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

TL;DR

This paper introduces the concept of best $\

Contribution

It demonstrates that convergents of $\\mathcal{F}_{p^l}$-continued fractions are exactly the best approximations under certain conditions, extending rational approximation theory.

Findings

Convergents of $\\mathcal{F}_{p^l}$-continued fractions are optimal approximations.

The paper defines and characterizes best $\\mathcal{X}$-approximations for a subset of rationals.

Establishes a link between continued fraction convergents and best approximation properties.

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 $x$.