also implies:

Theorem 1.13**.**

Let $\epsilon$ be a positive number and let $\xi$ be an irrational number not in $\Theta_{\alpha}$. If there exist infinitely many $\beta\in\Theta_{\alpha}$ such

[TABLE]

then $\xi$ is transcendental.

This is essentially Theorem 4.4 from [8].

This paper is structured as follows: In Section 2, we give preliminaries that will be used throughout this paper. In Sections 3 and 4, we give proofs of Theorems 1.6 and 1.7. In Section 5, we give proofs of Theorems 1.12 and 1.13.

2. preliminaries

In this paper, we write

[TABLE]

for the finite continued fraction expansion

[TABLE]

and write

[TABLE]

for the infinite continued fraction expansion

[TABLE]

where $a_{1},a_{2},\cdots,$ are positive integers and $a_{0}$ is an integer. An eventually periodic continued fraction is written as

[TABLE]

where $a_{0};a_{1},a_{2},\cdots,a_{k-1}$ is the preperiod and $a_{k},\cdots,a_{k+m-1}$ is the shortest period.

The sequence of convergents of

[TABLE]

is defined by

[TABLE]

[TABLE]

We have (cf.[11])

Lemma 2.1**.**

[TABLE]

[TABLE]

and

[TABLE]

for $m,n\geq 1.$

For any finite nonempty word of integers $B=b_{0}b_{1}b_{2}\cdots b_{n}$, set

[TABLE]

Then it is well-known that (cf.[10])

[TABLE]

Hence

[TABLE]

When $a_{0}=0$, by taking the conjugation, we get

[TABLE]

Hence

[TABLE]

A simple proof by induction shows that:

Lemma 2.2**.**

For two finite words of positive integers

[TABLE]

and

[TABLE]

[TABLE]

implies

[TABLE]

Another well-known result about continued fractions is that:

Lemma 2.3**.**

Let $\alpha$ and $\beta$ be two irrational real numbers. If there exists an $A\in PSL(2,\mathbb{Z})$ such that $\beta=A\alpha$. , then the continued fraction expansions of $\alpha$ and $\beta$ have the same tail.

As in [1, 4, 5, 6, 7] the proofs of Theorems 1.6 and 1.7 need the Schmidt subspace theorem [14].

Theorem 2.4**.**

Let $n\geq 1$ be an integer. For every

[TABLE]

set

[TABLE]

Let $L_{0}(\textbf{x}),\cdots,L_{n}(\textbf{x})$ be linearly independent linear forms in $n+1$ variables with algebraic coefficients. Then for any positive number $\epsilon$, the solutions $\textbf{x}\in\mathbb{Z}^{n+1}$ of the inequality

[TABLE]

lie in finitely many proper linear subspaces of $\mathbb{Q}^{n+1}$

We also need the following lemma which is contained in the proof of Theorem 3.1 from [7]. For the readers’ convenience, we give here a different proof.

Lemma 2.5**.**

Let $\alpha$ be an real algebraic number of degree higher than 2, and let $\{p_{n}/q_{n}\}_{n\geq 0}$ be the sequence of convergents of the continued fraction expansion of $\alpha$. Then there do not exist a nonzero element $(x_{1},x_{2},x_{3},x_{4})\in\mathbb{Q}^{4}$, and an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that

[TABLE]

for each $n\in\mathbb{N}^{\prime}$.

Proof.

Assume that there exist a nonzero element $(x_{1},x_{2},x_{3},x_{4})\in\mathbb{Q}^{4}$, and an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that

[TABLE]

for each $n\in\mathbb{N}^{\prime}$. First, we have $(x_{1},x_{2})\neq(0,0)$, otherwise dividing (3) by $q_{n}$ and letting $n$ tend to infinity along $\mathbb{N}^{\prime}$ would implies that $\alpha$ is rational. Similarly, $(x_{3},x_{4})\neq(0,0)$. Now we assume that $x_{1}\neq 0$ and define three linearly independent linear forms

[TABLE]

[TABLE]

By (3) and Lemma 2.1, we have

[TABLE]

for each $n\in\mathbb{N}^{\prime}$. It follows from Theorem 2.4 that there exist a nonzero element $(y_{1},y_{2},y_{3})\in\mathbb{Q}^{3}$, and an infinite subset $\mathbb{N}^{\prime\prime}$ of $\mathbb{N}^{\prime}$ such that

