Malaysian Journal of Mathematical Sciences, May 2015, Vol. 9, No. 2
Non-classical Study on the Simultaneous Rational Approximation
Bellaouar Djamel and Boudaoud Abdelmadjid
Corresponding Email: bellaouardj@yahoo.fr
Received date: -
Accepted date: -
Abstract:
This study is placed in the framework of Internal Set Theory (Nelson, 1977). Real numbers $(\xi_i)_{i=1,2,...,k}$ are called simultaneously approximable in the infinitesimal sense, if for every positive infinitesimal $\varepsilon$, there exist rational numbers $(\frac{p_i}{q})_{i=1,2,...,k}$ such that $$\left \{ \begin{matrix}
\xi_i=\frac{p_{i}}{q}+\varepsilon £_{i}; \: i=1,2,...,k \\
\varepsilon q \simeq 0
\end{matrix} \right.$$
where $(£_{i})_{i=1,2,...,k}$ are limited numbers. Let $(\xi_{1},\xi_{2},...,\xi_{\omega})$ be a system of reals, with $\omega$ unlimited. In this paper, we will give a necessary condition for which $(\xi_i)_{i=1,2,...,\omega}$ are simultaneously approximable in the infinitesimal sense. The converse of this condition is also discussed.
Keywords: Internal set theory, simultaneous rational approximation, infinitesimal sense
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