Linear independence of values of logarithms revisited

TL;DR

This paper establishes new lower bounds for linear forms in powers of logarithms of algebraic numbers, refining previous results and utilizing Hermite-Mahler Padé approximations in both complex and p-adic contexts.

Contribution

It provides a novel lower bound for linear forms in logarithmic values, improving upon previous bounds with new techniques involving Padé approximations.

Findings

Refined lower bounds for linear forms in logarithms.

Applicable to both complex and p-adic cases.

Enhances previous results by Nesterenko-Waldschmidt.

Abstract

Let $m\ge 2$ be an integer, $K$ an algebraic number field and $\alpha\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in $1,{\rm{log}}(1+\alpha),\ldots,{\rm{log}}^{m-1}(1+\alpha)$ with algebraic integer coefficients in both complex and $p$-adic cases (see Theorem $2.1$ and Theorem $2.4$). Especially, in the complex case, our result is a refinement of the result of Nesterenko-Waldschmidt on the lower bound of linear form in certain values of power of logarithms. The main integrant is based on Hermite-Mahler's Pad\'{e} approximation of exponential and logarithm functions.

Equations

$$\exp (-\Phi (m,H))\le |P(\theta )|.$$

$$|P(\log (\alpha ))|\ge \exp \left(-C{m}^2(\log (L)+m\log (m))(1+\log (m){)}^{-1}\right)$$

$$|\cdot {|}_p:{\mathbb{C}}_p⟶{\mathbb{R}}_{\ge 0},|p|=p^{-1}.$$

$$\mathrm{den}:\overline{\mathbb{Q}}⟶\mathbb{N},\alpha \mapsto \min \{n\in \mathbb{N}\mid n\alpha \text{is an algebraic integer}\}.$$

$$|p{|}_v:=p^{-\frac{[K_v:{\mathbb{Q}}_p]}{[K:\mathbb{Q}]}}\text{if}v\in M_K^f\text{and}v|p,$$

$$|x{|}_v:=|{\sigma }_{v}x{|}^{\frac{[K_v:\mathbb{R}]}{[K:\mathbb{Q}]}}\text{if}v\in M_K^{\infty },$$

$$\prod _{v\in M_K}|\xi {|}_v=1\text{for}\xi \in K\setminus \{0\}.$$

$$\mathrm{H}(\mathbf{\beta }):=\prod _{v\in M_K}\max \{1,|{\beta }_0{|}_v,\dots ,|{\beta }_m{|}_v\}.$$

$$\prod _{k=1}^{[K:\mathbb{Q}]}\max \{1,|{\beta }_0^{(k)}|,\dots ,|{\beta }_m^{(k)}|\}=\mathrm{H}(\mathbf{\beta }{)}^{[K:\mathbb{Q}]}.$$

$$(\mathrm{i}){\log }_p(1+z)=\sum _{k=1}^{\infty }\frac{(-1{)}^{k+1}{z}^k}{k}\text{if}|z{|}_p<1,$$

$$(\mathrm{ii}){\log }_p(xy)={\log }_p(x)+{\log }_p(y)\text{for}x,y\in {\mathbb{C}}_p\setminus \{0\},$$

$$(\mathrm{iii}){\log }_p(p)=0.$$

$$T(\alpha )=\exp \left(\frac{2|\log (1+\alpha )|}{1+\sqrt{1+4|\log (1+\alpha )|}}\right)(m-1)!,$$

$$T^{(k)}(\alpha )=\frac{2^m(1+|{\alpha }^{(k)}|)(m-1)!}{|{\alpha }^{(k)}|}\text{for}1\le k\le [K:\mathbb{Q}],$$

$${\mathcal{A}}^{(k)}(\alpha )=m(1+\log (2))+\log (\mathrm{den}(\alpha ))+\log (1+|{\alpha }^{(k)}|)\text{for}1\le k\le [K:\mathbb{Q}],$$

$$A(\alpha )=m\log \left(\frac{m}{|\log (1+\alpha )|}\right)-\left(\frac{m(1+\sqrt{1+4|\log (1+\alpha )|})}{2}+\frac{2|\log (1+\alpha )|}{1+\sqrt{1+4|\log (1+\alpha )|}}\right)-\log (\mathrm{den}(\alpha )),$$

$$\nu (\alpha ):=A(\alpha )+{\mathcal{A}}^{(1)}(\alpha ),$$

$$\delta (\alpha ):=A(\alpha )+{\mathcal{A}}^{(1)}(\alpha )-\frac{(m-1){\sum }_{k=1}^{[K:\mathbb{Q}]}{\mathcal{A}}^{(k)}(\alpha )}{[K_{\infty }:\mathbb{R}]},$$

$$m\left[\sqrt{\log n}\cdot \exp \left(\frac{(\sqrt{322}-\sqrt{546})\sqrt{\log n}}{\sqrt{515}}\right)\right]<\frac{[K_{\infty }:\mathbb{R}]{\delta }^2(\alpha )ϵ}{2(m-1)[K:\mathbb{Q}](2\nu (\alpha )+ϵ\delta (\alpha ))},$$

$$\log \left(T(\alpha )n^{\frac{m}{2}}{\left[\prod _{k=1}^{[K:\mathbb{Q}]}m!(T^{(k)}(\alpha ){)}^{m-1}{n}^{2m(m-1)}\right]}^{\frac{1}{[K_{\infty }:\mathbb{R}]}}\right)\le \frac{ϵ{\delta }^2(\alpha )n}{4(2\nu (\alpha )+ϵ\delta (\alpha ))}.$$

$$\log \left(\left|\sum _{i=0}^{m-1}{\beta }_i{\log }^i(1+\alpha )\right|\right)>-\left(\frac{[K:\mathbb{Q}]\nu (\alpha )}{[K_{\infty }:\mathbb{R}]\delta (\alpha )}+\frac{ϵ[K:\mathbb{Q}]}{2[K_{\infty }:\mathbb{R}]}\right)\log (H).$$

$$\nu (\alpha ):=$$

$$-\log (d)+m(1+\log (2))+\log (d)+\log (1+\left|c/d\right|).$$

$$\delta (\alpha ):=$$

$$-\log (d)-(m-2)\left(m(1+\log (2))+\log (d)+\log (1+\left|c/d\right|)\right).$$

$$m\left[\sqrt{\log n}\cdot \exp \left(\frac{(\sqrt{322}-\sqrt{546})\sqrt{\log n}}{\sqrt{515}}\right)\right]<\frac{{\delta }^2(\alpha )ϵ}{2(m-1)(2\nu (\alpha )+ϵ\delta (\alpha ))},$$

$$\log \left(2^{(2m+1)(m-1)}(m!{)}^{m+1}(m-1){!}^m{\left(\frac{(d+|c|)}{|c|}\right)}^{2(m-1)}\right)+\left(\frac{2|\log (1+\frac{c}{d})|}{1+\sqrt{1+4|\log (1+\frac{c}{d})|}}\right)$$

$$+\left(\frac{m}{2}+2m(m-1)\right)\log (n)<\frac{ϵ\delta (\alpha {)}^{2}n}{4(2\nu (\alpha )+ϵ\delta (\alpha ))}.$$