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime}$. We observe as before that $y_{1}\neq 0$. Now combining (3) and (4) implies that there exists a nonzero element $(a,b,c,d)\in\mathbb{Q}^{4}$ such that

[TABLE]

and

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime}$. Letting $n$ tend to infinity along $\mathbb{N}^{\prime\prime}$, we get

[TABLE]

As $\alpha$ is an algebraic number of degree higher than 2, (5) forces

[TABLE]

and then

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime}$. This contradicts (1). The case $x_{2}\neq 0$ can be treated similarly. ∎

3. Proof of Theorem 1.6

Proof of Theorem 1.6.

Assume that (i), (ii) and (iii) of Condition 1.4 are satisfied with three sequences of finite nonempty words $\{A_{n}\}_{n\geq 1}$, $\{A^{\prime}_{n}\}_{n\geq 1}$, $\{B_{n}\}_{n\geq 1}$. If $A_{n}=Ca$ and $A^{\prime}_{n}=C^{\prime}a$ for some $n$ and some positive integer $a$, we can replace $A_{n},A^{\prime}_{n},B_{n}$ with $C,C^{\prime},aB_{n}$ without violating (i), (ii) and (iii) of Condition 1.4. Hence we can further require that

Convention 3.1**.**

the last letter of $A_{n}$ and the last letter of $A^{\prime}_{n}$ are different if $\min(|A_{n}|,|A^{\prime}_{n}|)\geq 3$.

Set $k_{n}=|A_{n}|$, $l_{n}=|A^{\prime}_{n}|$, and $m_{n}=|B_{n}|$. Then we have

[TABLE]

and

[TABLE]

The above two identities immediately imply

[TABLE]

We define four linearly independent linear forms as follows:

[TABLE]

Set

[TABLE]

It follows from Lemma 2.1 and (3) that

[TABLE]

Set

[TABLE]

Then by Lemma 2.1 and (iii) of Condition 1.4

[TABLE]

where $\delta=\tfrac{\log 2}{2L\log M}$. Here and throughout, the constants implied in $\ll$ depend only on $\alpha$ and $\alpha^{\prime}$.

Now applying Theorem 2.4 implies that there exist a nonzero element $(x_{1},x_{2},x_{3},x_{4})\in\mathbb{Q}^{4}$, and an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that

[TABLE]

for each $n\in\mathbb{N}^{\prime}$.

The rest proof is divided into 3 cases.

Case 1: there exists an infinite subset $\mathbb{N}^{\prime\prime}$ of $\mathbb{N}^{\prime}$ such that both the sequences $\{k_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$, $\{l_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ are bounded. Then $a=a_{1}a_{2}\cdots$ and $\textrm{a}^{\prime}=a^{\prime}_{1}a^{\prime}_{2}\cdots$ have the same tail.

Case 2: there exists an infinite subset $\mathbb{N}^{\prime\prime}$ of $\mathbb{N}^{\prime}$ such that only one of the sequences $\{k_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$, $\{l_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ is bounded. Without loss of generality, we assume that $\{k_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ is bounded and $\{l_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ is unbounded. Then there exists an infinite subset $\mathbb{N}^{\prime\prime\prime}$ of $\mathbb{N}^{\prime\prime}$ and a positive integer $k$ such that $k_{n}=k$ for each $n\in\mathbb{N}^{\prime\prime\prime}$ and $\{l_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ tends to infinity. If $\alpha^{\prime}$ is a quadratic irrational number, then the continued fraction expansion

[TABLE]

is eventually periodic, and we can replace $A^{\prime}_{n}$ with a prefix of bounded length without violating (i),(ii) and (iii) of Condition 1.4, and the proof can be reduced to Case 1. If $\alpha^{\prime}$ is an algebraic number of degree higher than 2, then by (3) we have

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime\prime}$. This contradicts Lemma 2.5 since the matrix

[TABLE]

is nonsingular and hence

[TABLE]

Case 3: there exists an infinite subset $\mathbb{N}^{\prime\prime}$ of $\mathbb{N}^{\prime}$ such that both the sequences $\{k_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$, $\{l_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ are strictly increasing. If at least one of $\alpha$ and $\alpha^{\prime}$ is a quadratic irrational number, then the proof can be reduced to Case 2 as before. Hence we assume that both $\alpha$ and $\alpha^{\prime}$ are algebraic numbers of degree higher than 2.

