$b$-ary expansions of algebraic numbers

TL;DR

This paper generalizes previous results on the $b$-ary expansions of algebraic numbers, introduces new transcendence criteria, and explores the independence of expansions of algebraic numbers, leading to a generalized Borel conjecture.

Contribution

It extends earlier work on $b$-ary expansions, providing new transcendence criteria and insights into the independence of algebraic numbers' expansions.

Findings

New transcendence criteria for algebraic numbers

Linearly independent algebraic numbers have quite independent $b$-ary expansions

Proposal of a generalized Borel conjecture

Abstract

In this paper we give a generalization of the main results in \cite{ab,ab1} about $b$-ary expansions of algebraic numbers. As a byproduct we get a large class of new transcendence criteria. One of our corollaries implies that $b$-ary expansions of linearly independent irrational algebraic numbers are quite independent. Motivated by this result, we propose a generalized Borel conjecture.

Equations

$$\omega =[\omega ]+\sum _{i\ge 1}^{\infty }{a}_i{b}^{-i}=[\omega ]+0.a_1{a}_2\cdots ,$$

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

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

$$\underset{n\to +\infty }{\lim }p(\omega ,n)-n=+\infty .$$

$$\text{a}=a_1{a}_2{a}_3\cdots$$

$${\text{a}}^{\prime }=a_1^{\prime }{a}_2^{\prime }{a}_3^{\prime }\cdots$$

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

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

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

$$\text{a}=a_1{a}_2{a}_3\cdots$$

$${\text{a}}^{\prime }=a_1^{\prime }{a}_2^{\prime }{a}_3^{\prime }\cdots$$

$$a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n=0,$$

$$\alpha =0.a_1{a}_2{a}_3\cdots$$

$$\text{a}=a_1{a}_2{a}_3\cdots .$$

$$\lfloor \{b_{i,j}\}\rfloor =\underset{i,j}{\min }(b_{i,j}),$$

$$⟨\{b_{i,j}\}⟩=\underset{\genfrac{}{}{0ex}{}{1\le i\le n}{1\le j\le a_i-1}}{\min }(b_{i,j+1}-b_{i,j}),$$

$$⟨\overline{\{b_{i,j}\}}⟩=\underset{(i,j)\ne (i^{\prime },j^{\prime })}{\min }(|b_{i,j}-b_{i^{\prime },j^{\prime }}|).$$

$$f(D)=\sum _{i=1}^n{\alpha }_i{b}^{d_{i,1}}.$$

$$G.C.D([f(D)+P(D)],b^{\lfloor D\rfloor })<b^{ϵ\lfloor D\rfloor }.$$

$$1,{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n$$

$$f(D)=\sum _{i=1}^n\sum _{j=1}^{a_i}{\alpha }_i{c}_{i,j}{b}^{d_{i,j}}.$$

$$G.C.D([f(D)+P(D)],b^{\lfloor D\rfloor })<b^{ϵ\lfloor D\rfloor }.$$

$$f(D)=\sum _{i=1}^n{c}_i[\alpha b^{d_i}].$$

$$G.C.D([f(D)],b^{\lfloor D\rfloor })<b^{ϵ\lfloor D\rfloor }.$$

$$f(D)=\sum _{i=1}^n[{\alpha }_i{b}^{d_i}].$$

$$G.C.D(f(D),b^{\lfloor D\rfloor })<b^{ϵ\lfloor D\rfloor }.$$

$$\overline{s_m\cdots s_1{s}_0}=\sum _{i=0}^m{s}_i{b}^i.$$

$$\{A_i^1{\}}_{i\ge 1},\cdots ,\{A_i^m{\}}_{i\ge 1},\{B_i^1{\}}_{i\ge 1},\cdots ,\{B_i^m{\}}_{i\ge 1}$$

$$|B_i^1|=\cdots =|B_i^m|=k_i,$$

$$|A_i^1|<\cdots <|A_i^m|,$$

Full text

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

*  *Fuzhou, 350108, China;

  Email: linxianzu@126.com

Keywords: $b$-ary expansions, subspace theorem, irrational algebraic number.

