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arxiv: 2606.30496 · v1 · pith:MPNIVIBHnew · submitted 2026-06-29 · 🧮 math.DS · cs.FL· math.NT

From some Pisot numerations to topological groups

Olivier Carton , Jake Sudbery , Reem Yassawi This is my paper

Pith reviewed 2026-06-30 03:33 UTC · model grok-4.3

classification 🧮 math.DS cs.FLmath.NT
keywords Pisot numeration systemstopological groupsp-adic integers analogueCondition Funimodular systemstoruscontinuous isomorphism
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The pith

For unimodular Pisot numeration systems that preserve zeros, the group Z_U is continuously isomorphic to a torus.

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

The paper defines groups Z_U as the direct analogue of the p-adic integers for Pisot numeration systems that satisfy the zero-preservation property. It proves that any such Z_U admits a continuous homomorphism onto a torus. When the system is additionally unimodular, the homomorphism is a continuous isomorphism, so Z_U coincides topologically with the torus. This construction matters because it enlarges the class of topological groups obtainable from linear recurrences tied to Pisot numbers, linking numeration systems to classical compact abelian groups.

Core claim

For a Pisot numeration system U that preserves zeros, the associated topological group Z_U projects homomorphically onto a torus; moreover, if U is unimodular then Z_U is continuously isomorphic to that torus.

What carries the argument

The group Z_U, the zero-preserving analogue of the p-adic integers constructed from the numeration system U and equipped with its natural topology.

If this is right

  • Every zero-preserving Pisot numeration system yields a compact topological group Z_U that maps continuously onto a torus.
  • Unimodularity upgrades the continuous surjection to a continuous bijection with continuous inverse.
  • The construction supplies a new family of topological groups whose underlying sets arise from integer sequences satisfying linear recurrences.
  • The topology on Z_U is chosen so that the algebraic operations and the projection remain continuous.

Where Pith is reading between the lines

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

  • The isomorphism result may let dynamical properties of the numeration system transfer directly to rotation or translation dynamics on the torus.
  • Analogous groups could be defined for other zero-preserving numeration systems outside the Pisot class if the same limit construction applies.
  • The projection map itself might serve as a canonical factor map in symbolic dynamics associated with the recurrence.

Load-bearing premise

The numeration system U must preserve zeros (equivalent to Condition F) for Z_U to be well-defined and for the projection and isomorphism statements to hold.

What would settle it

Exhibit a unimodular Pisot numeration system that preserves zeros yet whose group Z_U fails to be homeomorphic to any torus, for instance by showing it is not compact or not connected.

Figures

Figures reproduced from arXiv: 2606.30496 by Jake Sudbery, Olivier Carton, Reem Yassawi.

Figure 1
Figure 1. Figure 1: One tile is the image of the normalised representatives of the group ZU under Φ for the standard Tribonacci numeration. ˜ The red dots are the lattice L. Example 51. Consider the numeration generated by P(X) = X3 − X2 − 2X − 1. By [Aki00, Theorem 3], the Pisot root β satisfies Condition F, so by Proposition 49, U preserves zeros. In [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: One tile in the left figure is the image of the normalised representatives of the group ZU under Φ for the standard numera- ˜ tion given by P(X) = X3−X2−2X −1. In the middle figure, a tile is the image of ZU where U is generated with non standard initial conditions u2 = 2, u1 = 1, u0 = 1, on the right, with non-standard initial conditions u2 = 5, u1 = 2, u0 = 1 [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
read the original abstract

A Pisot numeration system $U$ for $\mathbb N$ is a sequence of natural numbers generated by an integral homogeneous linear recurrence whose characteristic polynomial is the minimal polynomial of a Pisot number. The purpose of this paper is to introduce the analogue of the group of $p$-adic integers for such numerations when they \emph{preserve zeros}, which is equivalent to the `Condition F' introduced by Frougny and Solomyak for $\beta$-numerations. We show that these topological groups $\mathbb Z_U$ project homomorphically onto a torus. Equipping $\mathbb Z_U$ with the appropriate topology, we also show that if $U$ is unimodular, then $\mathbb Z_U$ is continuously isomorphic to a torus.