Set

[TABLE]

By Lemma 1, we have

[TABLE]

Set $M_{n}=\max(|a_{n}|,|b_{n}|,|c_{n}|,|d_{n}|).$

Claim 3.2**.**

[TABLE]

Proof.

If the Claim is invalid, we can choose an infinite subset $\mathbb{N}^{\prime\prime\prime}$ of $\mathbb{N}^{\prime\prime}$ such that

[TABLE]

is a constant matrix, when $n\in\mathbb{N}^{\prime\prime\prime}$. By (10), this means that

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime\prime}$. Applying Lemma 2.2, we get

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime\prime}$. But this contradicts Convention 3.1 ∎

Now we assume that $x_{1}\neq 0$, the other three cases can be reduced to the case $x_{1}\neq 0$ by replacing $\alpha$ with $1/\alpha$ and/or replacing $\alpha^{\prime}$ with $1/\alpha^{\prime}$. We can further assume that $x_{1}=-1$ without loss of generality.

Hence by (3) we have

[TABLE]

for $n\in\mathbb{N}^{\prime\prime}$.

From now on we assume that there exists an infinite subset $\mathbb{N}^{\prime\prime\prime}$ of $\mathbb{N}^{\prime\prime}$ such that

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime\prime}$; the other case can be treated similarly.

Now we define three linearly independent linear forms as follows:

[TABLE]

(If $q_{k_{n}}\geq q^{\prime}_{l_{n}}$ for infinitely many $n$, we can set $L^{\prime}_{2}(Y_{1},Y_{2},Y_{3})=\alpha^{\prime}Y_{2}-Y_{3}.$) Then by (3) and (13), we have

[TABLE]

for $n\in\mathbb{N}^{\prime\prime\prime}$.

Now applying Theorem 2.4 implies that there exist a nonzero element $(y_{1},y_{2},y_{3})\in\mathbb{Q}^{3}$, and an infinite subset $\mathbb{N}^{(4)}$ of $\mathbb{N^{\prime\prime\prime}}$ such that

[TABLE]

for each $n\in\mathbb{N}^{(4)}$.

Claim 3.3**.**

[TABLE]

and $d_{n}\neq 0$ if $n\in\mathbb{N}^{(4)}$ is sufficiently large.

Proof.

If $y_{1}=y_{2}=0$, we have $d_{n}=0$ for $n\in\mathbb{N}^{\prime\prime\prime}$. This force

[TABLE]

for $n\in\mathbb{N}^{(4)}$. Combining Claim 3.2, (12) and (16) immediately yields a contradiction. ∎

We assume that $y_{1}\neq 0$; the case $y_{2}\neq 0$ can be treated similarly. We can further assume that $y_{1}=-1$ without loss of generality. Hence

[TABLE]

for each $n\in\mathbb{N}^{(4)}$. Set

[TABLE]

First we point out that $L^{\prime\prime}_{1}(Z_{1},Z_{2})\neq 0$, otherwise, we would have

[TABLE]

which contradicts the fact that $\alpha$ is of degree higher than 2.

By (3) we have

[TABLE]

Now we get as above that there exist a $z\in\mathbb{Q}$, and an infinite subset $\mathbb{N}^{(5)}$ of $\mathbb{N}^{(4)}$ such that

[TABLE]

for each $n\in\mathbb{N}^{(5)}$. Combining (12), (17) and (18) implies that

[TABLE]

for each $n\in\mathbb{N}^{(5)}$, where $D$ is a constant matrix. By Claim 3.2, $|d_{n}|$ tends to infinity along $\mathbb{N}^{(5)}$. This contradicts (11) and finishes the proof of the first assertion of Theorem 1.6. Now assume that

[TABLE]

If one of $\alpha$ and $\alpha^{\prime}$ is of degree higher than 2, then applying the arguments in cases 2 and 3, we get a contradiction. ∎

4. Proof of Theorem 1.7

As the proof is similar to that of Theorem 1.6, many details are omitted.

Proof of Theorem 1.7.