Mathematics Subject Classification 2010: 11A63, 11K16.

1. Introduction

Let $b\geq 2$ be a fixed integer. Any real number $\omega$ has a unique $b$-ary expansion:

[TABLE]

where $a_{i}\in\{0,1,\cdots,b-1\}$ and the set $\{i\mid a_{i}\neq b-1\}$ is infinite. $\omega$ is called $normal$ to base $b$ if, for every $k\geq 1$, every block of $k$ digits from $\{0,1,\cdots,b-1\}$ occurs in the $b$-ary expansion of $\omega$ with frequency $1/b^{k}$.

A classical theorem of Borel [3] says that almost all real numbers are normal to base $b$. In [4] Borel made the conjecture that all irrational algebraic numbers are normal to base $b$. It seems that this conjecture is far from the reach of modern mathematics. Let $p(\omega,n)$ be the number of distinct blocks of length $n$ occurring in the $b$-ary expansion of $\omega$. It follows from Borel’s conjecture that $p(\omega,n)=b^{n}$ for any irrational algebraic numbers $\omega$. But even this corollary is too difficulty. A large breakthrough in this direction is due to Adamczewski and Bugeaud [1]. Before introducing their results, we need some preparations.

We say that an infinite word $\textbf{a}=a_{1}a_{2}\cdots$ of elements from $\{0,1,\cdots,b-1\}$ has $long\ repetition$ if the following condition is satisfied, where the length of a finite word $A$ is denoted by $|A|$.

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. $($ii$)$

the sequence $\{|B_{n}|\}_{n\geq 1}$ is strictly increasing; 3. $($iii$)$

there exists a positive constant $L$ such that

[TABLE]

for every $n\geq 1$.

One of the main results in [1] is the following:

Theorem 1.2**.**

The $b$-ary expansion of an irrational algebraic number has no long repetitions.

It follows directly from this theorem that

[TABLE]

where $\omega$ is an irrational algebraic numbers. This result, though far from the conjecture that $p(\omega,n)=b^{n}$, is indeed a great advance comparing with the previous result [6] that

[TABLE]

In [2], Adamczewski and Bugeaud further explored the independence of $b$-ary expansions of two irrational algebraic numbers $\alpha$ and $\beta$.

Let

[TABLE]

and

[TABLE]

be two infinite words of elements from $\{0,1,\cdots,b-1\}$. The following is a condition about the pair $(\textbf{a},\textbf{a}^{\prime})$:

Condition 1.3**.**

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 a and the word $A^{\prime}_{n}B_{n}$ is a prefix of $\textbf{a}^{\prime}$; 2. $($ii$)$

the sequence $\{|B_{n}|\}_{n\geq 1}$ is strictly increasing; 3. $($iii$)$

there exists a positive constant $L$ such that

[TABLE]

for every $n\geq 1$.

The main result in [2] is:

Theorem 1.4**.**

Let $\alpha$ and $\alpha^{\prime}$ be two irrational algebraic numbers. If their $b$-ary expansions

[TABLE]

and

[TABLE]

satisfy Condition 1.3, then the two infinite words

[TABLE]

and

[TABLE]

have the same tail.

In this paper, we show that for a fix irrational algebraic number $\alpha$, and a fix nontrivial linear equation

[TABLE]

where $a_{1},a_{2},\cdots,a_{n}\in\mathbb{Z}$, the $b$-ary expansion of $\alpha$ can not have $n$ disjoint long sub-words which are correlated by the above linear equation. When the linear equation is $x_{1}-x_{2}=0$, we recover Theorem 1.2. Similar result holds for several algebraic numbers $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ such that $1,\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ are linearly independent over $\mathbb{Q}$. Applying this result to a pair of algebraic numbers quickly implies Theorem 1.4. Our method is different from that of [1, 2]. In particular, we do not use rational approximations to algebraic numbers. Instead, we deduce all things from a single theorem about greatest common divisor of a big sum and a pow of $b$ (Theorem 2.1).