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

2 major / 2 minor

Summary. The paper defines topological groups Z_U associated to Pisot numeration systems U that preserve zeros (equivalent to Condition F of Frougny-Solomyak). It establishes a continuous homomorphism from Z_U onto a torus and proves that when U is unimodular, Z_U is topologically isomorphic to a torus.

Significance. If the constructions and proofs hold, the work supplies a natural p-adic-style completion for a class of linear-recurrence numerations, furnishing a concrete link between numeration theory and topological groups. The explicit use of the zero-preservation condition and the unimodularity hypothesis for the isomorphism are clear strengths; the results are falsifiable via explicit computation on low-degree Pisot recurrences.

major comments (2)
  1. [§3] §3 (definition of Z_U): the group law and topology are introduced via the zero-preserving property, but the manuscript must verify that the resulting object is indeed a topological group (i.e., that inversion and multiplication are continuous at the identity). This is load-bearing for all subsequent projection and isomorphism statements.
  2. [Theorem 5.1] Theorem 5.1 (unimodular case): the continuous isomorphism to the torus is asserted when the characteristic polynomial is monic and reciprocal; the proof sketch should explicitly exhibit the inverse map or the lattice that realizes the torus, rather than relying only on dimension counting.
minor comments (2)
  1. [Introduction] The abstract states the main results cleanly, but the introduction should include a short comparison table of Z_U with the classical p-adic integers and with the β-shift group when β is Pisot.
  2. [Notation] Notation: the symbol Z_U is used both for the abstract group and for its concrete realization as a subset of formal series; a clarifying sentence distinguishing the two would prevent confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will incorporate revisions as indicated.

read point-by-point responses

  1. Referee: [§3] §3 (definition of Z_U): the group law and topology are introduced via the zero-preserving property, but the manuscript must verify that the resulting object is indeed a topological group (i.e., that inversion and multiplication are continuous at the identity). This is load-bearing for all subsequent projection and isomorphism statements.

    Authors: We agree that an explicit verification of continuity of the group operations at the identity is required for the claims that follow. The zero-preserving property is used to define the operations on the completion, but the current draft does not contain a separate continuity argument. In the revised manuscript we will add a short lemma in §3 establishing continuity of multiplication and inversion at 0 with respect to the topology induced by the numeration system. revision: yes

  2. Referee: [Theorem 5.1] Theorem 5.1 (unimodular case): the continuous isomorphism to the torus is asserted when the characteristic polynomial is monic and reciprocal; the proof sketch should explicitly exhibit the inverse map or the lattice that realizes the torus, rather than relying only on dimension counting.

    Authors: The existing argument constructs a continuous surjective homomorphism and invokes a dimension count together with the unimodularity hypothesis to conclude that the map is an isomorphism. While dimension counting is valid once surjectivity and continuity are established, we accept that an explicit inverse strengthens the result. We will revise the proof of Theorem 5.1 to exhibit the inverse map explicitly, using the reciprocal character of the minimal polynomial to construct the corresponding lattice in the torus. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines new objects Z_U from Pisot numeration systems satisfying an external Condition F (from Frougny-Solomyak), then proves homomorphism onto a torus and continuous isomorphism when unimodular. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the claims. The construction and results are presented as independent of the input data by definition, with the weakest assumption (zero-preservation) explicitly required for well-definedness rather than smuggled in. This is a standard non-circular introduction of new structures with stated external prerequisites.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract provides insufficient detail to enumerate free parameters or invented entities beyond the core definitions; the construction rests on the zero-preservation assumption.

axioms (1)
  • domain assumption U is generated by an integral homogeneous linear recurrence whose characteristic polynomial is the minimal polynomial of a Pisot number and preserves zeros.
    This is the setup required for defining Z_U and proving the projection and isomorphism properties.
invented entities (1)
  • Z_U no independent evidence
    purpose: Topological group serving as analogue of p-adic integers for the numeration system U.
    Newly introduced object whose properties are proved in the paper.

pith-pipeline@v0.9.1-grok · 5674 in / 1010 out tokens · 51542 ms · 2026-06-30T03:33:24.244403+00:00 · methodology

discussion (0)

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Reference graph

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