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Some Aspects of Higher Continued Fractions

Etan Basser, Nicholas Ovenhouse, Anuj Sakarda

Abstract

We investigate some properties of the higher continued fractions defined recently by Musiker, Ovenhouse, Schiffler, and Zhang. We prove that the maps defining the higher continued fractions are increasing continuous functions on the positive real numbers. We also investigate some asymptotics of these maps.

Some Aspects of Higher Continued Fractions

Abstract

We investigate some properties of the higher continued fractions defined recently by Musiker, Ovenhouse, Schiffler, and Zhang. We prove that the maps defining the higher continued fractions are increasing continuous functions on the positive real numbers. We also investigate some asymptotics of these maps.
Paper Structure (5 sections, 18 theorems, 54 equations)

This paper contains 5 sections, 18 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Monotonicity
  3. Continuity
  4. Extension of Higher Continued Fractions to $\mathbb{R}$
  5. Asymptotics

Key Result

Theorem 1

(mosz) If $x = [a_1,a_2,\dots]$ is the continued fraction for an irrational number, and if $x_n = [a_1,\dots,a_n]$ are its rational convergents, then the sequence $r_{i,m}(x_n)$ converges.

Theorems & Definitions (46)

  • Definition 1
  • Remark 1
  • Example 1
  • Example 2
  • Theorem 1
  • Definition 2
  • Example 3
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 36 more