$$\log \left(\left|\sum _{i=0}^{m-1}{b}_i{\log }^i(1+\alpha )\right|\right)>-\left(\frac{\nu (\alpha )}{\delta (\alpha )}+\frac{ϵ}{2}\right)\cdot \log H.$$

$$\log \left(\left|\sum _{i=0}^{m-1}{b}_i{\log }^i(1+\alpha )\right|\right)\ge -C(\alpha )(m-1{)}^2(1+\log (m-1){)}^{-1}((m-1)\log (m-1)+\log (L)),$$

$$-C(\alpha )(m-1{)}^2\log (H)\ge -C(\alpha )(m-1{)}^2(1+\log (m-1){)}^{-1}((m-1)\log (m-1)+\log (L)),$$

$$\frac{\nu (\alpha )}{\delta (\alpha )}$$

$$T_p(\alpha )=\frac{(2m{)}^{m-1}}{|\alpha {|}_p},$$

$$A_p(\alpha )=-m\log (|\alpha {|}_p),$$

$${\nu }_p(\alpha )=A_p(\alpha ),$$

$${\delta }_p(\alpha )=A_p(\alpha )-\frac{(m-1){\sum }_{k=1}^{[K:\mathbb{Q}]}{\mathcal{A}}^{(k)}(\alpha )}{[K_{\infty }:\mathbb{R}]}.$$

$$\frac{1}{\log |\alpha {|}_p^{-1}}+\frac{1}{m}\le n,$$

$$mn\left[\sqrt{\log n}\cdot \exp \left(\frac{(\sqrt{322}-\sqrt{546})\sqrt{\log n}}{\sqrt{515}}\right)\right]\le \frac{ϵ{\delta }_p^2(\alpha )[K_p:{\mathbb{Q}}_p]n}{2(m-1)(2{\nu }_p(\alpha )+ϵ{\delta }_p(\alpha ))[K:\mathbb{Q}]},$$

$$\log \left(T_p(\alpha )n^{m-1}{\left[\prod _{k=1}^{[K:\mathbb{Q}]}m!(T^{(k)}(\alpha ){)}^{m-1}{n}^{2m(m-1)}\right]}^{\frac{1}{[K_p:{\mathbb{Q}}_p]}}\right)\le \frac{ϵ{\delta }_p^2(\alpha )n}{4(2{\nu }_p(\alpha )+ϵ{\delta }_p(\alpha )},$$

$$\log \left({\left|\sum _{i=0}^{m-1}{\beta }_i{\log }_p^i(1+\alpha )\right|}_p\right)>-\left(\frac{[K:\mathbb{Q}]{\nu }_p(\alpha )}{[K_p:{\mathbb{Q}}_p]{\delta }_p(\alpha )}+\frac{ϵ[K:\mathbb{Q}]}{2[K_p:{\mathbb{Q}}_p]}\right)\log (H).$$

Full text

1. 1 Introduction

2. 2 Notations and main results

3. 3 Padé approximations of formal power series

4. 4 Padé approximation of exponential functions

5. 5 Padé approximations of power of logarithm functions

6. 6 Estimations

7. 7 Proof of main theorems

8. 7.1 Non-vanishing of certain determinants

9. 7.2 Proof of Theorem 2.1

10. 7.3 Proof of Theorem 2.4 *** Key words and phrases. Padé approximation, Linear independence, Logarithm, Exponential function.

1 Introduction

Let $\theta$ be a transcendental complex number. The function $\Phi:\mathbb{N}\times\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0}$ is a transcendental measure for $\theta$ if, for any sufficiently large positive integer $m$, any sufficiently large positive real number $H$ and any nonzero polynomial $P(z)\in\mathbb{Z}[z]$ with ${\rm{deg}}(P)\leq m$ and ${\rm{H}}(P)\leq H$, we have

[TABLE]

Let $\alpha$ be an algebraic number different from [math] and $1$. Then the complex number ${\rm{log}}(\alpha)$ is transcendental. A great deal of work has already been done on finding transcendence measures for the values of logarithms. For example, Mahler [7],[8],[9], Gel’fond [4], Feldman [3], Cijsouw [2], Reyssat [14] and Waldschmidt [18]. In [12], Nesterenko-Waldschmidt gave the following transcendence measure of values of logarithm.

Theorem 1.1**.**

$($[12, Theorem $6.$ $1)$]$)$ Let $\alpha$ be an algebraic number, $\alpha\neq 0,1$. Then there exists a effectively computable positive number $C=C(\alpha)$, depending only on $\alpha$ and the determination of the logarithm of $\alpha$ such that if $P(z)\in\mathbb{Z}[z]\setminus\{0\}$, ${\rm{deg}}P\leq m,{\rm{L}}(P)\leq L$, then

[TABLE]

where ${\rm{L}}(P)=\sum_{i=0}^{m}|a_{i}|$ if $P(z)=\sum_{i=0}^{m}a_{i}z^{i}$.

The purpose of the present article is to give an improvement of Theorem 1.1 for algebraic numbers $\alpha$ which are sufficiently close to $1$ and give a $p$-adic version of the result. The main integrant of the proof of these results is Hermite-Padé approximation of exponential and logarithm functions.

2 Notations and main results

We collect some notations which we use throughout this article. For a prime number $p$, we denote the $p$-adic number field by $\mathbb{Q}_{p}$, the $p$-adic completion of a fixed algebraic closure of $\mathbb{Q}_{p}$ by $\mathbb{C}_{p}$ and the normalized $p$-adic valuation on $\mathbb{C}_{p}$ by

[TABLE]

We fix an algebraic closure of $\mathbb{Q}$ and denote it by $\overline{\mathbb{Q}}$. We define the denominator function by

[TABLE]

We fix embeddings $\sigma:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and $\sigma_{p}:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$. For an algebraic number field $K$, we consider $K$ as a subfield of $\overline{\mathbb{Q}}$ and denote the ring of integers of $K$ by $\mathcal{O}_{K}$. For $\alpha\in K$, we denote $\sigma(\alpha)=\alpha$, $\sigma_{p}(\alpha)=\alpha$ and the set of conjugates of $\alpha$ by $\{\alpha^{(k)}\}_{1\leq k\leq[K:\mathbb{Q}]}$ with $\alpha^{(1)}=\alpha$ and $\alpha^{(2)}$ is the complex conjugate of $\alpha$ if $\sigma(K)\not\subset\mathbb{R}$. We denote the set of places of $K$ (resp. infinite places, finite places) by $M_{K}$ (resp. $M^{\infty}_{K}$, $M^{f}_{K}$). For $v\in M_{K}$, we denote the completion of $K$ with respect to $v$ by $K_{v}$. For $v\in M_{K}$, we define the normalized absolute value $|\cdot|_{v}$ as follows:

[TABLE]

where $\sigma_{v}$ is the embedding $K\hookrightarrow\mathbb{C}$ corresponding to $v$. Then we have the product formula

[TABLE]

Let $m$ be a natural number and $\boldsymbol{\beta}:=(\beta_{0},\ldots,\beta_{m})\in K^{m+1}\setminus\{\mathbb{0}\}$. We define the absolute height of $\boldsymbol{\beta}$ by

[TABLE]

Note that, for $\boldsymbol{\beta}=(\beta_{0},\ldots,\beta_{m})\in\mathcal{O}^{m+1}_{K}\setminus\{\mathbb{0}\}$, we have $\mathrm{H}(\boldsymbol{\beta})=\prod_{v\in M^{\infty}_{K}}\max\{1,|\beta_{0}|_{v},\ldots,|\beta_{m}|_{v}\}$ and

[TABLE]

Let ${\rm{log}}:\mathbb{C}\setminus\mathbb{R}_{\leq 0}\longrightarrow\mathbb{C}$ be the principal value logarithm function and ${\rm{log}}_{p}:\mathbb{C}_{p}\setminus\{0\}\longrightarrow\mathbb{C}_{p}$ the $p$-adic logarithm function, i.e. ${\rm{log}}_{p}$ is a $p$-adic locally analytic function satisfying the following conditions:

[TABLE]

Under the above notations, we shall prove the following results.

