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

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 $x$.