Известия Российской академии наук. Серия математическая
RUS  ENG    ЖУРНАЛЫ   ПЕРСОНАЛИИ   ОРГАНИЗАЦИИ   КОНФЕРЕНЦИИ   СЕМИНАРЫ   ВИДЕОТЕКА   ПАКЕТ AMSBIB  
Общая информация
Последний выпуск
Скоро в журнале
Архив
Импакт-фактор
Правила для авторов
Загрузить рукопись

Поиск публикаций
Поиск ссылок

RSS
Последний выпуск
Текущие выпуски
Архивные выпуски
Что такое RSS



Изв. РАН. Сер. матем.:
Год:
Том:
Выпуск:
Страница:
Найти






Персональный вход:
Логин:
Пароль:
Запомнить пароль
Войти
Забыли пароль?
Регистрация


Известия Российской академии наук. Серия математическая, 2023, том 87, выпуск 2, страницы 56–68
DOI: https://doi.org/10.4213/im9285
(Mi im9285)
 

Эта публикация цитируется в 1 научной статье (всего в 1 статье)

On the transference principle and Nesterenko's linear independence criterion

O. N. Germanab, N. G. Moshchevitinab

a National Research University Higher School of Economics, Moscow
b Moscow Center for Fundamental and Applied Mathematics
Список литературы:
Аннотация: We consider the problem of simultaneous approximation of real numbers $\theta_1,\dots,\theta_n$ by rationals and the dual problem of approximating zero by the values of the linear form $x_0+\theta_1x_1+\dots+\theta_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation which proves their result. We also derive several previously unknown corollaries. In particular, we show that, together with German's inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and “one half” of the inequalities by Marnat and Moshchevitin. Moreover, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.
Bibliography: 19 titles.
Ключевые слова: Diophantine approximation, Diophantine exponents, transference inequalities, linear independence criterion.
Финансовая поддержка Номер гранта
Российский научный фонд 22-41-05001
Фонд развития теоретической физики и математики "БАЗИС"
This research was supported by the Russian Science Foundation (grant № 22-41-05001), https://rscf.ru/en/project/22-41-05001/.
The first author is a winner of the “Junior Leader” contest conducted by the Foundation for the Advancement of Theoretical Physics and Mathematics Advancement Foundation “BASIS” and would like to thank its sponsors and jury.
Поступило в редакцию: 08.11.2021
Исправленный вариант: 26.07.2022
Англоязычная версия:
Izvestiya: Mathematics, 2023, Volume 87, Issue 2, Pages 252–264
DOI: https://doi.org/10.4213/im9285e
Реферативные базы данных:
Тип публикации: Статья
УДК: 511.4
MSC: 11H06, 11J82
Язык публикации: английский

Dedicated to Yu. V. Nesterenko on the occasion of his 75-th birthday

§ 1. Introduction

This paper originated as a result of studying the prominent paper [1] by Nesterenko, where he proves his linear independence criterion.

Given an integer $n\geqslant 2$, we fix an arbitrary $n$-tuple $\boldsymbol \theta=(\theta_1,\dots,\theta_n)\in\mathbb{R}^n$. It is well known that the problem of approximating $\theta_1,\dots,\theta_n$ simultaneously by rational numbers with equal denominators is related to the problem of approximating zero by the values of the linear form $x_0+\theta_1x_1+\dots+\theta_nx_n$ at non-zero integer points. This relation is performed by the so-called transference principle discovered by Khintchine [2]. He formulated it in terms of Diophantine exponents.

Definition 1. The (regular) Diophantine exponent $\lambda=\lambda(\boldsymbol \theta)$ is defined as the supremum of real numbers $\gamma$ such that the system of inequalities

$$ \begin{equation} \smash{\max_{1\leqslant i\leqslant n}|x_0\theta_i-x_i|\leqslant t^{-\gamma}},\qquad 0<|x_0|\leqslant t, \end{equation} \tag{1.1} $$
admits solutions in $\mathbf x=(x_0,\dots,x_n)\in\mathbb{Z}^{n+1}$ for some arbitrarily large values of $t$.

The respective uniform Diophantine exponent $\widehat\lambda=\widehat \lambda(\boldsymbol\theta)$ is defined as the supremum of real numbers $\gamma$ such that (1.1) admits solutions in $\mathbf x=(x_0,\dots,x_n)\in \mathbb{Z}^{n+1}$ for every $t$ large enough.

Definition 2. The (regular) Diophantine exponent $\omega=\omega(\boldsymbol \theta)$ is defined as the supremum of real numbers $\gamma$ such that the system of inequalities

$$ \begin{equation} \smash[t]{|x_0+\theta_1x_1+\dots+\theta_nx_n|\leqslant t^{-\gamma}},\qquad \smash[b]{0<\max_{1\leqslant i\leqslant n}|x_i|\leqslant t} \end{equation} \tag{1.2} $$
admits solutions in $\mathbf x=(x_0,\dots,x_n)\in\mathbb{Z}^{n+1}$ for some arbitrarily large values of $t$.

The respective uniform Diophantine exponent $\widehat\omega=\widehat \omega(\boldsymbol\theta)$ is defined as the supremum of real numbers $\gamma$ such that (1.2) admits solutions in $\mathbf x=(x_0,\dots,x_n)\in \mathbb{Z}^{n+1}$ for every $t$ large enough.

It follows immediately from Dirichlet’s approximation theorem (or from Minkowski’s convex body theorem) that the Diophantine exponents satisfy the trivial relations $\lambda\geqslant \widehat\lambda\geqslant 1/n$ and $\omega\geqslant \widehat\omega\geqslant n$. It is also known (see [3]) that, unless all the $\theta_i$ are rational, we have a slightly less trivial inequality $\widehat\lambda\leqslant 1$.

Let us give a brief account on the existing non-trivial relations.

The aforementioned Khintchine’s transference principle was published in 1926. It can be formulated as follows:

$$ \begin{equation} \frac{1+\omega}{1+\lambda}\geqslant n,\qquad \frac{1+\omega^{-1}}{1+\lambda^{-1}}\geqslant \frac1n. \end{equation} \tag{1.3} $$
Later on, in 2007–10, Bugeaud and Laurent [4], [5] improved upon (1.3) by showing that
$$ \begin{equation} \frac{1+\omega}{1+\lambda}\geqslant \frac{n-1}{1-\widehat\lambda},\qquad \frac{1+ \omega^{-1}}{1+\lambda^{-1}}\geqslant \frac{1-\widehat\omega^{-1}}{n-1}, \end{equation} \tag{1.4} $$
provided that $1,\theta_1,\dots,\theta_n$ are linearly independent over $\mathbb{Q}$. It can be easily verified that (1.4) implies (1.3), as $\widehat\omega^{-1}\leqslant 1/n\leqslant \widehat\lambda\leqslant 1$.