Theorem 2.1**.**

Let $m\in\mathbb{Z}_{\geq 2}$, $K$ be an algebraic number field and $\alpha\in K\setminus\{0,-1\}$. We define the real numbers

[TABLE]

where $K_{\infty}$ is the completion of $K$ with respect to the fixed embedding $\sigma:K\hookrightarrow\mathbb{C}$. We assume $\dfrac{m}{|{\rm{log}}(1+\alpha)|}\geq 4$ and $\delta(\alpha)>0$, then the numbers $1,{\rm{log}}(1+\alpha),\ldots,{\rm{log}}^{m-1}(1+\alpha)$ are linearly independent over $K$. For any $\epsilon>0$, we take a natural number $n$ satisfying

[TABLE]

Then $H_{0}=\left(\dfrac{1}{2}{\rm{exp}}[{\delta(\alpha)n}]\right)^{\tfrac{[K_{\infty}:\mathbb{R}]}{[K:\mathbb{Q}]}}$ satisfies the following property$:$

For any $\boldsymbol{\beta}:=(\beta_{0},\ldots,\beta_{m-1})\in\mathcal{O}^{m}_{K}\setminus\{\mathbb{0}\}$ satisfying $H_{0}<\mathrm{H}(\boldsymbol{\beta})\leq H$, then we have

[TABLE]

We will prove Theorem 2.4 in Section $6.1$. In the case of $\alpha$ is a rational number, we obtain the following corollary.

Corollary 2.2**.**

Let $m\in\mathbb{Z}_{\geq 2}$, $\epsilon>0$ and $\alpha=c/d\in{\mathbb{Q}}\setminus\{0,-1\}$ with $(c,d)=1$ and $d>0$. Put

[TABLE]

Suppose $\delta(\alpha)>0$. Then we have

$({\rm{i}})$* The complex numbers $1,{\rm{log}}(1+\alpha),\ldots,{\rm{log}}^{m-1}(1+\alpha)$ are linearly independent over $\mathbb{Q}$.*

$({\rm{ii}})$* Let $n=n(\epsilon)$ be a natural number satisfying*

[TABLE]

Then for $H_{0}:=\tfrac{1}{2}{\rm{exp}}\left(\delta(\alpha)n\right)$ and $\mathbb{b}=(b_{0},b_{1},\ldots,b_{m-1})\in\mathbb{Z}^{m}\setminus\{\mathbb{0}\}$ of $H_{0}<\mathrm{H}(\mathbb{b})\leq H$, we have

[TABLE]

Remark 2.3**.**

We show that Corollary 2.2 gives an improvement of Theorem 1.1 for $1+\alpha\in\mathbb{Q}\setminus\{1,0\}$ which are sufficiently close to $1$ and $m\geq 3$. We compare the result of Theorem 1.1 with that of Corollary 2.2. Let $m\in\mathbb{Z}_{\geq 2}$, $\epsilon>0$ and $\alpha=c/d\in{\mathbb{Q}}\setminus\{0,-1\}$ with $(c,d)=1$ and $d>0$. For $\mathbb{b}=(b_{0},b_{1},\ldots,b_{m-1})\in\mathbb{Z}^{m}\setminus\{\mathbb{0}\}$, by Theorem 1.1, we obtain

[TABLE]

where $C(\alpha)$ is a positive number depending on $\alpha$ with $C(\alpha)>105500\cdot e^{{\rm{H}}(\alpha)}$ for $\mathrm{H}(\mathbb{b})\leq H$. Since we have

[TABLE]

we compare $C(\alpha)(m-1)^{2}$ and $\nu(\alpha)/\delta(\alpha)$. Since we have

[TABLE]

if $|\alpha|=|c|/d$ is sufficiently close to [math], Corollary 2.2 improves Theorem 1.1 for $1+\alpha\in\mathbb{Q}\setminus\{1,0\}$ which are sufficiently close to $1$ and $m\geq 3$.

Second, we introduce a $p$-adic version of Theorem 2.1.

Theorem 2.4**.**

Let $m\in\mathbb{Z}_{\geq 2}$, $p$ be a prime number, $K$ an algebraic number field and $\alpha\in K\setminus\{0,-1\}$ with $|\alpha|_{p}<1$. We use the same notations as in Theorem 2.1. We also define the real numbers

[TABLE]

We assume $\delta_{p}(\alpha)>0$, then the numbers $1,{\rm{log}}_{p}(1+\alpha),\ldots,{\rm{log}}^{m-1}_{p}(1+\alpha)$ are linearly independent over $K$. For any $\epsilon>0$, we take a natural number $n$ satisfying

[TABLE]

where $K_{p}$ is the completion of $K$ with respect to the fixed embedding $\sigma_{p}:K\hookrightarrow\mathbb{C}_{p}$. Then $H_{0}=\left(\dfrac{1}{2}{\rm{exp}}[{\delta_{p}(\alpha)n}]\right)^{\tfrac{[K_{p}:\mathbb{Q}_{p}]}{[K:\mathbb{Q}]}}$ satisfies the following property$:$

For any $\boldsymbol{\beta}:=(\beta_{0},\ldots,\beta_{m-1})\in\mathcal{O}^{m}_{K}\setminus\{\mathbb{0}\}$ satisfying $H_{0}<\mathrm{H}(\boldsymbol{\beta})\leq H$, then we have

[TABLE]

We will prove Theorem 2.4 in Section $6.2$.

3 Padé approximations of formal power series

In this section, we recall the definition and basic properties of Padé approximation of formal power series. In the following of this section, we use $K$ as a field with characteristic [math].

Lemma 3.1**.**

Let $m\in\mathbb{N}$ and $\mathbb{f}=(f_{1}(z),\ldots,f_{m}(z))\in K[[z]]^{m}$. For $\mathbb{n}:=(n_{1},\ldots,n_{m})\in\mathbb{Z}^{m}_{\geq 0}$, there exists a family of polynomials $(A_{1}(z),\ldots,A_{m}(z))\in K[z]^{m}$ satisfying the following properties$:$

[TABLE]

In this article, we call the polynomials $(A_{1}(z),\ldots,A_{m}(z))\in K[z]^{m}$ satisfying the conditions $({\rm{i}}),({\rm{ii}}),({\rm{iii}})$ in Lemma 3.1 as a weight $\mathbb{n}$ Padé approximants of $\mathbb{f}$. For a weight $\mathbb{n}$ Padé approximants of $\mathbb{f}$, $(A_{1}(z),\ldots,A_{m}(z))$, we call the formal power series $\sum_{j=1}^{m}A_{j}(z)f_{j}(z)$ as a weight $\mathbb{n}$ Padé approximation of $\mathbb{f}$.

