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arXiv · 2112.00367

Best Approximations by $\mathcal{F}_{p^l}$-Continued Fractions

S. Kushwaha·R. Sarma
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Abstract

In this article, for a certain subset $\mathcal{X}$ of the extended set of rational numbers, we introduce the notion of {\it best $\mathcal{X}$-approximations} of a real number. The notion of best $\mathcal{X}$-approximation is analogous to that of best rational approximation. We explore these approximations with the help of $\mathcal{F}_{p^l}$-continued fractions, where $p$ is a prime and $l\in\mathbb{N}$, we show that the convergents of the $\mathcal{F}_{p^l}$-continued fraction expansion of a real number $x$ satisfying certain maximal conditions are exactly the best $\mathcal{F}_{p^l}$-approximations of $x$.

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BibTeXRIS

S. Kushwaha, R. Sarma. 2021-12-01. Best Approximations by $\mathcal{F}_{p^l}$-Continued Fractions. https://arxiv.org/abs/2112.00367

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