This paper is structured as follows: In Section 2, we state the main results of this paper after some preparations. In Section 3, we supply all the proofs. In Sections 4, we propose a generalized Borel conjecture and some other questions.

2. Main results

Throughout this paper, let $b\geq 2$ be a fixed integer. All the irrational algebraic numbers we consider lie in the interval $(0,1)$. All the words considered in this paper mean words of elements from $\{0,1,\cdots,b-1\}$. For an irrational algebraic number $\alpha$, we always identify the $b$-ary expansions

[TABLE]

with the infinite word

[TABLE]

The length of a finite word $A$ is denoted by $|A|$. For any real number $a$, $[a]$ and $\{a\}$ denote respectively the integer part and the fractional part of $a$. Finally, for two integers $a$ and $b$, denote the greatest common divisor of $a$ and $b$ by $G.C.D(a,b)$.

Before giving the main result, we introduce some new definitions and notations.

Given $n$ positive integers $a_{1},a_{2},\cdots,a_{n}$, an array of nonzero rational numbers $\{b_{i,j}\}$, where $1\leq i\leq n$, $1\leq j\leq a_{i}$, will be called an $(a_{1},a_{2},\cdots,a_{n})$-$array$. For each $(a_{1},a_{2},\cdots,a_{n})$-array $\{b_{i,j}\}$, Set

[TABLE]

[TABLE]

and set

[TABLE]

Let $L$ be a positive real number. An $(a_{1},a_{2},\cdots,a_{n})$-array $\{b_{i,j}\}$ is $L$-$admissible$ if the following conditions are satisfied :

1. $($i$)$

$b_{i,j}$ is a positive integer for $1\leq i\leq n$, $1\leq j\leq a_{i}$; 2. $($ii$)$

$b_{i,j}<b_{i,j+1}$ for $1\leq i\leq n$, $1\leq j\leq a_{i}-1$; 3. $($iii$)$

$\max_{i,j}(b_{i,j})\leq L\min_{i,j}(b_{i,j})$.

Now we are in the position to state a theorem from which all the results of this paper can be derived:

Theorem 2.1**.**

Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ be $n$ irrational algebraic numbers and let $\epsilon$, $L$, and $M$ be fix positive numbers. Let $P$ be an real valued function defined on the set of all $L$-admissible $(\underbrace{1,\cdots,1}_{n})$-arrays such that $|P(D)|<M$ for every $D$. For each $L$-admissible $(\underbrace{1,\cdots,1}_{n})$-array $D=\{d_{i,1}\}$, set

[TABLE]

Then if both $\lfloor D\rfloor$ and $\langle\overline{D}\rangle$ are sufficiently large, we have

[TABLE]

A direct consequence of Theorem 2.1 is the following:

Theorem 2.2**.**

Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ be $n$ algebraic numbers, such that

[TABLE]

are linearly independent over $\mathbb{Q}$. Let $a_{1},a_{2},\cdots,a_{n}$ be $n$ positive integers, let $C=\{c_{i,j}\}$ be a fixed $(a_{1},a_{2},\cdots,a_{n})$-$array$, and let $\epsilon$, $L$ and $M$ be fixed positive numbers. Let $P$ be an real valued function defined on the set of all $L$-admissible $(a_{1},a_{2},\cdots,a_{n})$-arrays such that $|P(D)|<M$ for every $D$. For each $L$-admissible $(a_{1},a_{2},\cdots,a_{n})$-array $D=\{d_{i,j}\}$, set

[TABLE]

Then when both $\lfloor D\rfloor$ and $\langle D\rangle$ are sufficiently large, we have

[TABLE]

The following two special cases of Theorem 2.2 is more convenient for applications.