Definition 3.2**.**

Let $m\in\mathbb{Z}_{\geq 1}$ and $\mathbb{f}:=(f_{1}(z),\ldots,f_{m}(z))\in K[[z]]^{m}$.

$({\rm{i}})$ Let $\mathbb{n}=(n_{1},\ldots,n_{m})\in\mathbb{Z}^{m}_{\geq 0}$. We say $\mathbb{n}$ is normal with respect to $\mathbb{f}$ if for any weight $\mathbb{n}$ Padé approximation $R(z)$ of $\mathbb{f}$ satisfy the equality

[TABLE]

$({\rm{ii}})$ We call $\mathbb{f}$ is perfect if any indices $\mathbb{n}\in\mathbb{Z}^{m}_{\geq 0}$ are normal with respect to $\mathbb{f}$.

Remark 3.3**.**

Let $m\in\mathbb{N}$, $\mathbb{n}=(n_{1},\ldots,n_{m})\in\mathbb{Z}^{m}_{\geq 0}$ and $\mathbb{f}=(f_{j}(z):=\sum_{k=0}^{\infty}f_{j,k}z^{k})_{1\leq j\leq m}\in K[[z]]^{m}$. We put $N=\sum_{j=1}^{m}(n_{j}+1)$. For $r\in\mathbb{Z}_{\geq 0}$, we define a $N\times(r+1)$ matrix $A_{\mathbb{n},r}(\mathbb{f})$ by

[TABLE]

where $a_{j,k}=0$ if $k<0$ for $1\leq j\leq m$. Then we have the following bijection:

[TABLE]

Note that the indice $\mathbb{n}$ is normal with respect to $\mathbb{f}$ is equivalent to $A_{\mathbb{n},N-1}(\mathbb{f})\in{\rm{GL}}_{N}(K)$.

Lemma 3.4**.**

Let $m\in\mathbb{N}$, $\mathbb{f}:=(f_{1},\ldots,f_{m})\in K[[z]]^{m}$ and $\mathbb{n}:=(n_{1},\ldots,n_{m})\in\mathbb{N}^{m}$. Put $\mathbb{n}_{i}:=(n_{1},\ldots,n_{i-1},n_{i}+1,n_{i+1},\ldots,n_{m})\in\mathbb{N}^{m}$ for $1\leq i\leq m$. Suppose $\mathbb{n}$ is normal with respect to $\mathbb{f}$. Then we have

[TABLE]

for any weight $\mathbb{n}_{i}$ Padé approximants $(A_{1}(z),\ldots,A_{m}(z))$ of $\mathbb{f}$.

Proof..

Put $N:=\sum_{j=1}^{m}(n_{j}+1)$. Since $\mathbb{n}$ is normal with respect to $\mathbb{f}$, we have ${\rm{dim}}_{K}{\rm{ker}}(A_{\mathbb{n},N-1}(\mathbb{f}))=0$. Suppose there exist $1\leq i\leq m$ and a weight $\mathbb{n}_{i}$ Padé approximants $(A_{1}(z),\ldots,A_{m}(z))$ of $\mathbb{f}$ satisfying ${\rm{deg}}A_{i}<n_{i}+1$. Put

[TABLE]

Then we have

[TABLE]

This is a contradiction. This completes the proof of Lemma 3.4. ∎

4 Padé approximation of exponential functions

In this section, we recall some properties of Padé approximation of exponential functions. We quote some propositions for the Padé approximation of exponential functions in [6].

Proposition 4.1**.**

$($cf. [6, Theorem $1.2.1$]$)$ Let $n$ be a natural number and $\omega_{1},\ldots,\omega_{n}$ pairwise distinct complex numbers. Then the functions $e^{\omega_{1}z},\ldots,e^{\omega_{n}z}$ are perfect. Especially, for $l\in\mathbb{N}$, the functions $1,e^{z},\ldots,e^{lz}$ are perfect.

Let $\omega_{1},\ldots,\omega_{n}$ be pairwise distinct complex numbers. Explicit construction of Padé approximations of $e^{\omega_{1}z},\ldots,e^{\omega_{n}z}$ are given by Hermite as follows.

Proposition 4.2**.**

$(cf.$[6, p. $242$]$)$ Let $\mathbb{m}:=(m_{1},\ldots,m_{n})\in\mathbb{Z}^{n}_{\geq 0}$ and $\{a_{h,j}(\mathbb{m},\boldsymbol{\omega})\}_{1\leq h\leq n,1\leq j\leq m_{h}+1}$ be the family of complex numbers satisfying the following equality$:$

[TABLE]

Then the formal power series

[TABLE]

is a weight $\mathbb{m}$ Padé approximation of $e^{\omega_{1}z},\ldots,e^{\omega_{n}z}$.

5 Padé approximations of power of logarithm functions

In this section, we construct Padé approximations of $1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z)$ for $m\in\mathbb{Z}_{\geq 2}$ by using that of exponential functions obtained in Proposition 4.2.

Lemma 5.1**.**

Let $f(z)\in K[[z]]$. Suppose there exists $g(z)\in K[[z]]$ satisfying $f(g(z))=g(f(z))=z$. Put $\tilde{g}(z):=g(z)+1$ and assume $1,\tilde{g}(z),\ldots,\tilde{g}^{l}(z)$ are perfect for any $l\in\mathbb{N}$. Then the any indices $\mathbb{n}\in\{(n_{0},n_{1},\ldots,n_{m-1})\in\mathbb{Z}^{m}_{\geq 0}|\ n_{0}\geq n_{1}\geq\ldots\geq n_{m-1}\}$ are normal with respect to $(1,f(z),\ldots,f^{m-1}(z))$ for any $m\in\mathbb{Z}_{\geq 2}$.

Proof..

Denote the set $\{(n_{0},n_{1},\ldots,n_{m-1})\in\mathbb{Z}^{m}_{\geq 0}|\ n_{0}\geq n_{1}\geq\ldots\geq n_{m-1}\}$ by $\mathcal{X}_{m}$. Let

[TABLE]

where $(n_{i})_{r_{i}}=(n_{i},\ldots,n_{i})\in\mathbb{Z}^{r_{i}}_{\geq 0}$ for $0\leq i\leq s$. We put

[TABLE]

We define the $K$-isomorphism $\Psi$ by

[TABLE]

Note that $\Psi$ is an order preserving map, namely we have ${\rm{ord}}F(z)={\rm{ord}}\Psi(F(z))$ for $F(z)\in K[[z]]$. We prove that $\Psi$ induces the bijection $\Psi:V_{\mathbb{n}}\longrightarrow W_{\mathbb{m}}$. Let $R(z)=\sum_{j=0}^{m-1}A_{j}(z)f^{j}(z)\in V_{\mathbb{n}}$ and put $A_{j}(z)=\sum_{h=0}^{n_{0}}a_{h,j}(1+z)^{h}$ with for $0\leq j\leq m-1$. Then we obtain