As for the uniform exponents, Jarník [3] proved in 1938 that the following remarkable identity holds if $n= 2$ and $1,\theta_1,\theta_2$ are linearly independent over $\mathbb{Q}$:

$$ \begin{equation} \widehat\omega^{-1}+\widehat\lambda=1. \end{equation} \tag{1.5} $$
In 2012 it was shown by German [6], [7] that for arbitrary $n\geqslant 2$ we have
$$ \begin{equation} \widehat\omega\geqslant \frac{n-1}{1-\widehat\lambda},\qquad \widehat\lambda\geqslant \frac{1-\widehat\omega^{-1}}{n-1}. \end{equation} \tag{1.6} $$
Clearly, (1.6) turns into Jarník’s identity for $n= 2$.

In 2013 Schmidt and Summerer [8] showed that

$$ \begin{equation} \widehat\omega\leqslant \frac{1+\omega}{1+\lambda},\qquad \widehat\lambda\leqslant \frac{1+ \omega^{-1}}{1+\lambda^{-1}}, \end{equation} \tag{1.7} $$
provided that $1,\theta_1,\dots,\theta_n$ are linearly independent over $\mathbb{Q}$. An alternative proof of their result can be found in [9]. Clearly, (1.6) and (1.7) imply Bugeaud and Laurent’s inequalities (1.4).

Furthermore, in 1950–1954 Jarník [10], [11] proved for $n= 2$ that

$$ \begin{equation} \frac{\omega}{\widehat\omega}\geqslant \widehat\omega-1,\qquad \frac{\lambda}{\widehat \lambda}\geqslant \frac{\widehat\lambda}{1-\widehat\lambda}. \end{equation} \tag{1.8} $$
It is interesting to note that the right-hand sides in (1.8) coincide for $n= 2$, by (1.5). They are both equal to $\widehat\omega\widehat\lambda$ as well as to $(1-\widehat\omega^{-1})/(1-\widehat\lambda)$. We shall mention this fact in Section 3.1.

The inequalities (1.8) were recently generalized to the case of arbitrary $n\geqslant 2$ by Marnat and Moshchevitin [12]. They showed that

$$ \begin{equation} \frac{\omega}{\widehat\omega}\geqslant G_{\mathrm{lin}}(\widehat\omega),\qquad \frac{\lambda}{\widehat\lambda}\geqslant G_{\mathrm{sim}}(\widehat\lambda), \end{equation} \tag{1.9} $$
where $G_{\mathrm{lin}}(\widehat\omega)$ and $G_{\mathrm{sim}}(\widehat \lambda)$ are the largest roots of the polynomials
$$ \begin{equation} f(x)=\widehat\omega^{-1}x^n-x+(1-\widehat\omega^{-1}), \qquad g(x)=(1-\widehat \lambda)x^n-x^{n-1}+\widehat\lambda \end{equation} \tag{1.10} $$
respectively. An alternative proof of the second inequality in (1.9) can be found in [13].

Summing up, we can say that all non-trivial relations currently known follow from (1.6), (1.7), and (1.9). The main purpose of this paper is to present a geometric observation, a rather simple one, which proves (1.7) almost immediately. It appears that this observation also provides a quite simple proof of Nesterenko’s linear independence criterion.

The rest of the paper is organized as follows. In Section 2 we formulate and prove our main result and apply it to prove (1.7). In Section 3 we derive some corollaries of (1.7) concerning lower bounds for the ratios $\omega/\widehat\omega$, $\lambda/\widehat\lambda$ and show that (1.7) yields the weakest of the inequalities (1.9). Moreover, we analyse how (1.6) and (1.7) split Bugeaud and Laurent’s inequalities (1.4), and compare some of our corollaries with a recent result by Schleischitz [14]. Finally, Section 4 is devoted to Nesterenko’s linear independence criterion. We present a rather simple proof of his theorem, which is based on our main geometric observation described in Section 2. We also show that in order to prove the linear independence criterion itself, it suffices to use the first step of Nesterenko’s induction.

§ 2. Empty cylinder lemma

Let us introduce some notation. We write $\ell$ for the one-dimensional subspace generated by $(1,\theta_1,\dots,\theta_n)$ and let $\ell^\perp$ be its orthogonal complement. Given any $\mathbf x\in\mathbb{R}^{n+1}$, we denote the Euclidean distance from $\mathbf x$ to $\ell$ (resp. from $\mathbf x$ to $\ell^\perp$) by $r(\mathbf x)$ (resp. $h(\mathbf x)$). We also write $\langle\,\cdot\,,\cdot\,\rangle$ for the inner product in $\mathbb{R}^{n+1}$.

Our main geometric observation is described by the following statement.

Lemma 1. Let $t$, $\alpha$, $\beta$ be positive real numbers such that $t^\beta>2t^\alpha$. Suppose that $\mathbf v\in\mathbb{Z}^{n+1}$ satisfies

$$ \begin{equation} r(\mathbf v)=t^{\alpha-1-\beta},\qquad h(\mathbf v)=t^\alpha. \end{equation} \tag{2.1} $$
Consider the half-open cylinder
$$ \begin{equation} \mathcal{C}=\mathcal{C}(t,\alpha,\beta)=\{\mathbf x\in\mathbb{R}^{n+1} \mid r(\mathbf x)<t, \,\,t^{-\beta}\leqslant h(\mathbf x)\leqslant t^{-\alpha}-t^{-\beta}\}. \end{equation} \tag{2.2} $$
Then $\mathcal{C}\cap\mathbb{Z}^{n+1}=\varnothing$.

Proof. Consider an arbitrary $\mathbf x\in\mathcal{C}$. Let us show that $0< \langle \mathbf v,\mathbf x\rangle< 1$. To do this, we write $\mathbf y$ for the orthogonal projection of $\mathbf x$ to the two-dimensional subspace $\pi$ spanned by $\ell$ and $\mathbf v$. Then $r(\mathbf y)$ and $h(\mathbf y)$ can be interpreted as the absolute values of the coordinates of $\mathbf y$ in $\pi$ with respect to the coordinate axes $\ell$ and $\ell^\bot\cap \pi$ (see Fig. 1). Hence
$$ \begin{equation*} \langle\mathbf v,\mathbf x\rangle= \langle\mathbf v,\mathbf y\rangle\geqslant h(\mathbf v)h(\mathbf y)-r(\mathbf v)r(\mathbf y)> t^{\alpha-\beta}-t^{\alpha-\beta}=0 \end{equation*} \notag $$
and
$$ \begin{equation*} \langle\mathbf v,\mathbf x\rangle= \langle\mathbf v,\mathbf y\rangle\leqslant h(\mathbf v)h(\mathbf y)+r(\mathbf v)r(\mathbf y)< 1-t^{\alpha-\beta}+t^{\alpha-\beta}=1. \end{equation*} \notag $$
Thus, indeed, $0< \langle\mathbf v,\mathbf x\rangle< 1$, and $\mathbf x$ cannot be an integer point. Lemma 1 is proved.

