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

In this work we proof the following theorem which is, in addition to someother lemmas, our main result:\noindent $\mathbf{\text{theorem}}$. Let$\ X={ ( x_{1}$\text{, }$%t_{1}) $\text{, }$( x_{2}$\text{, }$t_{2}) \text{, …, }(x_{n}$\text{, }$t_{n})} $beafinitepartof$$\mathbb{R}$\times $\mathbb{R}$^{\ast +}$,thenthereexistafinitepart$R$of$$\mathbb{R}$%^{\ast +}$suchthatforall$\varepsilon > 0$thereexists$r\in R$suchthatif$0 < \varepsilon \leq r$thenthereexistrationalnumbers$( \dfrac{p_{i}}{q}) _{i=1,2,…,n}$ 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 $\epsilon q\le t\mathrm{\_}i$ for $$ is equivalent to \varepsilon q\leq t=\underset{i=1,2,...,n}{Min%} $(t\mathrm{\_}i)$.
Also, we have () for all $\epsilon$verifying $0 < \varepsilon \leq \varepsilon _{0}=\min R$.The previous theorem is the classical equivalent of the following one whichis formulated in the context of the nonstandard analysis ($[2]$%, $[5]$, $[6]$, $[8]$).\noindent $\mathbf{\text{theorem.}}$For every positive infinitesimal real $\epsilon$, there exists an unlimited integer $q$ depending only of $\epsilon$, such that $\forall ^{st}x \in $\mathbb{R}$
\exists p_{x} \in $\mathbb{Z}$$ $:{ {ccc}x & = & \dfrac{p_{x}}{q}+\varepsilon \phi \varepsilon q & \cong & 0.$ 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.