[TABLE]

Using $(\ref{R 1})$, we have

[TABLE]

Since $\Psi$ is order preserving map, the equality $(\ref{Psi R})$ shows that $\Psi(R)$ is a weight $\mathbb{m}$ Padé approximation of $\tilde{\mathbb{g}}$. Then we have $\Psi(V_{\mathbb{n}})\subseteq W_{\mathbb{m}}$. By the similar way, we also obtain $W_{\mathbb{m}}\subseteq\Psi(V_{\mathbb{n}})$. Then the map $\Psi:V_{\mathbb{n}}\longrightarrow W_{\mathbb{m}}$ is bijection. Since $\mathbb{m}$ is normal with respect to $\tilde{\mathbb{g}}$, we have ${\rm{ord}}S(z)=\sum_{i=0}^{s}(n_{i}+1)r_{i}-1$ for all $S(z)\in W_{\mathbb{m}}$. Since the bijection $\Psi:V_{\mathbb{n}}\longrightarrow W_{\mathbb{m}}$ is order preserving map, we also obtain ${\rm{ord}}R(z)=\sum_{i=0}^{s}(n_{i}+1)r_{i}-1$ for all $R(z)\in V_{\mathbb{n}}$. This shows that the indice $\mathbb{n}$ is normal with respect to $\mathbb{f}$. This completes the proof of Lemma 5.1. ∎

Proposition 5.2**.**

$($cf. [6, Theorem $1.2.3$]$)$ Let $m\in\mathbb{Z}_{\geq 2}$. Denote the set $\{(n_{0},\ldots,n_{m-1})\in\mathbb{Z}^{m}_{\geq 0}|\ n_{0}\geq n_{1}\geq\ldots\geq n_{m-1}\}$ by $\mathcal{X}_{m}$. Then any indices $\mathbb{n}\in\mathcal{X}_{m}$ are normal with respect to $(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$.

Proof..

By Proposition 4.1, we have $1,e^{z},\ldots,e^{lz}$ are perfect for $l\in\mathbb{N}$. Using Lemma 5.1 for $f(z):={\rm{log}}(1+z)$ and $g(z):=e^{z}-1$, any indices $\mathbb{n}\in\mathcal{X}_{m}$ are normal with respect to $(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$. This completes the proof of Proposition 5.2. ∎

Let $m\in\mathbb{Z}_{\geq 2}$ and $\mathbb{n}\in\mathcal{X}_{m}$. We obtain a weight $\mathbb{n}$ Padé approximation of $1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z)$ as follows:

Proposition 5.3**.**

Let $\{r_{i}\}_{0\leq i\leq s}\subset\mathbb{N}$ and $\{n_{i}\}_{0\leq i\leq s}\subset\mathbb{Z}_{\geq 0}$ satisfying $r_{0}+\ldots+r_{s}=m$ and $n_{0}>\ldots>n_{s}$. Put

[TABLE]

We define the family of rational numbers $\{a_{h,j}(\mathbb{m},\boldsymbol{\omega})\}_{0\leq h\leq n_{0}+1,1\leq j\leq m-1}$ as follows$:$

[TABLE]

Then the formal power series

[TABLE]

is a weight $\mathbb{n}$ Padé approximation of $(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$.

Proof..

We define a $\overline{\mathbb{Q}}$-isomorphism $\Psi$ by

[TABLE]

By Proposition 4.2, the formal power series

[TABLE]

is a weight $\mathbb{m}$ Padé approximation of $1,e^{z},\ldots,e^{n_{0}z}$. By the proof of Lemma 5.1, we have

[TABLE]

is a weight $\mathbb{n}$ Padé approximation of $(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$. Since the right hand side of the equality $(\ref{R})$ is the formal power series $R(z)$ defined in $(\ref{Pade log power})$, this completes the proof of Proposition 5.3. ∎

Remark 5.4**.**

Let $(n_{0},\ldots,n_{m})\in\mathcal{X}_{m}$ and $R(z)$ and $\mathcal{R}(z)$ be the formal power series defined in $(\ref{Pade log power})$ and $(\ref{explicit Pade e})$ respectively. Put $N:=\sum_{j=0}^{m-1}(n_{j}+1)$. We have $\mathcal{R}(z)=\dfrac{z^{N-1}}{(N-1)!}+\text{(higher order term)}$ (see p. $242$ [6]). Then by the definition of $R(z)$, we have

[TABLE]

On the other hand, in p. $245$ [6], Jager proved that the function

[TABLE]

where $C$ is a contour with positive orientation enclosing the set $\{0,1,\ldots,n_{0}\}$, is a weight $\mathbb{n}$ Padé approximation of $(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$ and $r(z)$ satisfies

[TABLE]

Since $\mathbb{n}$ is normal with respect to $(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$, a weight $\mathbb{n}$ Padé approximation of $(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$ is uniquely determined up to constant. Thus, by $(\ref{first term R})$ and $(\ref{first term r})$, we obtain

[TABLE]

6 Estimations

From this section to the last section, we use the following notations for $m\in\mathbb{Z}_{\geq 2}$, $n\in\mathbb{Z}_{\geq 0}$ and $1\leq i\leq m$:

[TABLE]

We define the set of rational numbers $\{a_{h,j}(\mathbb{m}_{i},\boldsymbol{\omega})\}_{1\leq i\leq m,0\leq h\leq n+1,1\leq j\leq m}$ satisfying the equality

[TABLE]

By Proposition 4.2 and Proposition 5.3, the formal power series

[TABLE]

are weight $\mathbb{m}_{i}$ Padé approximation of $1,e^{z},\ldots,e^{(n+1)z}$ and weight $\mathbb{n}_{i}$ Padé approximation of $1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z)$ respectively. We define

[TABLE]

Lemma 6.1**.**

$($cf. [8, Theorem $1(a)$]$)$ We use the notations as above. For any $1\leq i\leq m$, $0\leq j\leq m-1$ and $0\leq h\leq n+1$, we have

[TABLE]

Especially, for an algebraic number field $K$ and an element $\alpha\in K$, we have

[TABLE]

for $1\leq i\leq m$, $0\leq j\leq m-1$.

Proof..

Recall that, by the definition of $a_{h,j}(\mathbb{m}_{i},\boldsymbol{\omega})$, we have

[TABLE]

Fix a nonzero integer $\lambda$ satisfying $0\leq\lambda\leq n$. By the definition of $Q_{m,i,n+1}(x)$, we have the following equalities:

[TABLE]

Since we have

[TABLE]

then there exist a set of integers $\{c_{i,k}\}_{k\in\mathbb{Z}_{\geq 0}}$ satisfying

[TABLE]

where $t$ is an intermediate. Substituting $t=\dfrac{x-\lambda}{d_{n+1}}$ in the equality $(\ref{bekikyuusuu})$, we obtain

[TABLE]

Substituting $(\ref{bekiyuusuu2})$ for the equality $(\ref{lambda exp})$ and compare the equality $(\ref{fukusyu})$ and $(\ref{lambda exp})$, we have

[TABLE]

By the relation $(n+1)!^{m}\dfrac{1}{\lambda!^{m}(n-\lambda)!^{m}(n+1-\lambda)^{i}}\in\mathbb{Z}$ and the equality $(\ref{key kill denomi})$, we obtain

[TABLE]

In the case of $\lambda=n+1$, by using the same method as above, we also obtain

[TABLE]

This completes the proof of $(\ref{tisai denominator})$. The latter assertions are obtained by $(\ref{tisai denominator})$ and the definition of $A_{i,j,n+1}(z)$. This completes the proof of Lemma 6.1. ∎

Lemma 6.2**.**

$($cf. [8, Theorem $1(b)$]$)$ Let $\alpha$ be a complex number. Then we have

[TABLE]

for any $1\leq i\leq m$, $0\leq j\leq m-1$ and $n\in\mathbb{Z}_{\geq 0}$.