Now we can prove Schmidt and Summerer’s inequalities (1.7).

Proof of (1.7). We may assume that $\lambda>1/n$ and $\omega> n$. Indeed, by Khintchine’s inequalities (1.3) the equalities $\lambda=1/n$ and $\omega=n$ are equivalent, and if they hold, trivial inequalities yield that $\widehat\lambda=1/n$, $\widehat\omega=n$. With these values of the exponents both inequalities (1.7) are obviously true. So, let us suppose that $\lambda>1/n$ and $\omega> n$.

By the definition of $\lambda$ and $\omega$ we can choose a point $\mathbf v \in\mathbb{Z}^{n+1}$ and a positive real number $\gamma$ to satisfy any of the following two collections of conditions:

1) $h(\mathbf v)$ is arbitrarily large, $r(\mathbf v)=h(\mathbf v)^{-\gamma}$, $\gamma>1/n$, $\gamma$ is arbitrarily close to $\lambda$;

2) $h(\mathbf v)$ is arbitrarily small, $r(\mathbf v)=h(\mathbf v)^{-\gamma}$, $\gamma<1/n$, $\gamma$ is arbitrarily close to $1/\omega$.

We shall refer to the first choice as Case 1, and to the second one as Case 2.

In either case the main part of the argument is basically the same. So, let us choose $\mathbf v$ and $\gamma$ according to one of the two cases. Set $t=t(\mathbf v,\gamma)$ to be the smallest positive real number such that the cylinder

$$ \begin{equation} \mathcal{C}_{\mathbf v}=\{\mathbf x\in\mathbb{R}^{n+1}\mid r(\mathbf x)\leqslant t,\, h(\mathbf x)\leqslant t\cdot h(\mathbf v)^{-1-\gamma}\} \end{equation} \tag{2.3} $$
contains a non-zero integer point. Define also $\alpha=\alpha(\mathbf v,\gamma)$ and $\beta=\beta(\mathbf v,\gamma)$ by the relations
$$ \begin{equation} h(\mathbf v)=t^\alpha,\qquad \alpha=\frac{1+\beta}{1+\gamma}. \end{equation} \tag{2.4} $$
Then $r(\mathbf v)=h(\mathbf v)^{-\gamma}=t^{-\alpha\gamma}=t^{\alpha-1-\beta}$ and $t\cdot h(\mathbf v)^{-1-\gamma}=t^{-\beta}$. Hence $\mathbf v$ satisfies (2.1) and
$$ \begin{equation*} \mathcal{C}_{\mathbf v}=\{\mathbf x\in\mathbb{R}^{n+1}\mid r(\mathbf x)\leqslant t,\, h(\mathbf x)\leqslant t^{-\beta}\}. \end{equation*} \notag $$

In order to fulfill the hypothesis of Lemma 1, it remains to prove that $t^\beta>2t^\alpha$. We recall that in Case 1 $h(\mathbf v)$ is arbitrarily large and $\gamma>1/n$, and in Case 2 $h(\mathbf v)$ is arbitrarily small and $\gamma<1/n$. In both cases by Minkowski’s convex body theorem the volume of $\mathcal{C}_{\mathbf v}$ is bounded (by $2^{n+1}$), and so the product $t^{n+1}h(\mathbf v)^{-1-\gamma}=h(\mathbf v)^{(n+1)/\alpha-(1+\gamma)}$ is also bounded. Hence in Case 1 we have $\alpha\gamma\geqslant (n+1)\gamma/(1+ \gamma)> 1$ and, therefore, $\beta=\alpha+\alpha\gamma-1>\alpha$, whereas in Case 2 we have $\alpha\gamma\leqslant (n+1)\gamma/(1+\gamma)< 1$ and, therefore, $\beta=\alpha+\alpha\gamma-1<\alpha$. Thus, in both cases we have

$$ \begin{equation*} t^\beta= h(\mathbf v)^{\beta/\alpha}> 2h(\mathbf v)= 2t^\alpha. \end{equation*} \notag $$
Having the hypothesis of Lemma 1 fulfilled, we conclude that there are no integer points in the cylinder $\mathcal{C}=\mathcal{C}(t,\alpha,\beta)$ defined by (2.2). At the same time there are some non-zero integer points in $\mathcal{C}_{\mathbf v}$, but all of them lie on the boundary of $\mathcal{C}_{\mathbf v}$. Moreover, every such point satisfies the condition $r(\mathbf x)=t$ since
$$ \begin{equation*} \{\mathbf x\in\mathcal{C}_{\mathbf v}\mid r(\mathbf x)<t,\, h(\mathbf x)=t^{-\beta}\}= \mathcal{C}_{\mathbf v}\cap(\mathcal{C}\cup(-\mathcal{C})) \end{equation*} \notag $$
and $\mathcal{C}\cup(-\mathcal{C})$ is empty. Thus, there is an integer point satisfying
$$ \begin{equation*} r(\mathbf x)=t,\qquad h(\mathbf x)\leqslant t^{-\beta}, \end{equation*} \notag $$
and there are no non-zero integer points satisfying
$$ \begin{equation*} r(\mathbf x)<t,\qquad h(\mathbf x)\leqslant t^{-\alpha}-t^{-\beta}. \end{equation*} \notag $$

Suppose now that Case 1 holds. Then $\beta>\alpha$ and $t$ can be arbitrarily large. Hence $\omega\geqslant \beta$, $\widehat\omega\leqslant \alpha$, and

$$ \begin{equation*} \widehat\omega\leqslant \alpha= \frac{1+\beta}{1+\gamma}\leqslant \frac{1+\omega}{1+ \lambda-\varepsilon} \end{equation*} \notag $$
with $\varepsilon$ positive and arbitrarily small. This gives us the first inequality in (1.7).

Suppose that Case 2 holds. Then $\beta<\alpha$ and $t$ can be arbitrarily small. Hence $\lambda\geqslant 1/\beta$, $\widehat\lambda\leqslant 1/\alpha$, and

$$ \begin{equation*} \widehat\lambda\leqslant \alpha^{-1}= \frac{1+\gamma}{1+\beta}\leqslant \frac{1+\omega^{-1}+ \varepsilon}{1+\lambda^{-1}} \end{equation*} \notag $$
with $\varepsilon$ positive and arbitrarily small. This gives us the second inequality in (1.7).

