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The number system in rational base $3/2$ and the $3x+1$ problem

Shalom Eliahou, Jean-Louis Verger-Gaugry

TL;DR

This work uncovers a deep link between rational-base representations in base $3/2$ and the Collatz $3x+1$ problem. It defines the representation $n=\frac12(a_k(3/2)^k+...+a_0)$ with digits in $\{0,1,2\}$ and studies the associated language $L$ of admissible words, the odometer for constructing $n+1$ from $n$, and the interplay with the Collatz map $T$. The authors introduce saturated words and the related map $U$, showing that sat$(k)=U^{(k)}(0)$ and that the growth and prefix structure of saturated words encode Collatz-like dynamics. Open questions connect the behavior of $U$ to the Collatz conjecture, propose a normality conjecture for the limiting saturated word, and seek a deeper understanding of $T$ within the rational-base framework, offering a new arithmetic-dynamical perspective on both topics.

Abstract

The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz $3x+1$ function. Our purpose in this note is to expose these links and motivate further research into them.

The number system in rational base $3/2$ and the $3x+1$ problem

TL;DR

This work uncovers a deep link between rational-base representations in base and the Collatz problem. It defines the representation with digits in and studies the associated language of admissible words, the odometer for constructing from , and the interplay with the Collatz map . The authors introduce saturated words and the related map , showing that sat and that the growth and prefix structure of saturated words encode Collatz-like dynamics. Open questions connect the behavior of to the Collatz conjecture, propose a normality conjecture for the limiting saturated word, and seek a deeper understanding of within the rational-base framework, offering a new arithmetic-dynamical perspective on both topics.

Abstract

The representation of numbers in rational base was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case . Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz function. Our purpose in this note is to expose these links and motivate further research into them.

Paper Structure

This paper contains 10 sections, 10 theorems, 20 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Basics
  3. Construction of $\langle n \rangle$
  4. The odometer
  5. The Collatz conjecture
  6. Links with the Collatz function
  7. Extending admissible words
  8. Saturated words
  9. The function $U$
  10. Open questions

Key Result

Proposition 2.2

Let $n_0 \in \mathbb N \setminus \{0\}$. Denote $\langle n_0 \rangle=a_k\cdots a_0$ with $a_i \in \{0,1,2\}$ for all $i$. Then Moreover, we have $a_0=2n_0-3n_1$ and $a_0 \equiv n_1 \bmod 2$.

Theorems & Definitions (27)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Conjecture 2.4
  • Conjecture 2.5
  • Proposition 2.6
  • proof
  • Proposition 3.1
  • ...and 17 more