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Collatz Representations With Bounded Partial Quotients

Franciszek Kobus

TL;DR

The paper develops Collatz representations for rationals with odd numerator and denominator and proves a density/fractal structure for the subset whose representations use only $1$'s and $2$'s. It defines a Collatz map $C$ from $Q^{odd}$ to infinite sequences and shows a bijection between eventually periodic sequences and rationals with eventually periodic representations. By introducing the point set $P(A)$ and the family of graphs $P_n$, it uncovers self-similarity and proves that these bounded-quotient approximants densely approximate all $z\le -1$ via $C^{-1}([A,(1)])$, highlighting a fractal, tree-like organization of representations. The results yield constructive approximation methods, discuss limitations (not all infinite sequences are represented), and connect to reformulations of the Weak Collatz Conjecture, underscoring the interplay between number representation, fractal geometry, and dynamical systems.

Abstract

We define Collatz representations for a subset of rational numbers and prove that each real number \( x \notin (-1,1) \) can be approximated arbitrarily well by rational numbers which have only \( 2 \)'s and \( 1 \)'s in their Collatz representation.

Collatz Representations With Bounded Partial Quotients

TL;DR

The paper develops Collatz representations for rationals with odd numerator and denominator and proves a density/fractal structure for the subset whose representations use only 's and 's. It defines a Collatz map from to infinite sequences and shows a bijection between eventually periodic sequences and rationals with eventually periodic representations. By introducing the point set and the family of graphs , it uncovers self-similarity and proves that these bounded-quotient approximants densely approximate all via , highlighting a fractal, tree-like organization of representations. The results yield constructive approximation methods, discuss limitations (not all infinite sequences are represented), and connect to reformulations of the Weak Collatz Conjecture, underscoring the interplay between number representation, fractal geometry, and dynamical systems.

Abstract

We define Collatz representations for a subset of rational numbers and prove that each real number \( x \notin (-1,1) \) can be approximated arbitrarily well by rational numbers which have only 's and 's in their Collatz representation.

Paper Structure

This paper contains 4 sections, 7 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Collatz representation
  3. Bounded partial quotients
  4. Further Results

Key Result

Lemma 2.10

For any eventually periodic sequence $[a_1,...,a_m,(b_1,...,b_n)] \in S^{inf}$ there exists precisely one $a \in \mathbb{Q}^{odd}$ such that $C(a)=[a_1,...,a_m,(b_1,...,b_n)]$. We denote it as $C^{-1}([a_1,...,a_m,(b_1,...,b_n)])$.

Figures (3)

  • Figure 1: Finding approximation, starting with $A=\emptyset$.
  • Figure :
  • Figure :

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Lemma 2.10
  • ...and 19 more