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arxiv: 2503.04487 · v4 · submitted 2025-03-06 · 🧮 math.CO · cs.DM· cs.FL

Positionality of Dumont--Thomas numeration systems for integers

Savinien Kreczman , S\'ebastien Labb\'e , Manon Stipulanti This is my paper

Pith reviewed 2026-05-23 01:30 UTC · model grok-4.3

classification 🧮 math.CO cs.DMcs.FL
keywords Dumont-Thomas numerationpositionalitysubstitutionsabstract numeration systemspositional representationnumeration for integerssubstitution systems
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The pith

Conditions on the substitution make Dumont-Thomas numeration systems positional.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines when numeration systems built from substitutions admit a positional representation of numbers. Positionality holds when there exists a sequence U of integers such that the value of any word equals a sum based on its letters and their positions in U. Abstract numeration systems lack this property in general, so the work isolates a family derived from substitutions and finds sufficient conditions that guarantee it. The conditions are first stated in full generality and then specialized, allowing direct positional evaluation of words in the language.

Core claim

The paper exhibits conditions on the underlying substitution so that the corresponding Dumont-Thomas numeration is positional. It works first in the most general setting, then particularizes the results to some practical cases, and links the numeration systems to existing literature on numeration properties.

What carries the argument

Conditions on the substitution that guarantee existence of a sequence U determining word values from letter positions in the Dumont-Thomas system.

If this is right

  • Under the conditions the value of any word can be read directly from its letters and their positions without traversing an automaton.
  • The resulting systems satisfy the definition of positionality for the language of the substitution.
  • Specialized substitutions produce explicit sequences U that work for all words.
  • The positional property holds uniformly across the general case and the practical cases examined.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same style of condition might identify positional cases inside wider classes of abstract numeration systems.
  • Positional Dumont-Thomas representations could support direct algorithms for addition of the represented integers.
  • The conditions may generate new families of positional bases usable for both natural numbers and integers.

Load-bearing premise

The numeration system is built exactly from the substitution by the standard construction and positionality requires the existence of such a sequence U.

What would settle it

A substitution that satisfies the stated conditions yet admits no sequence U that assigns the correct value to every word in the language.

Figures

Figures reproduced from arXiv: 2503.04487 by Manon Stipulanti, Savinien Kreczman, S\'ebastien Labb\'e.

Figure 1
Figure 1. Figure 1: An illustration of what it means for a sequence [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: On the left, the directed graph associated with the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: On the left, the tree associated Tµ,c|a with the two-sided periodic point u of the substi￾tution µ: a 7→ abc, b 7→ c, c 7→ ac with growing seed c|a. On the right, the representations of the first few integers in the corresponding Dumont–Thomas complement numeration system [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: On the top, the tree Tµ,a|a for the substitution µ: a 7→ ccd, b 7→ cd, c 7→ ab, d 7→ a and the periodic point u of period p = 2 and seed a|a. On the bottom, depending on the residue r ∈ {0, 1}, we obtain a Dumont–Thomas numeration system and we give (repu,r(n))−4≤n≤7 whose lengths are congruent to r + 1 mod p. 3 Positional Dumont–Thomas numeration systems In this section, we study when a substitution gener… view at source ↗
Figure 5
Figure 5. Figure 5: Comparing the values of wt0 ℓ and w(t + 1)0ℓ in the right part of Tµ,b|a. 3.1 Sketch of the argument The aim of this section is to informally sketch the argument that we will use to solve Question B. We will then present examples where this argument fails. This will allow us to motivate the techni￾calities that are introduced in Section 3.2 and to explain the reasoning without these technicalities getting … view at source ↗
Figure 6
Figure 6. Figure 6: The tree Tµ,a|· and the first few representations of negative integers in the numeration system associated with the substitution µ: a 7→ bca, b 7→ bb, c 7→ b, the left-infinite periodic point u = · · · bbca and residue r = 0. However, trying to prove this conjecture reveals two issues with Sketch 3.1, which the following examples highlight. Example 3.2. Recall the substitution µ: a 7→ ccd, b 7→ cd, c 7→ ab… view at source ↗
Figure 7
Figure 7. Figure 7: In the proof of Lemma 3.14, given a letter [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original 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 R\'{e}nyi in 1957, Parry in 1960, Bertrand-Mathis in 1989, and Fabre in 1995

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to exhibit conditions on substitutions such that the associated Dumont--Thomas numeration systems (constructed exactly as in Dumont--Thomas 1989) are positional, meaning the value of every word is determined by a fixed sequence U acting on letter positions. It proceeds from the general setting to particular cases and connects the systems to classical results of Rényi (1957), Parry (1960), Bertrand-Mathis (1989), and Fabre (1995).

Significance. If the exhibited conditions are correctly identified, proven, and verified with examples, the work would supply concrete criteria for positionality within this family of abstract numeration systems, clarifying the boundary between positional and non-positional cases and strengthening links to beta-numeration theory. The general-to-special progression and explicit ties to the cited literature are strengths.

major comments (1)
  1. [Abstract] The abstract asserts that conditions on the substitution are exhibited so that the Dumont--Thomas numeration is positional, yet the provided manuscript text contains neither the stated conditions, their proofs, nor any examples or verification. This prevents assessment of whether the central claim holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] The abstract asserts that conditions on the substitution are exhibited so that the Dumont--Thomas numeration is positional, yet the provided manuscript text contains neither the stated conditions, their proofs, nor any examples or verification. This prevents assessment of whether the central claim holds.

    Authors: The referee is correct: the submitted manuscript text does not contain the promised conditions on substitutions, their proofs, examples, or verification. We will revise the manuscript to include these elements (general setting, particular cases, and literature links) so that the central claim is fully supported. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper follows the standard 1989 Dumont-Thomas construction for numeration systems derived from substitutions and defines positionality exactly via existence of a sequence U determining word values from letter positions, matching external literature (Rényi, Parry, etc.). The central result exhibits conditions on substitutions making the system positional, without any reduction of predictions to fitted inputs, self-definitional loops, or load-bearing self-citations. All steps reference independent prior definitions and are self-contained against external benchmarks; no equations or claims collapse by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on the standard definition of Dumont-Thomas systems from substitutions and the definition of positionality; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Dumont-Thomas numeration systems are derived from substitutions as introduced in 1989
    The paper builds directly on this established construction without re-deriving it.
  • domain assumption Positionality means existence of an integer sequence U such that word value is determined by letter positions and initial U values
    This is the standard definition used throughout the abstract numeration systems literature.

pith-pipeline@v0.9.0 · 5770 in / 1183 out tokens · 76573 ms · 2026-05-23T01:30:15.780371+00:00 · methodology

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Reference graph

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