Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuity of the continued fraction mapping revisited

Min Woong Ahn

TL;DR

The paper addresses the continuity behavior of the continued fraction mapping across the unit interval by embedding it into a topological framework. It introduces an extended encoding $f$ and a limit map $\varphi$ on a carefully constructed sequence space $\Sigma$, and then builds a robust metric and quotient-topology structure to analyze continuity. The main result is that $f$ is continuous at all irrationals and at the endpoints $0$ and $1$, while at every rational in $(0,1)$ the map is one-sidedly continuous, unifying the treatment of rational and irrational points. This topological viewpoint not only clarifies the CF mapping's behavior but also suggests streamlined approaches to related results, such as the continuity properties of the error-sum function discussed in prior work like RP00.$

Abstract

The continued fraction mapping maps a number in the interval $[0,1)$ to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space $\mathbb{R}$, the continued fraction mapping is a homeomorphism onto the product space $\mathbb{N}^{\mathbb{N}}$, where $\mathbb{N}$ is a discrete space. In this short note, we examine the continuity of the continued fraction mapping, addressing both irrational and rational points of the unit interval.

Continuity of the continued fraction mapping revisited

TL;DR

The paper addresses the continuity behavior of the continued fraction mapping across the unit interval by embedding it into a topological framework. It introduces an extended encoding and a limit map on a carefully constructed sequence space , and then builds a robust metric and quotient-topology structure to analyze continuity. The main result is that is continuous at all irrationals and at the endpoints and , while at every rational in the map is one-sidedly continuous, unifying the treatment of rational and irrational points. This topological viewpoint not only clarifies the CF mapping's behavior but also suggests streamlined approaches to related results, such as the continuity properties of the error-sum function discussed in prior work like RP00.$

Abstract

The continued fraction mapping maps a number in the interval to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space , the continued fraction mapping is a homeomorphism onto the product space , where is a discrete space. In this short note, we examine the continuity of the continued fraction mapping, addressing both irrational and rational points of the unit interval.
Paper Structure (3 sections, 18 theorems, 25 equations)

This paper contains 3 sections, 18 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of continued fractions
  3. Main results and proofs

Key Result

Lemma 2.1

For any $\sigma \in \mathbb{N}_\infty^{\mathbb{N}} \setminus \Sigma_\infty$, there exists an $n \in \mathbb{N}$ such that $\widetilde{\varphi}(\sigma) = \widetilde{\varphi}_n (\sigma)$.

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Proposition 2.2: See IK02
  • Proposition 2.3: See HW08IK02Khi97
  • Proposition 2.4
  • Proposition 3.1: Ahn23
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 20 more