Proof..

In our proof of Lemma 6.2, we refer some of the arguments of [8, Theorem $1(b)$]. First we remark that the $a_{\lambda,j}(\mathbb{m}_{i},\boldsymbol{\omega})$ can be represented as follows:

[TABLE]

Let $\lambda$ be a natural number satisfying $0\leq\lambda\leq n+1$. Then by the equality $(\ref{residue})$, we have the following inequality:

[TABLE]

Next, we estimate a lower bound of ${\rm{sup}}_{|z-\lambda|=1/2}\left|{Q_{m,i,n+1}(z)}\right|$. Since we have the inequality

[TABLE]

for natural number $h$ satisfying $0\leq h\leq n+1$ and $h\neq\lambda$ and $z\in\{z\in\mathbb{C}|\ |z-\lambda|=\tfrac{1}{2}\}$, we obtain the following inequalities:

[TABLE]

In the case of $0\leq\lambda\leq n$, using the inequality $(\ref{ineq Q})$, we obtain

[TABLE]

Combining the inequality (see Proof of [8, Theorem $1$, p. $376$])

[TABLE]

and $(\ref{zero en})$, we obtain

[TABLE]

In the case of $\lambda=n+1$, using the inequality $(\ref{ineq Q})$, we obtain

[TABLE]

By the same arguments as above, from $(\ref{n+1})$, we obtain

[TABLE]

Using the inequalities $(\ref{a1})$, $(\ref{lambda conclusion})$ and $(\ref{n+1 2})$, we obtain

[TABLE]

for $0\leq\lambda\leq n+1$, $1\leq j\leq m$ and $1\leq i\leq m$. By the definition of $P_{i,j,n+1}(z)$ and use the inequalities $(\ref{a conclusion})$, we obtain

[TABLE]

for $\alpha\in\mathbb{C}$. This completes the proof of Lemma 6.2. ∎

Lemma 6.3**.**

Let $m$ be a natural number $m\geq 2$. Let $\alpha\in\mathbb{C}\setminus\{0,-1\}$ satisfying $2\leq m/|{\rm{log}}(1+\alpha)|$. Then we have

[TABLE]

Proof..

This proof is based on that of [8, Theorem $1$]. By $(\ref{integral rep R})$, we have

[TABLE]

where $C_{\rho}$ is a circle in the $x$-plane of center $x=0$ and radius $\rho>n+1.$ In the following, we take a positive real number $\rho$ satisfying $\rho\geq 2(n+1)$. For $x\in C_{\rho}$, we have

[TABLE]

Since

[TABLE]

and

[TABLE]

we have

[TABLE]

By $(\ref{integ rep R i n+1})$ and the above inequality, we obtain

[TABLE]

Put $f(x)=x|{\rm{log}}(1+\alpha)|+\dfrac{m(n+1)(n+2)}{x}-m(n+1){\rm{log}}(x)$ for $x>0$. Then $f(x)$ takes the minimal value at

[TABLE]

Since $n+2<m(n+1)$, we take $\rho=\dfrac{m(n+1)(1+\sqrt{1+4|{\rm{log}}(1+\alpha)|})}{2|{\rm{log}}(1+\alpha)|}$. Note that, by the assumption $2\leq m/|{\rm{log}}(1+\alpha)|$, we have $2(n+1)\leq\rho$. By $(\ref{upper jyouyo 1})$, we obtain the desire inequality. This completes the proof of Lemma 6.3. ∎

Next, we give a $p$-adic version of Lemma 6.3.

Lemma 6.4**.**

Let $\alpha\in\mathbb{C}_{p}$ satisfying $|\alpha|_{p}<1$. Then we have

[TABLE]

for any natural number $n$ satisfying $1/{\rm{log}}|\alpha|^{-1}_{p}+1/m\leq n.$

Proof..

Since $R_{i,n+1}(z)$ is a wight $\mathbb{n}_{i}$ Padé approximation of $(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$, we have ${\rm{ord}}R_{i,n+1}(z)=m(n+1)+i-1$. Put $E_{n+1}:=d^{m}_{n+1}(n+1)!^{m}(m-1)!$ and

[TABLE]

First we prove the inequalities

[TABLE]

By Lemma 6.1, we have $E_{n+1}P_{i,j,n+1}(z)\in\mathbb{Z}[z]$ for $1\leq i\leq m$, $0\leq j\leq m-1$. Using the equality

[TABLE]

and the definition of ${\rm{log}}(1+z)$, we have ${\rm{den}}(r_{i,k,n+1})\leq k^{m-1}$ for $m(n+1)+i-1\leq k$. Then we obtain the inequalities $(\ref{abs val coeff R p})$. Using the inequalities $(\ref{abs val coeff R p})$, we have

[TABLE]

Since we have $\max_{m(n+1)+i-1\leq k}|r_{i,k,n+1}\alpha^{k}|_{p}\leq(m(n+1)+i-1)^{m-1}|\alpha|^{m(n+1)+i-1}_{p}$ for any natural number $n$ satisfying $1/{\rm{log}}|\alpha|^{-1}_{p}+1/m\leq n$, we have the desire inequalities. This completes the proof of Lemma $\ref{p upper bound jyouyo}.$ ∎

7 Proof of main theorems

In this section, we give the proofs of Theorem 2.1 and Theorem 2.4.

7.1 Non-vanishing of certain determinants

Lemma 7.1**.**

$($cf. [6, Theorem $1.2.3$] $)$ Let $K$ be a field with characteristic [math] and $\mathbb{f}=(1,f_{1},\ldots,f_{m-1})\in K[[z]]^{m}$. Let $\mathbb{n}=(n_{1},\ldots,n_{m})\in\mathbb{Z}^{m}_{\geq 0}$. Put $\mathbb{n}_{i}=(n_{1}+1,n_{2}+1,\ldots,n_{i}+1,n_{i+1},\ldots,n_{m})\ \text{for}\ 1\leq i\leq m.$ Let $(A_{i,1}(z),\ldots,A_{i,m}(z))\in K[z]^{m}$ be a weight $\mathbb{n}_{i}$ Padé approximants of $\mathbb{f}$. We define a polynomial $\Delta(z)$ by

[TABLE]

Then there exists $\gamma\in K$ satisfying

[TABLE]

where $N=\sum_{j=1}^{m}(n_{j}+1)$. Moreover, if the set of indices $\{\mathbb{n}\}\cup\{\mathbb{n}_{i}\}_{1\leq i\leq m-1}$ are normal with respect to $\mathbb{f}$, we have $\Delta(z)\neq 0$, i.e. $\gamma\neq 0$.

