Diophantine approximation with improvement of the simultaneous control of the error and of the denominator
Résumé
In this work we proof the following theorem which is, in addition to some
other lemmas, our main result:
\noindent \textbf{theorem}. Let$\ X=\left\{ \left( x_{1}\text{, }%
t_{1}\right) \text{, }\left( x_{2}\text{, }t_{2}\right) \text{, ..., }\left(
x_{n}\text{, }t_{n}\right) \right\} $ be a finite part of $\mathbb{R}\times
\mathbb{R}^{\ast +}$, then there exist a finite part $R$ of $\mathbb{R}%
^{\ast +}$ such that for all $\varepsilon >0$ there exists $r\in R$ such
that if $0<\varepsilon \leq r$ then there exist rational numbers $\left(
\dfrac{p_{i}}{q}\right) _{i=1,2,...,n}$ such that:
\begin{equation}
\left\{
\begin{array}{c}
\left\vert x_{i}-\dfrac{p_{i}}{q}\right\vert \leq \varepsilon t_{i} \\
\varepsilon q\leq t_{i}%
\end{array}%
\right\vert \text{, }i=1,2,...,n\text{.} \tag{*}
\end{equation}
\noindent It is clear that the condition $\varepsilon q\leq t_{i}$ for $%
i=1,2,...,n$ is equivalent to $\varepsilon q\leq t=\underset{i=1,2,...,n}{Min%
}$ $\left( t_{i}\right) $.\ Also, we have (*) for all $\varepsilon $
verifying $0<\varepsilon \leq \varepsilon _{0}=\min R$.
The previous theorem is the classical equivalent of the following one which
is formulated in the context of the nonstandard analysis ($\left[ 2\right] $%
, $\left[ 5\right] $, $\left[ 6\right] $, $\left[ 8\right] $).
\noindent \textbf{theorem. }For every positive infinitesimal real $%
\varepsilon $, there exists an unlimited integer $q$\ depending only of $%
\varepsilon $, such that\textit{\ }$\forall ^{st}x\in \mathbb{R}$ $\exists $
$p_{x}\in \mathbb{Z}$:
\begin{equation*}
\left\{
\begin{array}{ccc}
x & = & \dfrac{p_{x}}{q}+\varepsilon \phi \\
\varepsilon q & \cong & 0%
\end{array}%
\text{ .}\right.
\end{equation*}
For this reason,\ to prove the nonstandard version of the main result and to
get its classical version\ we place ourselves in the context of the
nonstandard analysis.
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)
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