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

TL;DR

This paper establishes a new theorem in Diophantine approximation, improving simultaneous control over approximation error and denominator size, using nonstandard analysis techniques to extend classical results.

Contribution

The paper introduces a novel theorem that enhances simultaneous Diophantine approximation by incorporating nonstandard analysis methods for better control of error and denominator.

Findings

Existence of finite sets of denominators controlling approximation error

Extension of classical Diophantine approximation results using nonstandard analysis

Improved bounds on approximation error relative to denominator size

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.