Assume that (i’), (ii) and (iii) of Condition 1.4 is satisfied with three sequences of finite nonempty words $\{A_{n}\}_{n\geq 1}$, $\{A^{\prime}_{n}\}_{n\geq 1}$, $\{B_{n}\}_{n\geq 1}$. Set $k_{n}=|A_{n}|$, $l_{n}=|A^{\prime}_{n}|$, and $m_{n}=|B_{n}|$. Then we have

[TABLE]

and

[TABLE]

The above two identities immediately implies

[TABLE]

Evaluating the linear forms

[TABLE]

on the quadruple

[TABLE]

we get

[TABLE]

By (4) we have

[TABLE]

Now applying Theorem 2.4 as before implies that there exist a nonzero element $(x_{1},x_{2},x_{3},x_{4})\in\mathbb{Q}^{4}$, and an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that

[TABLE]

for each $n\in\mathbb{N}^{\prime}$.

At this point the proof is divided into 3 cases as before.

Case 1: there exists an infinite subset $\mathbb{N}^{\prime\prime}$ of $\mathbb{N}^{\prime}$ such that both the sequences $\{k_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$, $\{l_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ are bounded. Without loss of generality, we assume that $k_{n}=l_{n}=0$ for each $n\in\mathbb{N}^{\prime\prime}$. Then by (4) and (20), we have

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime}$.

By Lemma 2.5 and (4), both $\alpha$ and $\alpha^{\prime}$ are quadratic irrationals. Assume that the shortest periods of the continued fraction expansions of $\alpha$ and $\alpha^{\prime}$ are $A$ and $B$ respectively. Then (i’) of Condition 1.4 implies that $A$ and $B$ are cycle mirror symmetry.

Case 2: there exists an infinite subset $\mathbb{N}^{\prime\prime}$ of $\mathbb{N}^{\prime}$ such that only one of the sequences $\{k_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$, $\{l_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ is bounded. Without loss of generality, we assume that $\{k_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ is bounded. Then there exists an infinite subset $\mathbb{N}^{\prime\prime\prime}$ of $\mathbb{N}^{\prime\prime}$ and a nonnegative integer $k$ such that $k_{n}=k$ for each $n\in\mathbb{N}^{\prime\prime\prime}$. If $\alpha^{\prime}$ is a quadratic irrational number, then the continued fraction expansion

[TABLE]

is eventually periodic, and we can replace $A^{\prime}_{n}$ by a prefix of bounded length without violating (i’), (ii) and (iii) of Condition 1.4, and the proof can be reduced to Case 1. If $\alpha^{\prime}$ is an algebraic number of degree higher than 2, then by (20) we have

[TABLE]

for each $n\in\mathbb{N}^{\prime\prime\prime}$. This contradicts Lemma 2.5.

Case 3: there exists an infinite subset $\mathbb{N}^{\prime\prime}$ of $\mathbb{N}^{\prime}$ such that both the sequences $\{k_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$, $\{l_{n}\}_{n\in\mathbb{N}^{\prime\prime}}$ are strictly increasing. If at least one of $\alpha$ and $\alpha^{\prime}$ is a quadratic irrational number, then the proof can be reduced to Case 2. Hence we assume that both $\alpha$ and $\alpha^{\prime}$ are algebraic numbers of degree higher than 2. Without loss of generality we can further assume that $\alpha,\alpha^{\prime}\in(0,1)$. Hence we have $p_{n},p^{\prime}_{n}>0$ for each $n$. Set

[TABLE]

Then it is easy to see that

[TABLE]

Now the rest proof proceeds in the same way as in Case 3 of the proof of Theorem 1.6. ∎

5. proofs of Theorems

1.12 and 1.13

This section is devoted to the proofs of Theorems 1.12 and 1.13.

First we need two auxiliary lemmas, the first of which follows directly from [8, Lemma 2.2] and its proof.

Lemma 5.1**.**

Let

[TABLE]

and

[TABLE]

be the continued fraction expansions of two real numbers, and let $\{\frac{p_{n}}{q_{n}}\}_{n\geq 0}$ the sequence of convergents of $\beta$. Let $n$ be a nonnegative integer such that $a_{i}=b_{i}$ for $i=1,\cdots,n-1$, and $a_{n}\neq b_{n}$. Then we have

[TABLE]

Lemma 5.2**.**

Let

[TABLE]

be the continued fractional expansion of an irrational real algebraic number, and let $\{p_{n}/q_{n}\}_{n\geq 0}$ be the sequence of convergents. Let $k$ be a positive integer and let $\epsilon$ be a positive number. Then there can not exist an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that

[TABLE]

for each $n\in\mathbb{N}^{\prime}$.

