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

Abdelmadjid Boudaoud

Published 2016-05-09Version 1

In this work we proof the following theorem which is, in addition to someother lemmas, our main result:\noindent \textbf{theorem}. Let$\ X=\{ ( x\_{1}\text{, }%t\_{1}) \text{, }( x\_{2}\text{, }t\_{2}) \text{, ..., }(x\_{n}\text{, }t\_{n})\} $ 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 $( \dfrac{p\_{i}}{q}) \_{i=1,2,...,n}$ such that:\{c}| x\_{i}-\dfrac{p\_{i}}{q}| \leq \varepsilon t\_{i} \varepsilon q\leq t\_{i}|\text{, }i=1,2,...,n\text{.} \tag{*}\noindent It is clear that the condition $\varepsilon q\leq t\_{i}$ for $%i=1,2,...,n$ is equivalent to $\varepsilon q\leq t=\underset{i=1,2,...,n}{Min%}$ $( t\_{i}) $.\ 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 whichis formulated in the context of the nonstandard analysis ($[ 2] $%, $[ 5] $, $[ 6] $, $[ 8] $).\noindent \textbf{theorem. }For every positive infinitesimal real $\varepsilon$, there exists an unlimited integer $q$ depending only of $\varepsilon $, such that $\forall ^{st}x \in \mathbb{R}\ \exists p_{x} \in \mathbb{Z}$ $:\{ \{ccc}x & = \& \dfrac{p_{x}}{q}+\varepsilon \phi \varepsilon q \& \cong \& 0.$ 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.