§ 3. Corollaries of Schmidt and Summerer’s inequalities

3.1. Lower bounds for the ratios $\omega/\widehat\omega$ and $\lambda/\widehat\lambda$

The inequalities (1.7) can be rewritten as

$$ \begin{equation} \frac{\omega}{\widehat\omega}\geqslant \frac{1+\lambda}{1+\omega^{-1}},\qquad \frac{\lambda}{\widehat\lambda}\geqslant \frac{1+\lambda}{1+\omega^{-1}}, \end{equation} \tag{3.1} $$
giving thus new lower bounds in the problem of estimating the ratio between the regular and uniform exponents.

Note that, as in Jarník’s inequalities (1.8) for $n= 2$, the right-hand sides in (3.1) coincide. However, the lower estimate for $\omega/\widehat\omega$ and $\lambda/\widehat\lambda$ provided by (3.1) is a function of $\omega$ and $\lambda$, whereas it has been more traditional to think in this context of functions of $\widehat\omega$ and $\widehat \lambda$. The right-hand side $(1+\lambda)/(1+\omega^{-1})$ can be weakened to $(1-\widehat\omega^{-1})/(1-\widehat\lambda)$, thus generalizing Jarník’s inequalities (1.8) to arbitrary dimension.

Proposition 1. Suppose that $1,\theta_1,\dots,\theta_n$ are linearly independent over $\mathbb{Q}$. Then

$$ \begin{equation} \frac{1+\lambda}{1+\omega^{-1}}\geqslant \frac{1-\widehat\omega^{-1}}{1-\widehat\lambda}. \end{equation} \tag{3.2} $$

Proof. Set $B=(1+\lambda)/(1+\omega^{-1})$. Then (3.1) reads as $\omega\geqslant \widehat\omega B$ and $\lambda\geqslant \widehat\lambda B$. Inserting these inequalities into the definition of $B$, we get
$$ \begin{equation*} B\geqslant \frac{1+\widehat\lambda B}{1+\widehat\omega^{-1} B^{-1}} \end{equation*} \notag $$
or, equivalently,
$$ \begin{equation*} B\geqslant \frac{1-\widehat\omega^{-1}}{1-\widehat\lambda}. \end{equation*} \notag $$
The proposition is proved.

Corollary 1. Suppose that $1,\theta_1,\dots,\theta_n$ are linearly independent over $\mathbb{Q}$. Then

$$ \begin{equation} \frac{\omega}{\widehat\omega}\geqslant \frac{1-\widehat\omega^{-1}}{1-\widehat \lambda},\qquad \frac{\lambda}{\widehat\lambda}\geqslant \frac{1-\widehat \omega^{-1}}{1-\widehat\lambda}. \end{equation} \tag{3.3} $$

The inequalities (3.3) coincide with Jarník’s inequalities (1.8) for $n= 2$. Another generalization of (1.8) to arbitrary dimension is provided by (1.9). An important difference between (1.9) and (3.3) is that the lower bounds $G_{\mathrm{lin}}(\widehat \omega)$ and $G_{\mathrm{sim}}(\widehat\lambda)$ in (1.9) are functions of only one exponent, whereas (3.3) requires both.

If $G_{\mathrm{lin}}(\widehat\omega)=G_{\mathrm{sim}}(\widehat\lambda)$, then (1.9) is equivalent to (3.3), but if $G_{\mathrm{lin}}(\widehat \omega)\neq G_{\mathrm{sim}}(\widehat\lambda)$, then (3.3) is stronger than the weakest of (1.9) and weaker than the strongest.

Proposition 2. Suppose that $1,\theta_1,\dots,\theta_n$ are linearly independent over $\mathbb{Q}$. Then either

$$ \begin{equation} G_{\mathrm{lin}}(\widehat\omega)\leqslant (\widehat\omega\widehat\lambda)^{1/(n-1)}\leqslant \frac{1-\widehat\omega^{-1}}{1-\widehat \lambda}\leqslant G_{\mathrm{sim}}(\widehat\lambda), \end{equation} \tag{3.4} $$
or
$$ \begin{equation} G_{\mathrm{sim}}(\widehat\lambda)\leqslant \frac{1-\widehat\omega^{-1}}{1-\widehat \lambda}\leqslant (\widehat\omega\widehat \lambda)^{1/(n-1)}\leqslant G_{\mathrm{lin}}(\widehat\omega). \end{equation} \tag{3.5} $$
Moreover, if any two of the four quantities under comparison coincide, then so do all of them.

Proof. Let $f(x)$ and $g(x)$ be the polynomials defined by (1.10). Then, given $t\geqslant 1$, we have
$$ \begin{equation*} t>G_{\mathrm{lin}}(\widehat\omega)\ \Longleftrightarrow \ f(t)>0,\qquad t>G_{\mathrm{sim}}(\widehat\lambda)\ \Longleftrightarrow \ g(t)>0. \end{equation*} \notag $$
Since $\widehat\omega\geqslant n$ and $\widehat\lambda\geqslant 1/n$, both $(\widehat \omega\widehat\lambda)^{1/(n-1)}\geqslant 1$ and $(1-\widehat\omega^{-1})/(1-\widehat \lambda)\geqslant 1$. By simple substitution and regrouping it is easily verified that
$$ \begin{equation*} g\biggl(\frac{1-\widehat\omega^{-1}}{1-\widehat\lambda}\biggr)\geqslant 0\ \Longleftrightarrow \ \frac{1-\widehat\omega^{-1}}{1-\widehat\lambda}\leqslant (\widehat\omega\widehat\lambda)^{1/(n-1)}\ \Longleftrightarrow \ f\bigl((\widehat \omega\widehat\lambda)^{1/(n-1)}\bigr)\leqslant 0 \end{equation*} \notag $$
and that the respective equalities are equivalent. This proves the statement of the proposition.

Corollary 2. Denote

$$ \begin{equation*} B(\omega,\lambda)=\frac{1+\lambda}{1+\omega^{-1}},\qquad A(\widehat\omega,\widehat \lambda)=\frac{1-\widehat\omega^{-1}}{1-\widehat\lambda}. \end{equation*} \notag $$
Then, assuming that $1,\theta_1,\dots,\theta_n$ are linearly independent over $\mathbb{Q}$, we have
$$ \begin{equation} \frac{\omega}{\widehat\omega}\geqslant \max\bigl(B(\omega,\lambda),G_{\mathrm{lin}}(\widehat \omega)\bigr),\qquad \frac{\lambda}{\widehat\lambda}\geqslant \max \bigl(B(\omega,\lambda),G_{\mathrm{sim}}(\widehat\lambda)\bigr). \end{equation} \tag{3.6} $$
Moreover, $B(\omega,\lambda)\geqslant A(\widehat\omega,\widehat\lambda)\geqslant \min \bigl(G_{\mathrm{lin}}(\widehat\omega),G_{\mathrm{sim}}(\widehat \lambda)\bigr)$.