Proof.

Assume that there exists an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that

[TABLE]

for each $n\in\mathbb{N}^{\prime}$. Set $1+\varepsilon=\sqrt[k]{1+\epsilon}$. Then for any $n\in\mathbb{N}^{\prime}$, there exists a $0\leq i_{n}<k$ such that

[TABLE]

Now we have

[TABLE]

for each $n\in\mathbb{N}^{\prime}$. This contradicts Roth’s theorem [13]. ∎

Proof of Theorem 1.12.

The proof is divided into several claims. Assume that there exist a sequence $\{\beta_{n}\}_{n\geq 0}$ of distinct elements from $\Theta_{\alpha}$ such that

[TABLE]

Claim 5.3**.**

[TABLE]

and

[TABLE]

Proof.

Otherwise, we can choose an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that

[TABLE]

and the determinent

[TABLE]

is fixed when $n\in\mathbb{N}^{\prime}$ where $c$ and $d$ are constants. For distinct $\beta_{n_{1}}$ and $\beta_{n_{2}}$, we have

[TABLE]

We note that $c$ and $d$ are co-prime and

[TABLE]

Hence

[TABLE]

is a nonzero integer and $|\beta_{n_{1}}-\beta_{n_{2}}|\geq 1$. This contradicts the fact that

[TABLE]

and that $\{\beta_{n}\}_{n\geq 0}$ consists of distinct elements. ∎

From now on, we assume without loss of generality that $\alpha,\beta_{n}$ and $\xi$ all lie in the interval $(0,1)$. Now by Lemma 2.3, we can assume that

[TABLE]

and

[TABLE]

where $k_{n},l_{n}\geq 0$ and $a^{(n)}_{l_{n}}\neq a_{k_{n}}$.

Let $\{p_{k}/q_{k}\}_{k\geq 1}$ and $\{p^{(n)}_{k}/q^{(n)}_{k}\}_{k\geq 1}$ be respectively the sequence of convergents of $\alpha$ and $\beta_{n}$. Then we have

[TABLE]

Claim 5.4**.**

[TABLE]

Proof.

Otherwise there would exist an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ and two constants $k$ and $l$ such that $l_{n}=l$ and $k_{n}=k$ for each $n\in\mathbb{N}^{\prime}$. As

[TABLE]

$a^{(n)}_{1},a^{(n)}_{2},\cdots,a^{(n)}_{l}$ will be fixed when $n\in\mathbb{N}^{\prime}$ is sufficiently large. This implies $\xi\in\Theta_{\alpha}$ which contradicts our assumption. ∎

Let $\epsilon_{1}$ be another positive number.

Claim 5.5**.**

There exists an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that $\|\beta_{n}\|\geq(\overline{q^{(n)}_{l_{n}-1}}q_{k_{n}})^{1-\epsilon_{1}}$ for each $n\in\mathbb{N}^{\prime}$, where $\overline{x}=\max(1,x).$

Proof.

The proof is divided into three cases.

Case 1, there exists an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that $k_{n}=k$ is a constant for $n\in\mathbb{N}^{\prime}$. Then by (25) there exist a positive constant $M$ such that

[TABLE]

when $n\in\mathbb{N}^{\prime}$ is sufficiently large.

Case 2, there exists an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that $l_{n}=0$ for each $n\in\mathbb{N}^{\prime}$. Then by (25) we have

[TABLE]

when $n\in\mathbb{N}^{\prime}$. On the other hand, by Lemma 5.2, we have

[TABLE]

when $n\in\mathbb{N}^{\prime}$ is sufficiently large.