Proof..

Denote the formal power series $A_{i,1}(z)+A_{i,2}(z)f_{1}(z)+\dots+A_{i,m}(z)f_{m-1}(z)$ by $R_{i}(z)$ for $1\leq i\leq m$. Note that we have

[TABLE]

By adding the $i$-th column of the matrix $(\ref{det})$ multiplied by $f_{i-1}(z)$ to the first column of the matrix $(\ref{det})$ for all $2\leq i\leq m$, we obtain the following equality:

[TABLE]

For $1\leq t,u\leq m$, we denote the $(t,u)$-th cofactor of the matrix in $(\ref{equal determ 1})$ by $\Delta_{t,u}(z)$. Then, we obtain

[TABLE]

Using the inequalities $(\ref{order lower bound})$ and the equality $(\ref{decomp det})$, we have

[TABLE]

On the other hand, by using the equality $(\ref{det})$, we obtain

[TABLE]

Combining the inequalities $(\ref{order lower bound delta})$ and $(\ref{upper bound det})$, we obtain the equality $(\ref{calculation of det})$. If the set of indicies $\{\mathbb{n}\}\cup\{\mathbb{n}_{i}\}_{1\leq i\leq m}$ are normal with respect to $\mathbb{f}$ then, using Lemma 3.4, we have ${\rm{deg}}A_{i,i}(z)=n_{i}+1$ for $1\leq i\leq m$. Then we have

[TABLE]

This completes the proof of Lemma 7.1. ∎

Using Lemma 7.1 for $\mathbb{f}:=(1,{\rm{log}}(1+z),\ldots,{\rm{log}}^{m-1}(1+z))$ and $\mathbb{n}=(n,\ldots,n)\in\mathbb{N}^{m}$, we obtain the following corollary.

Corollary 7.2**.**

Let $\{A_{i,j,n+1}(z)\}_{1\leq i\leq m,0\leq j\leq m-1}\subset\mathbb{Q}[z]$ be the set of polynomials defined in $(\ref{coeff polynomial})$. Put $\Delta^{(n+1)}(z):={\rm{det}}(A_{i,j,n+1}(z))_{1\leq i\leq m,0\leq j\leq m-1}.$ Then there exists some $\gamma\in\mathbb{Q}\setminus\{0\}$ satisfying

[TABLE]

Especially, we have

[TABLE]

7.2 Proof of Theorem 2.1

Before starting to prove Theorem 2.1, we introduce a sufficient condition to obtain a lower bound of linear forms of complex numbers with integer coefficients. For $\boldsymbol{\beta}:=(\beta_{0},\ldots,\beta_{m})\in K^{m+1}\setminus\{\mathbb{0}\}$ and $\theta_{0},\theta_{1},\ldots,\theta_{m}\in\mathbb{C}$, we denote $\sum_{i=0}^{m}\beta_{i}\theta_{i}$ by $\Lambda(\boldsymbol{\beta},\boldsymbol{\theta})$.

Proposition 7.3**.**

Let $K$ be an algebraic number field and fix an embedding of $\sigma:K\hookrightarrow\mathbb{C}$. We denote the completion of $K$ by the fixed embedding $\sigma$ by $K_{\infty}$. Let $m\in\mathbb{N}$ and $\theta_{0}:=1,\theta_{1},\ldots,\theta_{m}\in\mathbb{C}^{*}$. Suppose there exist a set of matrices

[TABLE]

positive real numbers

[TABLE]

and a function $f:\mathbb{N}\longrightarrow\mathbb{R}_{\geq 0}$ satisfying

[TABLE]

and

[TABLE]

for $n\geq N$. Put

[TABLE]

Suppose $\delta>0$, then the numbers $\theta_{0},\ldots,\theta_{m}$ are linearly independent over $K$ and, for any $\epsilon>0$, there exists a constant $H_{0}$ depending on $\epsilon$ and the given data such that the following property holds. For any $\boldsymbol{\beta}:=(\beta_{0},\ldots,\beta_{m})\in\mathcal{O}^{m+1}_{K}\setminus\{\mathbb{0}\}$ satisfying $\mathrm{H}(\boldsymbol{\beta})\geq H_{0}$, then we have

[TABLE]

Proof..

Since ${\rm{det}}(a_{i,j,n})_{0\leq i,j\leq m}\neq 0$ for all $n\in\mathbb{N}$, there exists $0\leq I_{n}\leq m$ satisfying

[TABLE]

where the vector $(\beta_{0},\ldots,\beta_{m})$ is placed in the $I_{n}$-th line of the matrix in the definition of $\Theta_{\boldsymbol{\beta},n}$. Then by the product formula, we have

[TABLE]

where “\ {}^{\prime}\” in ${\prod_{k}}^{\prime}$ means $k$ runs $2\leq k\leq[K:\mathbb{Q}]$ if $K_{\infty}=\mathbb{R}$ and $3\leq k\leq[K:\mathbb{Q}]$ if $K_{\infty}=\mathbb{C}$. In the following, we denote the $(s,t)$-th cofactor of the matrix in the definition of $\Theta_{\boldsymbol{\beta},n}$ by $\Theta_{\boldsymbol{\beta},n,s,t}$. First we give an upper bound of $|\Theta^{(1)}_{\boldsymbol{\beta},n}|$.

[TABLE]

Secondly, we give an upper bound of $|\Theta^{(k)}_{\boldsymbol{\beta},n}|$ for $2\leq k\leq[K:\mathbb{Q}]$.

[TABLE]

Substituting the inequalities $(\ref{upper iota})$ and $(\ref{tau part infty})$ to the inequality $(\ref{upper infty})$ and taking the $\tfrac{1}{[K_{\infty}:\mathbb{R}]}$-th power, we obtain

[TABLE]

where

[TABLE]

Let $\epsilon>0$ and $0<\tilde{\epsilon}<\delta$ satisfying

[TABLE]

Write $\tilde{\delta}:=\delta-\tilde{\epsilon}$. By the assumptions $\delta>0$ and $(\ref{fn})$, there exists a natural number ${n^{*}}$ satisfying

[TABLE]

for all $n\geq n^{*}$. Consequently, using $(\ref{conclusion 2})$, we obtain

[TABLE]

Now for this fix $n^{*}$, we consider $H_{0}>1$ such that $e^{-\tilde{\delta}n^{*}}H^{\tfrac{[K:\mathbb{Q}]}{[K_{\infty}:\mathbb{R}]}}_{0}\geq\tfrac{1}{2}$. Then we have $e^{-\tilde{\delta}n^{*}}\mathrm{H}(\boldsymbol{\beta})^{\tfrac{[K:\mathbb{Q}]}{[K_{\infty}:\mathbb{R}]}}\geq\tfrac{1}{2}$ for all $\boldsymbol{\beta}\in\mathcal{O}^{m+1}_{K}\setminus\{\mathbb{0}\}$ satisfying $\mathrm{H}(\boldsymbol{\beta})\geq H_{0}$. Take $\boldsymbol{\beta}\in\mathcal{O}^{m+1}_{K}$ satisfying $\mathrm{H}(\boldsymbol{\beta})\geq H_{0}$. Let $\tilde{n}=\tilde{n}(\mathrm{H}(\boldsymbol{\beta}))\in\mathbb{N}$ be the least positive integer satisfying $e^{-\tilde{\delta}\tilde{n}}\mathrm{H}(\boldsymbol{\beta})^{\tfrac{[K:\mathbb{Q}]}{[K_{\infty}:\mathbb{R}]}}<\tfrac{1}{2}$. Note that $\tilde{n}>n^{*}$. Using inequality $(\ref{conclusion 3})$ for $\tilde{n}$, we have