Remark 1. In case $n= 2$ all the four quantities in Proposition 2 coincide giving rise to the relation

$$ \begin{equation*} \widehat\omega\widehat\lambda= \frac{1-\widehat\omega^{-1}}{1-\widehat\lambda}, \end{equation*} \notag $$
which recovers Jarník’s identity (1.5). Thus, Proposition 2 provides a generalization of this identity to arbitrary dimension in the following form:
$$ \begin{equation*} (\widehat\omega\widehat\lambda)^{1/(n-1)}= \frac{1-\widehat \omega^{-1}}{1-\widehat\lambda}. \end{equation*} \notag $$
However, this identity holds if and only if $G_{\mathrm{lin}}(\widehat \omega)=G_{\mathrm{sim}}(\widehat\lambda)$, which happens quite rarely. Nevertheless, if a strict inequality holds, then the sign of this inequality determines uniquely which of the quantities $G_{\mathrm{lin}}(\widehat \omega)$, $G_{\mathrm{sim}}(\widehat\lambda)$ is larger.

3.2. The inequalities by Bugeaud and Laurent

The inequalities (1.4) obviously follow from (1.7) and (1.6):

$$ \begin{equation} \frac{1+\omega}{1+\lambda}\geqslant \widehat\omega\geqslant \frac{n-1}{1-\widehat \lambda},\qquad \frac{1+\omega^{-1}}{1+\lambda^{-1}}\geqslant \widehat\lambda\geqslant \frac{1-\widehat\omega^{-1}}{n-1}. \end{equation} \tag{3.7} $$
Bugeaud and Laurent proved (1.4) with the help of the so-called intermediate Diophantine exponents by splitting Khintchine’s inequalities (1.3) into “going up” and “going down” chains of inequalities between consecutive intermediate exponents and improving on the first inequality in each chain with the respective uniform exponent. An analogous splitting of German’s inequalities (1.6) can be found in [6]. It would be interesting to establish a suitable splitting of Schmidt and Summerer’s inequalities (1.7). Such a splitting, combined with the existing splitting of (1.6), would provide a splitting of (1.4), alternative to the one of Bugeaud and Laurent.

As for (3.7) itself, the way it splits (1.4) with the help of the respective fourth exponent gives much additional information in the case when one of the relations (1.4) is an equality. In this case the respective pair of the inequalities (3.7) becomes a pair of equalities and we get a whole range of equalities for the intermediate exponents mentioned above. It is worth mentioning in this context that there is an explicit description (obtained recently by Schleischitz [15]) of triples of exponents for which either the first pair of the inequalities (3.7), or the second one, is a pair of equalities.

3.3. The inequalities by Schleischitz

Analysis of the quantity $(\widehat\omega\widehat\lambda)^{1/(n{-}1)}$ appearing in Proposition 2 provides the following observation.

Proposition 3. Suppose that $1,\theta_1,\dots,\theta_n$ are linearly independent over $\mathbb{Q}$. Then

$$ \begin{equation} \widehat\lambda\leqslant \frac{\omega^{n-1}}{\widehat\omega^n},\qquad \lambda\leqslant \frac{\omega^n}{\widehat\omega^{n+1}}. \end{equation} \tag{3.8} $$

Proof. Applying (3.6) and (3.2) in both cases (3.4) and (3.5), we get
$$ \begin{equation*} \frac{\omega}{\widehat\omega}\geqslant \max\biggl(\frac{1+\lambda}{1+ \omega^{-1}},\,G_{\mathrm{lin}}(\widehat\omega)\biggr)\geqslant \max\biggl(\frac{1-\widehat \omega^{-1}}{1-\widehat\lambda},\,G_{\mathrm{lin}}(\widehat\omega) \biggr)\geqslant (\widehat\omega\widehat\lambda)^{1/(n-1)}. \end{equation*} \notag $$
Hence the first inequality in (3.8) follows immediately.

Furthermore,1 by the first inequality in (1.7) we have

$$ \begin{equation*} \lambda\leqslant \frac{\omega-\widehat\omega+1}{\widehat\omega}= \frac{\omega^n}{\widehat \omega^{n+1}}-f\biggl(\frac{\omega}{\widehat\omega}\biggr), \end{equation*} \notag $$
where $f(x)$ is as in (1.10). Since $f\bigl(\omega/\widehat\omega \bigr)\geqslant 0$ by (1.9), we get the second inequality in (3.8). The proposition is proved.

In [14], Schleischitz proves two inequalities that very much resemble (3.8). They are of the form

$$ \begin{equation} \widehat\omega\leqslant \frac{\lambda^{n-1}}{\widehat\lambda^n}+\psi(\lambda,\widehat \lambda),\qquad \omega\leqslant \frac{\lambda^n}{\widehat\lambda^{n+1}}+ \chi(\lambda,\widehat\lambda) \end{equation} \tag{3.9} $$
with some non-negative $\psi(\lambda,\widehat\lambda)$ and $\chi(\lambda,\widehat \lambda)$ turning into zero only if $\lambda/\widehat\lambda=G_{\mathrm{sim}}(\widehat \lambda)$. One might wonder whether $\psi(\lambda,\widehat\lambda)$ or $\chi(\lambda,\widehat\lambda)$ can be removed from (3.9), so that it would resemble (3.8) closer. However, the answer is negative since it follows from the results of Kleinbock, Moshchevitin, and Weiss [16] that there exist many $\boldsymbol\theta$ with small $\lambda$ and infinite $\widehat\omega$. Their argument can be modified to show that, preserving $\lambda$ small, we can make $\widehat\omega$ finite, but however large. Thus, it is impossible to substitute the extra summands with zero in the general case.

§ 4. Nesterenko’s linear independence criterion

In 1985 Nesterenko published his famous linear independence criterion. His original proof was rather involved. A simpler argument can be found in [17] and [18]; see also [19]. Our Lemma 1 provides an even simpler proof.

As before, let us fix $\boldsymbol\theta=(\theta_1,\dots,\theta_n)\in \mathbb{R}^n$. Let $\ell$, $\ell^\perp$, $r(\,\cdot\,)$, $h(\,\cdot\,)$ be defined as at the beginning of Section 2.

Given any subspace $\mathcal{L}$ of $\mathbb{R}^{n+1}$, we write $\varphi(\mathcal{L})$ for the tangent of the angle between $\ell$ and $\mathcal{L}$. If $\mathcal{L}$ is defined over $\mathbb{Q}$, we write $H(\mathcal{L})$ for its height, i. e., the covolume of the lattice $\mathcal{L}\cap \mathbb{Z}^{n+1}$.