Theorem 2.3**.**

Let $\alpha$ be an irrational algebraic number. Let $n$ a positive integer. Let $C=\{c_{i}\}$ be a fixed $(n)$-$array$, and let $\epsilon$ and $L$ be fixed positive numbers. For each $L$-admissible $(n)$-array $D=\{d_{i}\}$, set

[TABLE]

Then when both $\lfloor D\rfloor$ and $\langle D\rangle$ are sufficiently large, we have

[TABLE]

Theorem 2.4**.**

Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ be $n$ algebraic numbers, such that $1,\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ are linearly independent over $\mathbb{Q}$. Let $\epsilon$ and $L$ be fixed positive numbers. For each $L$-admissible $(\underbrace{1,\cdots,1}_{n})$-array $D=\{d_{i}\}$, set

[TABLE]

Then when $\lfloor D\rfloor$ is sufficiently large, we have

[TABLE]

We need some preliminaries before giving more specific corollaries of Theorem 2.2.

Let $a_{1},a_{2},\cdots a_{m}$ be $m$ fixed non-zero integers. Let a be an infinite word of elements from $\{0,1,\cdots,b-1\}$. For any finite non-empty word $s_{m}\cdots s_{1}s_{0}$, set

[TABLE]

The following is a condition about a and the numbers $a_{1},a_{2},\cdots a_{m}$:

Condition 2.5**.**

There exist $2m$ sequences of finite nonempty words

[TABLE]

such that:

1. $($i$)$

for each $i\geq 1,j\leq m$, the word $A^{j}_{i}B^{j}_{i}$ is a prefix of the word a ; 2. $($ii$)$

for each $i\geq 1$,

[TABLE]

and the sequence $\{k_{i}\}_{i\geq 1}$ is strictly increasing; 3. $($iii$)$

for each $i\geq 1$,

[TABLE]

and for each $j<m$ the sequence $\{|A^{j+1}_{i}|-|A^{j}_{i}|\}_{i\geq 1}$ is strictly increasing; 4. $($iv$)$

for each $i\geq 1$,

[TABLE] 5. $($v$)$

there exists a positive constant $L$ such that

[TABLE]

for $i\geq 1,j\leq m$.

Let $\textbf{a}^{1},\cdots,\textbf{a}^{m}$ be $m$ infinite words of elements from $\{0,1,\cdots,b-1\}$. The second condition is about $\textbf{a}^{1},\cdots,\textbf{a}^{m}$ and the numbers $a_{1},a_{2},\cdots a_{m}$:

Condition 2.6**.**

There exist $2m$ sequences of finite nonempty words

[TABLE]

such that:

1. $($i$)$

for each $i\geq 1,j\leq m$, the word $A^{j}_{i}B^{j}_{i}$ is a prefix of the word $\textbf{a}^{j}$ ; 2. $($ii$)$

for each $i\geq 1$,

[TABLE]

and the sequence $\{k_{i}\}_{i\geq 1}$ is strictly increasing; 3. $($iii$)$

for each $i\geq 1$,

[TABLE] 4. $($iv$)$

there exists a positive constant $L$ such that

[TABLE]

for $i\geq 1,j\leq m$.

Theorem 2.7**.**

Let $\alpha$ be an irrational algebraic number, and let $a_{1},a_{2},\cdots a_{m}$ be $m$ fixed non-zero integers. Then the $b$-ary expansion of $\alpha$ does not satisfy Condition 2.5.

When $m=2$, $a_{1}=-a_{2}=1$, the congruence

[TABLE]

in Condition 2.5 forces $B^{1}_{i}=B^{2}_{i}$. In this way we recover Theorem 1.2 immediately.

Theorem 2.7 and its variants immediately yield a large class of new transcendence criteria. The following are two simplest examples:

Let $\Phi$ be the set of finite words of length $\geq 2$ on the alphabet $\{0,1,\cdots,b-1\}$. Let $s$ be a nonzero integer. For any $\textbf{a}\in\Phi$ of length $k$, let $\textbf{a}^{1}$ and $\textbf{a}^{2}$ be the two sub-words of a such that $\textbf{a}^{1}\textbf{a}^{2}$ is the prefix of a and

[TABLE]

Let $\textbf{a}^{3}$ be the unique word of length $[k/2]$ in $\Phi$ satisfying

[TABLE]

and let $\textbf{a}^{4}$ be the unique word of length $k$ in $\Phi$ satisfying

[TABLE]

We define two operations $f$ and $g$ on $\Phi$ by

[TABLE]

and

[TABLE]

Let $F(\textbf{a})$ be the limit of $f^{n}(\textbf{a})$, and let $G(\textbf{a})$ be the limit of $g^{n}(\textbf{a})$ for $n\rightarrow+\infty$. Now Theorem 2.7 immediately implies the following two transcendence criteria.

