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Dumont-Thomas complement numeration systems for $\mathbb{Z}$

Sébastien Labbé, Jana Lepšová

TL;DR

This work extends Dumont-Thomas numeration from ${\mathbb N}$ to all of ${\mathbb Z}$ by leveraging two-sided periodic points of substitutions, yielding a canonical complement numeration system. It proves that these two-sided periodic points produce automatic sequences and provides a total-order characterization on the recognized language, enabling a clean bijection between ${\mathbb Z}$ and representations. The framework naturally extends to ${\mathbb Z}^d$ via padding and unifies classical representations: the two's complement and its Fibonacci analogue appear as special cases. Overall, the paper links substitution dynamics, automatic sequences, and multi-dimensional numeration with potential applications to tilings and indexing in higher dimensions.

Abstract

We extend the well-known Dumont--Thomas numeration systems to $\mathbb{Z}$ using an approach inspired by the two's complement numeration system. Integers in $\mathbb{Z}$ are canonically represented by a finite word (starting with $\mathtt{0}$ when nonnegative and with $\mathtt{1}$ when negative). The systems are based on two-sided periodic points of substitutions as opposed to the right-sided fixed points. For every periodic point of a substitution, we construct an automaton which returns the letter at position $n\in\mathbb{Z}$ of the periodic point when fed with the representation of $n$ in the corresponding numeration system. The numeration system naturally extends to $\mathbb{Z}^d$. We give an equivalent characterization of the numeration system in terms of a total order on a regular language. Lastly, using particular periodic points, we recover the well-known two's complement numeration system and the Fibonacci analogue of the two's complement numeration system.

Dumont-Thomas complement numeration systems for $\mathbb{Z}$

TL;DR

This work extends Dumont-Thomas numeration from to all of by leveraging two-sided periodic points of substitutions, yielding a canonical complement numeration system. It proves that these two-sided periodic points produce automatic sequences and provides a total-order characterization on the recognized language, enabling a clean bijection between and representations. The framework naturally extends to via padding and unifies classical representations: the two's complement and its Fibonacci analogue appear as special cases. Overall, the paper links substitution dynamics, automatic sequences, and multi-dimensional numeration with potential applications to tilings and indexing in higher dimensions.

Abstract

We extend the well-known Dumont--Thomas numeration systems to using an approach inspired by the two's complement numeration system. Integers in are canonically represented by a finite word (starting with when nonnegative and with when negative). The systems are based on two-sided periodic points of substitutions as opposed to the right-sided fixed points. For every periodic point of a substitution, we construct an automaton which returns the letter at position of the periodic point when fed with the representation of in the corresponding numeration system. The numeration system naturally extends to . We give an equivalent characterization of the numeration system in terms of a total order on a regular language. Lastly, using particular periodic points, we recover the well-known two's complement numeration system and the Fibonacci analogue of the two's complement numeration system.
Paper Structure (12 sections, 19 theorems, 56 equations, 3 figures, 1 table)

This paper contains 12 sections, 19 theorems, 56 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dumont--Thomas numeration system for $\mathbb{N}$
  4. Some extensions of Dumont--Thomas results
  5. Dumont--Thomas complement numeration systems for $\mathbb{Z}$ based on periodic points
  6. More examples
  7. Periodic points as Automatic Sequences
  8. Numeration systems for $\mathbb{Z}^d$ based on periodic points
  9. A Total Order
  10. Relation with existing complement numeration systems
  11. Two's complement numeration system
  12. Fibonacci analogue of the two's complement numeration system

Key Result

Theorem 3.2

MR1020484 Let $a\in A$ and let $\eta:A^*\to A^*$ be a substitution. Let $u=\eta(u)$ be a right-infinite fixed point of $\eta$ with growing seed $u_0 = a$. For every integer $n\geq1$, there exists a unique integer $k=k(n)$ and a unique sequence $(m_i,a_i)_{i=0,\dots,k}$ such that

Figures (3)

  • Figure 1: The graph associated to the substitution $a\mapsto abc, b\mapsto c, c\mapsto ac$.
  • Figure 2: The set of paths starting in state $a$ in the directed graph provide a canonical representation of the nonnegative integers after removing leading zeroes.
  • Figure 3: Automata $\mathcal{A}_{\varphi,a}$, $\mathcal{A}_{\varphi,b}$ and $\mathcal{A}_{\varphi,s}$ for $\varphi:a\mapsto ab$, $b\mapsto a$ and $s=b|a$.

Theorems & Definitions (47)

  • Definition 3.1: admissible sequence
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Remark 3.7
  • proof
  • Lemma 3.8
  • proof
  • ...and 37 more