Theorem 1 (Nesterenko, 1985). Let $\alpha$, $\beta$, $c_1$, $c_2$, $\varepsilon$ be positive real numbers, $\beta\geqslant \alpha$. Let $(t_k)_{k\in\mathbb{N}}$ be an increasing sequence of positive real numbers such that

$$ \begin{equation} \lim_{k\to\infty}t_k=\infty,\qquad \limsup_{k\to\infty}\frac{\log(t_{k+1})}{\log(t_k)}=1. \end{equation} \tag{4.1} $$
Suppose that for every integer $k$ large enough there exists $\mathbf x\in \mathbb{Z}^{n+1}$ such that
$$ \begin{equation} r(\mathbf x)\leqslant t_k,\qquad c_1t_k^{-\beta}\leqslant h(\mathbf x)\leqslant c_2t_k^{-\alpha}. \end{equation} \tag{4.2} $$
Then, for every $d$-dimensional subspace $\mathcal{L}$ of $\mathbb{R}^{n+1}$ defined over $\mathbb{Q}$, there is a positive number $c_3=c_3(d,\varepsilon)$ such that
$$ \begin{equation} \varphi(\mathcal{L})\geqslant c_3H(\mathcal{L})^{-\delta-\varepsilon},\qquad \delta=\delta(d)=\frac{1+\beta}{1+\beta-d(1+\beta-\alpha)}, \end{equation} \tag{4.3} $$
provided $d<(1+\beta)/(1+\beta-\alpha)$.

4.1. A simple proof of Nesterenko’s theorem

Let us derive Theorem 1 from Lemma 1. We split our argument into three steps. Suppose that the hypothesis of Theorem 1 holds.

Step 1: Adjusting the hypothesis. Let $\varepsilon'$ be a positive real number sufficiently small with respect to $\varepsilon$. Set $\alpha'=(1-\varepsilon')\alpha$, $\beta'=(1+\varepsilon')\beta$. Let us show that for every real $t$ large enough there exists $\mathbf x\in \mathbb{Z}^{n+1}$ such that

$$ \begin{equation} r(\mathbf x)<t,\qquad t^{-\beta'}<h(\mathbf x)<t^{-\alpha'}-t^{-\beta'}. \end{equation} \tag{4.4} $$
For every large $t$ choose $k$ so that $t_k<t\leqslant t_{k+1}$. It follows from (4.1) that $t_{k+1}<t_k^{1+\varepsilon'/2}$ if $k$ is large enough. Hence for $\mathbf x\in\mathbb{Z}^{n+1}$ satisfying (4.2) we have $r(\mathbf x)\leqslant t_k< t$ and
$$ \begin{equation*} t^{-\beta'}< c_1t_k^{-\beta}\leqslant h(\mathbf x)\leqslant c_2t_k^{-\alpha}< c_2t^{-\alpha(1-\varepsilon'/2)}< t^{-\alpha'}-t^{-\beta'} \end{equation*} \notag $$
if $k$ and $t$ are large enough. Thus (4.4) holds.

Step 2: Proof for $d= 1$. Let $\mathcal{L}$ be the one-dimensional subspace generated by a primitive vector $\mathbf v\in\mathbb{Z}^{n+1}$ (primitive means that the coordinates of $\mathbf v$ are coprime). Then $\varphi(\mathcal{L})=r(\mathbf v)/h(\mathbf v)$ and $H(\mathcal{L})=\sqrt{r(\mathbf v)^2+h(\mathbf v)^2}$. Let us suppose that $h(\mathbf v)$ is large and that

$$ \begin{equation*} r(\mathbf v)<h(\mathbf v)^{1-\delta'},\qquad \delta'=\frac{1+\beta'}{\alpha'}. \end{equation*} \notag $$
Set $t=h(\mathbf v)^{1/\alpha'}$. Then $h(\mathbf v)=t^{\alpha'}$ and $r(\mathbf v)=t^{\alpha'-1-\beta''}$ with some $\beta''>\beta'$. By Lemma 1 there are no integer points $\mathbf x$ such that
$$ \begin{equation*} r(\mathbf x)<t,\qquad t^{-\beta''}\leqslant h(\mathbf x)\leqslant t^{-\alpha'}-t^{-\beta''}. \end{equation*} \notag $$
This contradicts the existence of an integer point $\mathbf x$ satisfying (4.4) if $t$ is large enough. Thus, if $h(\mathbf v)$ is large enough, we have
$$ \begin{equation} r(\mathbf v)\geqslant h(\mathbf v)^{1-\delta'}. \end{equation} \tag{4.5} $$
This yields (4.3) for $d= 1$ since $\varepsilon'$ can be arbitrarily small.

Step 3: Proof for $d\geqslant 2$. Suppose that $2\leqslant d<(1+\beta')/(1+\beta'-\alpha')$. Then, in particular,

$$ \begin{equation} \beta'>\alpha'>1,\qquad \smash[t]{1<\delta'<\frac{d}{d-1}}. \end{equation} \tag{4.6} $$
Given a $d$-dimensional subspace $\mathcal{L}$ defined over $\mathbb{Q}$, let $\ell'$ be the one-dimensional subspace generated by the point in $\mathcal{L}$ closest to the point $(1,\theta_1,\dots,\theta_n)$. Then the angle between $\ell$ and $\mathcal{L}$ equals the angle between $\ell$ and $\ell'$. Furthermore, given any $\mathbf x\in\mathcal{L}$, we write $r'(\mathbf x)$ and $h'(\mathbf x)$ for the Euclidean distances from $\mathbf x$ to $\ell'$ and $\mathcal{L}\cap\ell^\perp$ respectively.