Theorem 2.8**.**

For any $\textbf{a}\in\Phi$, set

[TABLE]

and

[TABLE]

Then neither

[TABLE]

nor

[TABLE]

can be irrational algebraic number.

Remark 2.9**.**

In fact, both

[TABLE]

and

[TABLE]

in the above theorem are transcendental except for some extreme cases.

Now we consider simultaneous $b$-ary expansions of several algebraic numbers.

Theorem 2.10**.**

Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$ be $m$ algebraic numbers, such that

[TABLE]

are linearly independent over $\mathbb{Q}$. Let $a_{1},a_{2},\cdots a_{m}$ be $m$ fixed non-zero integers. Then Condition 2.6 is not satisfied by the $b$-ary expansions of $\alpha$ $\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$.

When $m=2$, $a_{1}=-a_{2}=1$, Condition 2.6 reduces to Condition 1.3.

It can be derived directly from Theorem 2.2 that Theorems 2.7 and 2.10 have a common generalization of mixed type. It is a little tedious to write down the full result. A simplest case says that:

Theorem 2.11**.**

Let $a_{1},a_{2},a_{3}$ be three fixed non-zero integers. Let $\alpha$ and $\alpha^{\prime}$ be two algebraic numbers, such that $1,\alpha,\alpha^{\prime}$ are linearly independent over $\mathbb{Q}$. Then their $b$-ary expansions

[TABLE]

and

[TABLE]

can not satisfy Condition 2.12

Condition 2.12**.**

There exist six sequences of finite nonempty words

[TABLE]

such that:

1. $($i$)$

for each $i\geq 1$, both $A^{1}_{i}B^{1}_{i}$ and $A^{2}_{i}B^{2}_{i}$ are prefixes of $a_{1}a_{2}\cdots$ ; 2. $($ii$)$

for each $i\geq 1$, $A^{3}_{i}B^{3}_{i}$ is a prefix of $a^{\prime}_{1}a^{\prime}_{2}\cdots$ ; 3. $($iii$)$

for each $i\geq 1$,

[TABLE]

and the sequence $\{k_{i}\}_{i\geq 1}$ is strictly increasing; 4. $($iv$)$

for each $i\geq 1$,

[TABLE] 5. $($v$)$

for each $i\geq 1$, $|A^{1}_{i}|<|A^{2}_{i}|,$ and the sequence $\{|A^{2}_{i}|-|A^{1}_{i}|\}_{n\geq 1}$ is strictly increasing; 6. $($vi$)$

there exists a positive constant $L$ such that

[TABLE]

for $i\geq 1,j\leq 3$.

3. proofs of the main results

As in [1, 2], our proofs are dependent upon the following $p$-adic version of the Schmidt subspace theorem [8, 9]. Let $|\cdot|_{p}$ is the $p$-adic absolute value on $\mathbb{Q}$, normalized by $|p|_{p}=p^{-1}$. we pick an extension of $|\cdot|_{p}$ to $\overline{\mathbb{Q}}$.

Theorem 3.1**.**

Let $n\geq 1$ be an integer, and let $S$ be a finite set of places on $\mathbb{Q}$ containing the infinite place. For every

[TABLE]

set

[TABLE]

For every $p\in S$, let $L_{0,p}(\textbf{x}),\cdots,L_{n,p}(\textbf{x})$ 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 $\mathbb{Q}^{n+1}$

Proof of Theorem 2.1.

