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From some Pisot numerations to topological groups

This paper introduces a topological group analogue of the $p$ -adic integers for Pisot numeration systems that preserve zeros, demonstrating that these groups project homomorphically onto a torus and are continuously isomorphic to a torus when the numeration is unimodular.

Original authors: Olivier Carton, Jake Sudbery, Reem Yassawi

Published 2026-06-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Olivier Carton, Jake Sudbery, Reem Yassawi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a very special way of counting numbers, different from the usual base-10 system we use every day. Instead of powers of 10 (1, 10, 100...), you use a sequence of numbers that grows according to a specific rule, like the famous Fibonacci sequence (1, 1, 2, 3, 5, 8...). This is called a Pisot numeration system.

In our normal world, if you add two numbers together, any "carries" (like when 5 + 5 = 10, and you carry the 1 over) only move to the left. But in these special counting systems, carries can get confused and move both left and right. This makes doing math with them tricky and breaks the usual rules of how we build number systems.

The Main Idea: Building a "Number City"

The authors of this paper wanted to build a mathematical "city" (a topological group) for these special counting systems, similar to how mathematicians have built a city of p-adic integers for standard base-p systems.

Think of the p-adic integers as a city where every address is an infinite string of digits. In this city, two numbers are considered "close" if they share a long tail of identical digits at the end. This creates a very neat, organized structure.

The authors asked: Can we build a similar city for these special Fibonacci-like counting systems?

The Problem: The "Carry" Chaos

In standard math, if you add two numbers, the result is predictable. In these special systems, because carries can jump around, the usual way of defining "closeness" doesn't work. If you try to add two numbers, the result might look very different from the inputs in a way that breaks the smooth structure of the city.

To fix this, the authors introduced a concept called "preserving zeros."

The Analogy: Imagine you are trying to stack blocks. In a normal system, if you remove a block from the bottom, the whole tower stays stable. In these special systems, sometimes removing a block causes the whole tower to wobble or collapse.
The Solution: The authors found that if the system "preserves zeros" (meaning the wobbling is controlled and doesn't get out of hand), they can build a stable city. They call this new city $Z_U$ .

The Big Discovery: The City is a Donut

Once they built this city $Z_U$ and gave it the right rules for "closeness," they discovered something beautiful about its shape.

They proved that this city is actually topologically the same as a torus (a donut shape).

The Analogy: Imagine taking a long, infinite strip of paper (representing all the possible numbers in this system) and rolling it up into a tube, then connecting the ends to make a donut. Even though the numbers go on forever, the "shape" of the whole collection of numbers wraps around itself to form a donut.
The Result: If the rules of the counting system are "unimodular" (a specific mathematical property that keeps things balanced), the city $Z_U$ is perfectly identical to this donut shape. If the rules aren't perfectly balanced, the city still maps onto the donut, but it might wrap around it a few times or cover it in a more complex way.

Why This Matters (According to the Paper)

The paper connects three different worlds of mathematics:

Counting Systems: How we write numbers using special sequences (like Fibonacci).
Topology: The study of shapes and spaces (like donuts).
Algebra: The study of groups and operations.

The authors showed that the "shape" of the set of all numbers in these special systems is a donut. They also proved that this only works if the system satisfies a specific condition called "Condition F" (a rule discovered by other mathematicians Frougny and Solomyak). If the system follows this rule, the "carry chaos" is tamed, and the donut shape emerges.

A Visual Example

The paper uses the Tribonacci system (where each number is the sum of the previous three) as an example.

They took the "standard" way of writing numbers in this system.
They mapped these numbers into a geometric space.
The result was a single, beautiful tile that looks like a fractal (a shape that repeats itself).
When you put all these tiles together, they perfectly fill a space without gaps or overlaps, forming the "donut" shape described above.