[TABLE]

By the definition of $\tilde{n}$, we have $e^{-(\tilde{n}-1)\tilde{\delta}}\mathrm{H}(\boldsymbol{\beta})^{\tfrac{[K:\mathbb{Q}]}{[K_{\infty}:\mathbb{R}]}}\geq\tfrac{1}{2}$ and then $e^{\tilde{n}}\leq(2\mathrm{H}(\boldsymbol{\beta})^{\tfrac{[K:\mathbb{Q}]}{[K_{\infty}:\mathbb{R}]}})^{\tfrac{1}{\tilde{\delta}}}e$. Finally, we obtain

[TABLE]

Note that the last inequality is obtained by the inequality $(\ref{condition tilde mu})$. This completes the proof of Proposition 7.3. ∎

Remark 7.4**.**

In this remark, we explain that how to take the positive number $H_{0}$ in Proposition 7.3. Let $\epsilon>0$. At first we take $\tilde{\epsilon}:=\dfrac{\epsilon\delta^{2}}{(2\nu+\epsilon\delta)}$. Then we have

[TABLE]

Since $f(n)=o(n)(n\to\infty)$, there exists $\tilde{n}^{*}\in\mathbb{N}$ satisfying

[TABLE]

We take $n^{*}\geq\tilde{n}^{*}$ satisfying

[TABLE]

then we have the inequalities $(\ref{cond 1})$ and $(\ref{cond 2})$ for any $n\geq n^{*}$. At last, we take $H_{0}$ by

[TABLE]

Then by the proof of Proposition 7.3, the positive number $H_{0}$ satisfies the following property$:$

[TABLE]

for any $\boldsymbol{\beta}:=(\beta_{0},\ldots,\beta_{m})\in\mathcal{O}^{m+1}_{K}\setminus\{\mathbb{0}\}$ satisfying $\mathrm{H}(\boldsymbol{\beta})\geq H_{0}$.

Proof of Theorem 2.1.

We use the same notations as in Section $5$. Let $R_{i,n+1}(z)$ be the formal power series defined in $(\ref{exp pade S})$ for $1\leq i\leq m$ and $n\in\mathbb{N}$. Let $K$ be an algebraic number field. We fix an element $\alpha\in K\setminus\{0,-1\}$ satisfying the assumption in Theorem 2.1. Then we have

[TABLE]

Put

[TABLE]

Then by Lemma 6.1 and Corollary 7.2, we have

[TABLE]

Define the set of positive real numbers

[TABLE]

Define $g(n):=n\left[\sqrt{{\rm{log}}n}\cdot{\rm{exp}}\left(-\sqrt{({\rm{log}}n)/R}\right)\right]$ with $R:=\tfrac{515}{(\sqrt{546}-\sqrt{322})^{2}}$. Note that, in [15], Rosser-Schoenfeld gives an estimate of $d_{n}$ of the form

[TABLE]

Put $f(n):=mg(n)$. Then we see $f(n)=o(n)\ (n\to\infty)$. By Lemma 6.2 and Lemma 6.3, we have

[TABLE]

for all $n\in\mathbb{N}$. We use Proposition 7.3 for $\theta_{1}={\rm{log}}(1+\alpha),\ldots,\theta_{m-1}={\rm{log}}^{m-1}(1+\alpha)$ and the above datum $\{\mathcal{A}^{(k)}(\alpha)\}_{1\leq k\leq[K:\mathbb{Q}]}$, $\{c^{(k)}_{l}\}_{\begin{subarray}{c}0\leq l\leq m-1\\ 1\leq k\leq[K:\mathbb{Q}]\end{subarray}}$, $\{T^{(k)}(\alpha)\}_{1\leq k\leq[K:\mathbb{Q}]}$, $A(\alpha)$, $c(\alpha)$, and $T(\alpha)$ then we obtain the assertion of Theorem 2.1. ∎

7.3 Proof of Theorem 2.4

Before proving Theorem 2.4, we introduce a $p$-adic version of Proposition 7.3.

Proposition 7.5**.**

Let $K$ be an algebraic number field and fix an embedding $\iota_{p}:K\longrightarrow\mathbb{C}_{p}$. We denote the completion of $K$ by the fixed embedding $\sigma_{p}$ by $K_{p}$. Let $m\in\mathbb{N}$ and $\theta_{0}:=1,\theta_{1},\ldots,\theta_{m-1}\in\mathbb{C}_{p}$.

Suppose, for all $n\in\mathbb{N}$, there exist a set of matrices

[TABLE]

positive real numbers

[TABLE]

and a function $f:\mathbb{N}\longrightarrow\mathbb{R}_{\geq 0}$ satisfying

[TABLE]

and

[TABLE]

for all $n\geq N$. Put

[TABLE]

For $\epsilon>0$ and ${n}^{*}\in\mathbb{N}$ satisfying $n^{*}\geq N$ and

[TABLE]

for all $n\geq n^{*}$. Suppose $\delta_{p}>0$, then the numbers $\theta_{0},\ldots,\theta_{m}$ are linearly independent over $K$ and the positive number $H_{0}:=\left(\dfrac{1}{2}{\rm{exp}}\left[\delta_{p}n^{*}\right]\right)^{\tfrac{[K_{p}:\mathbb{Q}_{p}]}{[K:\mathbb{Q}]}}$ satisfies the following property$:$

For any $\boldsymbol{\beta}:=(\beta_{0},\ldots,\beta_{m})\in\mathcal{O}^{m+1}_{K}\setminus\{\mathbb{0}\}$ satisfying $H_{0}\leq\mathrm{H}(\boldsymbol{\beta})$, then we have

[TABLE]

Since Proposition 7.5 can be proved by the same argument of that of Proposition 7.3, we omit the proof.

Proof of Theorem 2.4.

We use the same notations as in the proof of Theorem 2.1. Let $K$ be an algebraic number field. We fix an element $\alpha\in K\setminus\{0,-1\}$ satisfying the assumption in Theorem 2.4. Put

[TABLE]

By Lemma 6.2 and Lemma 6.4, we obtain

[TABLE]

for all natural number $n$ satisfying $1/{\rm{log}}(|\alpha|^{-1}_{p})+1/m\leq n$. Using Proposition 7.5 for $\theta_{1}={\rm{log}}_{p}(1+\alpha),\ldots,\theta_{m-1}={\rm{log}}^{m-1}_{p}(1+\alpha)$, we obtain the assertion of Theorem 2.4. ∎

Acknowledgements. The author warmly thank Noriko Hirata-Khono for her comments on the earlier version of this manuscript.