For the infinite place $\infty$ and for each prime $p\mid b$, we will introduce $n+1$ linearly independent linear forms as follows:

For every

[TABLE]

set

[TABLE]

and for each $1\leq i\leq n$, set

[TABLE]

For each prime $p\mid b$ and each $0\leq i\leq n$, set

[TABLE]

Assume that there exists an infinite sequence $\{D_{k}\}_{k\geq 1}=\{\{d_{i,1,k}\}_{i}\}_{k\geq 1}$ of $L$-admissible $(\underbrace{1,\cdots,1}_{n})$-arrays, such that both $\lfloor D_{k}\rfloor$ and $\langle\overline{D_{k}}\rangle$ tend to infinity, and

[TABLE]

where

[TABLE]

and

[TABLE]

Set

[TABLE]

Then we have

[TABLE]

Direct estimation shows that

[TABLE]

for $k$ sufficiently large. By product formula (cf.[7, p.99]),

[TABLE]

for $i=1,\cdots,n$. Hence

[TABLE]

for $k$ sufficiently large. Now by Theorem 3.1, there exist a nonzero element $(e_{0},e_{1},\cdots,e_{n})\in\mathbb{Z}^{n+1}$, and an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that

[TABLE]

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

As $\langle\overline{D_{k}}\rangle$ tends to infinity, we can choose an infinite subset $\mathbb{N}^{\prime\prime}$ of $\mathbb{N}^{\prime}$, and a permutation $\{s_{1},\cdots,s_{n}\}$ of $\{1,\cdots,n\}$, satisfying:

1. $($i$)$

for each $k\in\mathbb{N}^{\prime\prime}$,

[TABLE] 2. $($ii$)$

for each $i<n$,

[TABLE]

Now dividing Formula (1) by $b^{d_{s_{1},1,k}}$ and letting $k$ tend to infinity along $\mathbb{N}^{\prime\prime}$, we obtain

[TABLE]

Hence, by the irrationality of $\alpha_{1}$,

[TABLE]

Finally, dividing Formula (1) by $b^{d_{s_{2},1,k}},\cdots,b^{d_{s_{n},1,k}}$ in turn, we get

[TABLE]

This concludes the proof of the theorem. ∎

Remark 3.2**.**

The above proof uses only the irrationality of $\alpha_{s_{1}}$. Thus if we require all the $L$-admissible $(\underbrace{1,\cdots,1}_{n})$-arrays $\{d_{i,1}\}$ to satisfy

[TABLE]

then Theorem 2.1 still holds if we only assume the irrationality of $\alpha_{1}$.

Proof of Theorem 2.2.

Assume that there exists an infinite sequence

[TABLE]

of $L$-admissible $(a_{1},a_{2},\cdots,a_{n})$-arrays, such that both $\lfloor D_{k}\rfloor$ and $\langle D_{k}\rangle$ tend to infinity, and

[TABLE]

where we set

[TABLE]

We can choose an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$, a partition of the set

[TABLE]

and a fixed $(i_{s},j_{s})\in U_{s}$ for each $s$, such that for each $k\in\mathbb{N}^{\prime}$,

1. $($i$)$

$\min_{(i,j)\in U_{s}}(d_{i,j,k})=d_{i_{s},j_{s},k}$, $0<s\leq t$; 2. $($ii$)$

$d_{i,j,k}-d_{i_{s},j_{s},k}$ is independent of $k$, when $0<s\leq t$ and $(i,j)\in U_{s}$; 3. $($iii$)$

the sequence $\{\mid d_{i,j,k}-d_{i_{s},j_{s},k}\mid\}_{k\in\mathbb{N}^{\prime}}$ tends to infinity, when $0<s\leq t$ and $(i,j)\notin U_{s}$.