By Minkowski’s convex body theorem, for each positive $t$, there is a non-zero integer point $\mathbf x$ in $\mathcal{L}$ such that

$$ \begin{equation} h'(\mathbf x)\leqslant t,\qquad \smash[t]{r'(\mathbf x)\leqslant c_d \biggl(\frac{H(\mathcal{L})}{t}\biggr)^{1/(d-1)}} \end{equation} \tag{4.7} $$
with some positive number $c_d$ depending only on $d$. Set
$$ \begin{equation} t_0=\bigl((4c_d)^{d-1}H(\mathcal{L})\bigr)^{1/(d-(d-1)\delta')}. \end{equation} \tag{4.8} $$
Choose a non-zero integer point $\mathbf v$ in $\mathcal{L}$ (see Fig. 2), so that, in accordance with (4.7),
$$ \begin{equation} h'(\mathbf v)\leqslant t_0,\qquad r'(\mathbf v)\leqslant c_d \biggl(\frac{H(\mathcal{L})}{t_0}\biggr)^{1/(d-1)}. \end{equation} \tag{4.9} $$
It follows from (4.6) that $t_0/H(\mathcal{L})\to \infty$ and $r'(\mathbf v)\to 0$ as $H(\mathcal{L})\to \infty$. We may also assume that $\varphi(\mathcal{L})\to 0$ as $H(\mathcal{L})\to \infty$, neglecting most of subspaces. Then $h'(\mathbf v)\to \infty$ as $H(\mathcal{L})\to \infty$ and
$$ \begin{equation} h(\mathbf v)\leqslant h'(\mathbf v)< 2h(\mathbf v). \end{equation} \tag{4.10} $$
Having thus chosen $\mathbf v$ and assuming $H(\mathcal{L})$ to be large enough, we apply (4.5), (4.6), (4.8)(4.10) and estimate $\varphi(\mathcal{L})$ as follows:
$$ \begin{equation*} \begin{aligned} \, \varphi(\mathcal{L}) &\geqslant \frac{r(\mathbf v)}{h(\mathbf v)}-\frac{r'(\mathbf v)}{h(\mathbf v)}> h(\mathbf v)^{-\delta'}-2\frac{r'(\mathbf v)}{h'(\mathbf v)}\geqslant h'(\mathbf v)^{-\delta'}-\frac{2c_d(H(\mathcal{L}))^{1/(d-1)}}{t_0^{1/(d-1)}h'(\mathbf v)} \\ &\geqslant h'(\mathbf v)^{-\delta'}\biggl(1-\frac{2c_d(H(\mathcal{L}))^{1/(d-1)}}{t_0^{1/(d-1)}h'(\mathbf v)^{1-\delta'}}\biggr) \geqslant t_0^{-\delta'}\biggl(1-\frac{2c_d(H(\mathcal{L}))^{1/(d-1)}}{t_0^{-\delta'+d/(d-1)}}\biggr) \\ &=\frac{t_0^{-\delta'}}2= \frac12 \bigl((4c_d)^{d-1}H(\mathcal{L})\bigr)^{-\delta'/(d-(d-1)\delta')}. \end{aligned} \end{equation*} \notag $$
Taking into account that $\delta'/(d-(d-1)\delta')=(1+\beta')/(1+\beta'-d(1+ \beta'-\alpha'))$, we get (4.3), as $\varepsilon'$ can be arbitrarily small.

4.2. Concerning the linear independence criterion itself

Theorem 1 yields the estimate

$$ \begin{equation} \smash[t]{\dim_{\mathbb{Q}}(\mathbb{Q}+\mathbb{Q}\theta_1+\dots+\mathbb{Q}\theta_n)\geqslant \frac{1+\beta}{1+\beta-\alpha}}. \end{equation} \tag{4.11} $$
However, this estimate easily follows from (4.3) with $d= 1$, i. e., we may actually avoid Step 3 if (4.11) is our aim. Indeed, if $\ell$ is contained in a $d$-dimensional subspace $\mathcal{L}$ defined over $\mathbb{Q}$, then Dirichlet’s approximation theorem guarantees that there are infinitely many integer points $\mathbf x$ in $\mathcal{L}$ such that
$$ \begin{equation*} r(\mathbf x)\leqslant c(d,H(\mathcal{L}))h(\mathbf x)^{-1/(d-1)} \end{equation*} \notag $$
with some positive $c(d,H(\mathcal{L}))$. But if $d<(1+\beta')/(1+\beta'-\alpha')$, we have $\delta'<d/(d-1)$, where $\alpha'$, $\beta'$, $\delta'$ are as in Section 4.1, and Step 2 tells us that the opposite holds for every integer point $\mathbf x$ with $h(\mathbf x)$ large enough:
$$ \begin{equation*} r(\mathbf x)\geqslant h(\mathbf x)^{1-\delta'}> c(d,H(\mathcal{L}))h(\mathbf x)^{-1/(d-1)}. \end{equation*} \notag $$
Thus, there is no such $\mathcal{L}$, and we get (4.11).

4.3. A slight refinement in the case of linear independence

Theorem 1 is mainly applied to prove linear independence of the numbers $1,\theta_1,\dots,\theta_n$ over $\mathbb{Q}$. But if it is already known that they are linearly independent, the estimate (4.3) in Theorem 1 can be improved for $d=n$. In this case, the restriction on $\alpha$ and $\beta$ in Theorem 1 implies that $\alpha>n-1$. For $\alpha> n$ the following statement provides a stronger inequality than (4.3). It also provides a lower bound for $\omega=\omega(\boldsymbol\theta)$.

Proposition 4. Under the hypothesis of Theorem 1, suppose additionally that $1,\theta_1,\dots,\theta_n$ are linearly independent over $\mathbb{Q}$ and that

$$ \begin{equation} \frac{(\alpha-1)(1+\beta)}{\alpha(1+\beta-\alpha)}>n-1. \end{equation} \tag{4.12} $$
Let $\mathcal{L}$ be an $n$-dimensional subspace of $\mathbb{R}^{n+1}$ defined over $\mathbb{Q}$. Then
$$ \begin{equation} \omega\leqslant \frac{(n-1)\alpha(1+\beta-\alpha)}{(\alpha-1)(1+\beta)-(n-1)\alpha(1+ \beta-\alpha)} \end{equation} \tag{4.13} $$
and
$$ \begin{equation} \varphi(\mathcal{L})\geqslant cH(\mathcal{L})^{-\delta-\varepsilon},\qquad \delta=\frac{(\alpha-1)(1+\beta)}{(\alpha-1)(1+\beta)-(n-1)\alpha(1+\beta-\alpha)}, \end{equation} \tag{4.14} $$
with some positive $c$ depending only on $n$ and $\varepsilon$.

Proof. It follows from (4.4) and (4.5) of Steps 1 and 2 that
$$ \begin{equation*} \widehat\omega\geqslant \alpha,\qquad \lambda\leqslant \frac{1+\beta}{\alpha}-1. \end{equation*} \notag $$
By (1.4) we also have
$$ \begin{equation*} \frac{1+\omega^{-1}}{1+\lambda^{-1}}\geqslant \frac{1-\widehat\omega^{-1}}{n-1}. \end{equation*} \notag $$
Hence
$$ \begin{equation*} \omega\leqslant \frac{n-1}{(1-\widehat\omega^{-1})(1+\lambda^{-1})-(n-1)}\leqslant \frac{(n-1)\alpha(1+\beta-\alpha)}{(\alpha-1)(1+\beta)-(n-1)\alpha(1+ \beta-\alpha)}. \end{equation*} \notag $$
Here we have used the fact that (4.12) implies positiveness of the denominators in both fractions above. This proves (4.13).

