Diophantine approximation with improvement of the simultaneous control of the error and of the denominator

Abstract : 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.
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Submitted on : Saturday, May 7, 2016 - 6:47:41 PM
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  • HAL Id : hal-01312603, version 1
  • ARXIV : 1605.02538

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Abdelmadjid Boudaoud. Diophantine approximation with improvement of the simultaneous control of the error and of the denominator. 2016. 〈hal-01312603〉

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