As $\langle D_{k}\rangle$ tend to infinity, we see that $(i,j),(i^{\prime},j^{\prime})\in U_{s}$ and $(i,j)\neq(i^{\prime},j^{\prime})$ imply $i\neq i^{\prime}$. Hence by the assumption on $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$,

[TABLE]

is an irrational algebraic number when $k\in\mathbb{N}^{\prime}$. For $k\in\mathbb{N}^{\prime}$ we have

[TABLE]

Set $E_{k}=\{d_{i_{s},j_{s},k}\}_{s}$. Then $\lfloor E_{k}\rfloor=\lfloor D_{k}\rfloor$ when $k\in\mathbb{N}^{\prime}$. Hence both $\lfloor E_{k}\rfloor$ and $\langle\overline{E_{k}}\rangle$ tend to infinity along $\mathbb{N}^{\prime}$. Now applying Theorems 2.1 to $\beta_{1},\cdots,\beta_{t}$ and $\{E_{k}\}_{k\in\mathbb{N}^{\prime}}$ implies

[TABLE]

when $k\in\mathbb{N}^{\prime}$ is sufficiently large. This contradicts inequality (2). ∎

Proofs of Theorems 2.3 and 2.4.

Theorem 2.3 follows by applying Theorem 2.2 to the case $n=1$ and

[TABLE]

Theorem 2.4 follows by applying Theorems 2.2 to the case

[TABLE]

and

[TABLE]

∎

Proofs of Theorems 2.7 and 2.10.

We prove Theorem 2.7 first. Assume that the $b$-ary expansion of $\alpha$ satisfies Condition 2.5. Set $d_{i,j}=|A^{j}_{i}B^{j}_{i}|$. Then each $D_{i}=\{d_{i,j}\}_{j}$ is an $(L+1)$-admissible $(m)$-array by (v) of Condition 2.5. It follows from (ii) and (iii) of Condition 2.5 that both the sequences $\{\lfloor D_{i}\rfloor\}_{i\geq 1}$ and $\{\langle D_{i}\rangle\}_{i\geq 1}$ are strictly increasing. (iv) of Condition 2.5 implies

[TABLE]

is divisible by $b^{k_{i}}$, hence

[TABLE]

for each $i\geq 1$. On the other hand, applying Theorem 2.3 to $f(D_{i})$ and the sequence $\{D_{i}\}_{i\geq 1}$ implies

[TABLE]

when $i$ is sufficiently large. This concludes the proof of Theorem 2.7. Theorem 2.10 follows from Theorem 2.4 in exactly the same way. ∎

Proof of Theorem 1.4.

Assume that the $b$-ary expansions of $\alpha$ and $\alpha^{\prime}$ satisfy Condition 1.3. It follows from Theorem 2.10 that $1,$ $\alpha$ and $\alpha^{\prime}$ are not linearly independent over $\mathbb{Q}$. Thus there exists a nonzero element $(x,y,z)$ in $\mathbb{Z}^{3}$, such that

[TABLE]

The above equality implies $y,z\neq 0$. Set $|A_{n}B_{n}|=l_{n},$ $|A^{\prime}_{n}B_{n}|=l^{\prime}_{n},$ and $|B_{n}|=k_{n}.$ Without loss of generality, we can choose an infinite subset $\mathbb{N}^{\prime}$ of $\mathbb{N}$ such that either $l_{n}-l^{\prime}_{n}$ is a fixed integer $s$, or $\{l_{n}-l^{\prime}_{n}\}$ tends to $+\infty$ along $\mathbb{N}^{\prime}$. In the second case, applying Theorem 2.1 to the sum $[\alpha b^{l_{n}}]-[\alpha^{\prime}b^{l^{\prime}_{n}}]$ implies

[TABLE]

when $n\in\mathbb{N}^{\prime}$ is sufficiently large; this contradicts (i) of Condition 1.3. Hence we assume that $l_{n}-l^{\prime}_{n}$ is a fixed positive integer $s$ for $n\in\mathbb{N}^{\prime}$. We have

[TABLE]

when $n\in\mathbb{N}^{\prime}$. If

[TABLE]

we can apply Theorem 2.1 to get a contradiction as above. Hence

[TABLE]

Now (i) of Condition 1.3 implies that

[TABLE]

is divisible by $b^{k_{n}}$ when $n\in\mathbb{N}^{\prime}$ is sufficiently large, where $|\theta_{n}|\leq 3$. This implies the $b$-ary expansion of the rational number $\tfrac{x}{z}$ has arbitrary long block of zero or $b-1$. On the other hand, it is well-know that the $b$-ary expansion of a rational number is eventually periodic. This forces $\tfrac{x}{z}b^{i}$ to be integer for some positive integer $i$. This fact combining with equalities $(\ref{for7})$ and $(\ref{for8})$ implies that the $b$-ary expansions of $\alpha$ and $\alpha^{\prime}$ have the same tail. ∎

4. a generalized Borel

conjecture and some other questions

In this section, we collect some problems and open questions for future work.

