Positionality of Dumont--Thomas numeration systems for integers

TL;DR

This work analyzes when Dumont--Thomas abstract numeration systems, derived from substitutions, yield positional representations of integers. It formalizes a general criterion: a Dumont--Thomas complement system is positional if and only if, for each residue class $j$ modulo the period $p$ of a two-sided periodic point, the letter-weights $|\mu^\ell(c)|$ stabilize across a class $E_j$ for all $\ell$ with $\ell+j\equiv r \pmod p$, and a supplementary condition (C) holds; the resulting weights are $U_\ell=|\mu^\ell(c)|$ and $V_\ell=|\mu^\ell(b)|$. This yields a bridge between symbolic substitutions and classical Bertrand numeration systems, with corollaries for primitive substitutions and Fabre-like substitutions, and clarifies when two different Fabre-like substitutions can produce the same weight sequence. The results illuminate structural criteria for positionality, linking growth lengths in substitution images to the existence of a global base-like weight sequence. These findings contribute to the broader understanding of when abstract numeration frameworks admit greedy, base-like representations and how they relate to historical numeration systems in the literature (Rényi, Parry, Bertrand-Mathis, Fabre).

Abstract

Introduced in 2001 by Lecomte and Rigo, abstract numeration systems provide a way of expressing natural numbers with words from a language $L$ accepted by a finite automaton. As it turns out, these numeration systems are not necessarily positional, i.e., we cannot always find a sequence $U=(U_i)_{i\ge 0}$ of integers such that the value of every word in the language $L$ is determined by the position of its letters and the first few values of $U$. Finding the conditions under which an abstract numeration system is positional seems difficult in general. In this paper, we thus consider this question for a particular sub-family of abstract numeration systems called Dumont--Thomas numeration systems. They are derived from substitutions and were introduced in 1989 by Dumont and Thomas. We exhibit conditions on the underlying substitution so that the corresponding Dumont--Thomas numeration is positional. We first work in the most general setting, then particularize our results to some practical cases. Finally, we link our numeration systems to existing literature, notably properties studied by Rényi in 1957, Parry in 1960, Bertrand-Mathis in 1989, and Fabre in 1995.

Figures (7)

• Figure 1: An illustration of what it means for a sequence $((m_i,a_i))_{0\le i \le 2}$ to be $a$-admissible with respect to a substitution $\mu$.
• Figure 2: On the left, the directed graph associated with the Tribonacci substitution $\tau\colon a\mapsto ab, b\mapsto ac, c\mapsto a$. On the right, the tree $\mathcal{T}_{\tau,a}$ displays all $a$-admissible sequences of length at most $3$ for the Tribonacci substitution $\tau$ and growing letter $a$.
• Figure 3: On the left, the tree associated $\mathcal{T}_{\mu,c|a}$ with the two-sided periodic point $u$ of the substitution $\mu\colon a\mapsto abc, b\mapsto c, c\mapsto ac$ with growing seed $c|a$. On the right, the representations of the first few integers in the corresponding Dumont--Thomas complement numeration system.
• Figure 4: On the top, the tree $\mathcal{T}_{\mu,a|a}$ for the substitution $\mu \colon a\mapsto ccd, b \mapsto cd, c \mapsto ab, d\mapsto a$ and the periodic point $u$ of period $p=2$ and seed $a|a$. On the bottom, depending on the residue $r\in\{0,1\}$, we obtain a Dumont--Thomas numeration system and we give $(\mathop{\mathrm{rep}}\nolimits_{u,r}(n))_{-4\le n\le 7}$ whose lengths are congruent to $r+1 \bmod p$.
• Figure 5: Comparing the values of $wt0^\ell$ and $w(t+1)0^\ell$ in the right part of $\mathcal{T}_{\mu,b|a}$.

Theorems & Definitions (34)

• theorem 2.1
• definition 1
• theorem 2.2
• theorem 2.3
• definition 2
• theorem 2.4
• lemma 1
• lemma 2
• proof : Proof of \ref{['thm: unique admin for right infinite periodic len r']}
• theorem 2.5