Case 3, there exists an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that the sequence $\{k_{n}\}_{n\in\mathbb{N}^{\prime}}$ is strictly increasing and $l_{n}>0$ for each $n\in\mathbb{N}^{\prime}$. Then by (25), it suffices to show that

[TABLE]

By (2), $\frac{q^{(n)}_{l_{n}}q_{k_{n}-1}}{q^{(n)}_{l_{n}-1}q_{k_{n}}}$ is the quotient of the two continued fractions $\overline{\beta_{n}}=[a^{(n)}_{l_{n}};a^{(n)}_{l_{n}-1},\cdots,a^{(n)}_{1}]$ and $\overline{\alpha}=[a_{k_{n}};a_{k_{n}-1},\cdots,a_{1}]$. By Lemma 5.2, we have

[TABLE]

when $n\in\mathbb{N}^{\prime\prime}$ is sufficiently large. Now it follows from Lemma 5.1 and (26) that

[TABLE]

when $n\in\mathbb{N}^{\prime\prime}$ is sufficiently large. ∎

Fix an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ satisfying Claim 5.5. Let

[TABLE]

and let $\{p^{\prime}_{k}/q^{\prime}_{k}\}_{k\geq 1}$ be the sequence of convergents.

Claim 5.6**.**

When $n\in\mathbb{N}^{\prime}$ is sufficiently large, we have $b_{s}=a^{(n)}_{s}$ for $1\leq s\leq l_{n}$. Moreover, when $n\in\mathbb{N}^{\prime}$ is sufficiently large, let $m_{n}$ be the nonnegative integer such that $b_{l_{n}+s}=a_{k_{n}+s}$ for $0\leq s\leq m_{n}$ and $b_{l_{n}+m_{n}+1}\neq a_{k_{n}+m_{n}+1}$. Then

[TABLE]

Proof.

Assume that there exists a positive integer $t_{n}<l_{n}$ such that $b_{s}=a^{(n)}_{s}$ for $1\leq s<t_{n}$ and $b_{t_{n}}\neq a^{(n)}_{t_{n}}$. Then by Claim 5.3 we have

[TABLE]

Now by Lemma 5.1 and Claim 5.5,

[TABLE]

when $n\in\mathbb{N}^{\prime}$. We can choose $\epsilon_{1}$ such that $(-2-\epsilon)(1-\epsilon_{1})<-2$. But then, by Lemma 5.2, (28) is impossible when $n$ is sufficiently large. The same proof shows that

[TABLE]

∎

From now on, we always assume that $n$ lies in $\mathbb{N}^{\prime}$ and is sufficiently large.

Claim 5.7**.**

There exist two positive numbers $\delta$ and $L$ such that

[TABLE]

Proof.

Set

[TABLE]

Then we have

[TABLE]

[TABLE]

and

[TABLE]

Hence

[TABLE]

By Lemma 5.1 and Claim 5.5, we have

[TABLE]

Lemma 5.2 implies that for any small positive integer $\epsilon_{2}$, there exists a positive number $M$ such that

[TABLE]

and

[TABLE]

Combining (29), (30), (31) and (32) and choosing $\epsilon_{1}$ and $\epsilon_{2}$ small enough imply the Claim. ∎

We are now in the position to prove Theorem 1.12. Set

[TABLE]

[TABLE]

and

[TABLE]

Then by Claims 5.6 and 5.7, the three sequences $\{A_{n}\}_{n\geq 1}$, $\{A^{\prime}_{n}\}_{n\geq 1}$, $\{B_{n}\}_{n\geq 1}$ satisfy Condition 1.9, and (i) and (ii) of Condition 1.4. Now applying Theorem 1.7 and Remark 1.8 implies that $\xi$ is transcendental and finishes the proof. ∎

Proof of Theorem 1.13.

Assume that there exist a sequence $\{\beta_{n}\}_{n\geq 0}$ of distinct elements from $\Theta_{\alpha}$ such that

[TABLE]

Then the above proof and notations can be directly applied. In the quadratic case, by replacing $\{\beta_{n}\}_{n\geq 0}$ with a subsequence, we can assume that $k_{n}=0$,

[TABLE]

and

[TABLE]

where $a^{(n)}_{l_{n}}\neq a_{k}$. Hence

[TABLE]

and

[TABLE]

It is well-known that the Galois conjugate of $\alpha$ is

[TABLE]

Hence by Lemma 5.1, we have

[TABLE]

Hence there exists a positive constant $M$ (depends on $\alpha$) such that

[TABLE]

Now the rest proof proceeds exactly as before. ∎

We close this paper with a question. In [3], Theorem 1.5 was generalized to the case of several irrational real algebraic numbers. What can we say about the continued fraction expansions of several irrational real algebraic numbers?