1. (i)

Theorem 2.10 implies that $b$-ary expansions of linearly independent irrational algebraic numbers are quite independent. Motivated by this result, we propose a generalized Borel conjecture.

Let $\xi=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$ be $n$-tuple of real numbers and let

[TABLE]

be their $b$-ary expansions. For two positive integers $m,N$ and each $n\times m$-matrix $D$ of elements from $\{0,1,\cdots,b-1\}$, set

[TABLE]

where

[TABLE]

The $n$-tuple of real numbers $\xi=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$ is called $normal$ to base $b$ if for each $m\geq 1$, each $n\times m$-matrix $D$ of elements from $\{0,1,\cdots,b-1\}$ occurs in the $b$-ary expansion of $\xi$ with the frequency $\tfrac{1}{b^{mn}}$, that is, if

[TABLE]

for each $m\geq 1$, and each $n\times m$-matrix $D$ of elements from $\{0,1,\cdots,b-1\}$.

The proof of Borel’theorem by the law of large numbers in [5, p.110] can be used directly to show that:

Theorem 4.1**.**

Almost all $n$-tuples of real numbers are normal to base $b$.

Motivated by this result and Theorem 2.10, we propose the following generalization of Borel conjecture.

Conjecture 4.2** **(Generalized Borel

Conjecture).

Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ be $n$ algebraic numbers, such that $1,\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ are linearly independent over $\mathbb{Q}$. Then $(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$ is normal to base $b$.*

It is easy to check that the above conjecture is invalid if $1,\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ are linearly dependent over $\mathbb{Q}$. 2. (ii)

The proof of Theorem 2.1 and Remark 3.2 implies the following theorem:

Theorem 4.3**.**

Let $k$ be a positive integer, let $\epsilon$ be a positive number, and let

[TABLE]

be a polynomial with real algebraic coefficients, where $a_{k}$ is irrational. Then we have

[TABLE]

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

It is reasonable to expect that the following more general result also holds:

Conjecture 4.4**.**

Let $\epsilon$ be a positive number, let $k$, $l$ be two positive integers,and let

[TABLE]

and

[TABLE]

be polynomials with real algebraic coefficients, where $a_{k}$ and $c_{l}$ are linearly independent over $\mathbb{Q}$. Then we have

[TABLE]

when $n\in\mathbb{N}$ is sufficiently large. 3. (iii)

All the congruences in Conditions 2.5, 2.6 and 2.12 are linear. We ask that wether similar results held for some nonlinear congruences. According to Theorems 2.1-2.4, this amounts to similar estimation of the upper bound of greatest common divisor of more general power sums. It is too difficult to give a general formulation of such question. Following are three simplest illustrations.

Conjecture 4.5**.**

Let $\alpha$, $\beta$ be two irrational algebraic numbers, and let $\epsilon$ be a positive number. For nonnegative integers $k$ and $m$, set

[TABLE]

Then we have

[TABLE]

when $m$ is sufficiently large.

Conjecture 4.6**.**

Let $\alpha$, $\beta$ and $\epsilon$ be as before. For two nonnegative integers $k$ and $m$, set

[TABLE]

Then we have

[TABLE]

when $m$ is sufficiently large.

Conjecture 4.7**.**

Let $\alpha$, $\beta$ and $\epsilon$ be as before. For positive integer $m$, set

[TABLE]

Then we have

[TABLE]

when $m$ is sufficiently large.

Conjectures 4.5 and 4.6, if valid, would provide new evidences for the Generalized Borel Conjecture.