Diophantine approximation with improvement of the simultaneous control of the error and of the denominator (1605.02538v1)

Abstract: 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.