Furthermore, let $\mathbf w$ be the integer normal of $\mathcal{L}$, i. e., the unique (up to sign) non-zero primitive integer vector orthogonal to $\mathcal{L}$. Bounds for $\varphi(\mathcal{L})$ and $\omega$ are obviously related, as $H(\mathcal{L})=\sqrt{h(\mathbf w)^2+r(\mathbf w)^2}$ and the angle between $\mathcal{L}$ and $\ell$ is equal to the angle between $\mathbf w$ and $\ell^\perp$. Hence

$$ \begin{equation*} \varphi(\mathcal{L})= \frac{h(\mathbf w)}{r(\mathbf w)}\geqslant cH(\mathcal{L})^{-\omega-1-\varepsilon}\geqslant cH(\mathcal{L})^{-\delta-\varepsilon} \end{equation*} \notag $$
with some positive $c$ depending only on $n$ and $\varepsilon$. Thus, (4.14) is also proved. The proposition is proved.

Acknowledgements

The authors are grateful to Johannes Schleischitz for a series of useful comments, especially, for drawing the authors’ attention to Schmidt and Summerer’s paper [8] once again.

The first author is a winner of the “Junior Leader” contest conducted by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” and would like to thank its sponsors and jury.

Bibliography

1. Ю. В. Нестеренко, “О линейной независимости чисел”, Вестн. Моск. ун-та. Сер. 1. Матем., мех., 1985, № 1, 46–49  mathnet  mathscinet  zmath; англ. пер.: Yu. V. Nesterenko, “On the linear independence of numbers”, Moscow Univ. Math. Bull., 40:1 (1985), 69–74
2. A. Khintchine, “Über eine Klasse linearer Diophantischer Approximationen”, Rend. Circ. Mat. Palermo, 50 (1926), 170–195  crossref  zmath
3. V. Jarník, “Zum Khintchineschen “Übertragungssatz””, Trav. Inst. Math. Tbilissi, 3 (1938), 193–216  zmath
4. Y. Bugeaud, M. Laurent, “Exponents of Diophantine approximation”, Diophantine geometry, CRM Series, 4, Ed. Norm., Pisa, 2007, 101–121  mathscinet  zmath
5. Y. Bugeaud, M. Laurent, “On transfer inequalities in Diophantine approximation. II”, Math. Z., 265:2 (2010), 249–262  crossref  mathscinet  zmath
6. O. N. German, “Intermediate Diophantine exponents and parametric geometry of numbers”, Acta Arith., 154:1 (2012), 79–101  crossref  mathscinet  zmath
7. O. N. German, “On Diophantine exponents and Khintchine's transference principle”, Mosc. J. Comb. Number Theory, 2:2 (2012), 22–51  mathscinet  zmath
8. W. M. Schmidt, L. Summerer, “Diophantine approximation and parametric geometry of numbers”, Monatsh. Math., 169:1 (2013), 51–104  crossref  mathscinet  zmath
9. O. N. German, N. G. Moshchevitin, “A simple proof of Schmidt–Summerer's inequality”, Monatsh. Math., 170:3-4 (2013), 361–370  crossref  mathscinet  zmath
10. V. Jarník, “Une remarque sur les approximations diophantiennes linéaires”, Acta Sci. Math. (Szeged), 12 B (1950), 82–86  mathscinet  zmath
11. V. Jarník, “Contribution à la théorie des approximations diophantiennes linéaires et homogènes”, Czechoslovak Math. J., 4:79 (1954), 330–353  mathscinet  zmath
12. A. Marnat, N. G. Moshchevitin, “An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation”, Mathematika, 66:3 (2020), 818–854  crossref  mathscinet
13. Ngoc Ai Van Nguyen, A. Poëls, D. Roy, “A transference principle for simultaneous rational approximation”, J. Théor. Nombres Bordeaux, 32:2 (2020), 387–402  crossref  mathscinet  zmath
14. J. Schleischitz, “On geometry of numbers and uniform rational approximation to the Veronese curve”, Period. Math. Hung., 83:2 (2021), 233–249  crossref  mathscinet  zmath; arXiv: 1911.03215
15. J. Schleischitz, “Optimality of two inequalities for exponents of Diophantine approximation”, J. Number Theory, 244 (2023), 169–203  crossref  mathscinet; arXiv: 2102.03154
16. D. Kleinbock, N. Moshchevitin, B. Weiss, “Singular vectors on manifolds and fractals”, Israel J. Math., 245:2 (2021), 589–613  crossref  mathscinet  zmath; arXiv: 1912.13070
17. S. Fischler, W. Zudilin, “A refinement of Nesterenko's linear independence criterion with applications to zeta values”, Math. Ann., 347:4 (2010), 739–763  crossref  mathscinet  zmath
18. A. Chantanasiri, “On the criteria for linear independence of Nesterenko, Fischler and Zudilin”, Chamchuri J. Math., 2:1 (2010), 31–46  mathscinet  zmath
19. S. Fischler, T. Rivoal, “Irrationality exponent and rational approximations with prescribed growth”, Proc. Amer. Math. Soc., 138:3 (2010), 799–808  crossref  mathscinet  zmath

Образец цитирования: O. N. German, N. G. Moshchevitin, “On the transference principle and Nesterenko's linear independence criterion”, Изв. РАН. Сер. матем., 87:2 (2023), 56–68; Izv. Math., 87:2 (2023), 252–264
Цитирование в формате AMSBIB
\RBibitem{GerMos23}
\by O.~N.~German, N.~G.~Moshchevitin
\paper On the transference principle and Nesterenko's linear independence criterion
\jour Изв. РАН. Сер. матем.
\yr 2023
\vol 87
\issue 2
\pages 56--68
\mathnet{http://mi.mathnet.ru/im9285}
\crossref{https://doi.org/10.4213/im9285}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4634760}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2023IzMat..87..252G}
\transl
\jour Izv. Math.
\yr 2023
\vol 87
\issue 2
\pages 252--264
\crossref{https://doi.org/10.4213/im9285e}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001054286300002}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85168164155}
Образцы ссылок на эту страницу:
  • https://www.mathnet.ru/rus/im9285
  • https://doi.org/10.4213/im9285
  • https://www.mathnet.ru/rus/im/v87/i2/p56
  • Эта публикация цитируется в следующих 1 статьяx:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
    Статистика просмотров:
    Страница аннотации:395
    PDF русской версии:33
    PDF английской версии:82
    HTML русской версии:198
    HTML английской версии:129
    Список литературы:32
    Первая страница:12
     
      Обратная связь:
     Пользовательское соглашение  Регистрация посетителей портала  Логотипы © Математический институт им. В. А. Стеклова РАН, 2024