Summary in a Nutshell

The paper takes a weird, complex way of counting numbers where math carries get messy. It proves that if you follow a specific set of rules to keep the mess under control, the entire collection of numbers forms a perfect, smooth shape: a donut. This bridges the gap between how we count and the geometric shapes those numbers create.

Technical Summary: From Some Pisot Numerations to Topological Groups

Problem Statement
The additive group of $p$ -adic integers, $\mathbb{Z}_p$ , is classically constructed as the inverse limit of finite groups $\mathbb{Z}/p^n\mathbb{Z}$ or as the topological completion of the natural numbers $\mathbb{N}$ under a $p$ -adic metric derived from base- $p$ expansions. A key property of this construction is that carries propagate strictly to the left, satisfying the non-Archimedean inequality $\nu_p(x+y) \geq \min(\nu_p(x), \nu_p(y))$ .

This paper addresses the construction of an analogous topological group for Pisot numeration systems. In such systems, the powers of a base $p$ are replaced by a sequence of natural numbers $(u_n)_{n \geq 0}$ generated by a linear recurrence whose characteristic polynomial is the minimal polynomial of a Pisot number $\beta$ (an algebraic integer $>1$ whose conjugates lie strictly inside the unit disk). Unlike base- $p$ systems, Pisot numerations (e.g., Zeckendorf/Fibonacci) exhibit "non-standard" carry propagation, where carries can move both left and right. Consequently, the classical $p$ -adic valuation properties fail, and the resulting group structure is not a profinite group with a zero-dimensional topology. The authors seek to define a group $\mathbb{Z}_U$ that serves as the analogue of $\mathbb{Z}_p$ for these systems and to characterize its topological structure.

Methodology
The authors construct the group $\mathbb{Z}_U$ through the following steps:

Group Construction via Quotients:
- Let $G$ be the abelian group of bounded integer-valued sequences $(g_n)_{n \geq 0}$ under component-wise addition.
- A "pseudo-valuation" $\nu_U$ is defined on finite sequences based on the order of the first non-zero digit in the normalised $U$ -expansion of the represented integer.
- The concept of zero preservation is introduced: A numeration $U$ preserves zeros if there exists a constant $K_c$ such that for any sequence $g$ with bounded entries, $\nu_U(g) \geq \text{ord}(g) - K_c$ . This condition ensures that carries do not propagate arbitrarily far to the right.
- A subgroup $H_U \subset G$ is defined as the set of sequences $g$ where the valuation of the prefixes tends to infinity.
- The group of $U$ -adic integers is defined as the quotient $\mathbb{Z}_U := G/H_U$ .
Topological Structure and Mapping:
- A metric topology is defined on $G$ using a norm weighted by the conjugates of $\beta$ , specifically $\|g\|_U = \sum |g_n| \gamma^n$ , where $\gamma < 1$ is the maximum modulus of the conjugates of $\beta$ .
- A homomorphism $\tilde{\Phi}: G \to \mathbb{C}^k \times \mathbb{R}^\ell$ is constructed, mapping a sequence to a sum involving the conjugates of $\beta$ .
- The image of the subgroup $H_U$ under $\tilde{\Phi}$ is shown to lie within a specific lattice $L$ generated by the powers of the conjugates.
- This induces a continuous group homomorphism $\Phi: \mathbb{Z}_U \to (\mathbb{C}^k \times \mathbb{R}^\ell)/L$ , where the codomain is a torus $\mathbb{T}^{d-1}$ .
Equivalence with Condition F:
- The paper establishes a critical link between the algebraic condition of "zero preservation" for the numeration $U$ and Condition F (introduced by Frougny and Solomyak) for the associated $\beta$ -numeration. Condition F requires that $\mathbb{Z}[\beta] \subset \text{Fin}(\beta)$ , meaning every element of the ring $\mathbb{Z}[\beta]$ has a finite $\beta$ -expansion.
- The authors prove that $U$ preserves zeros if and only if the associated $\beta$ satisfies Condition F.

Key Contributions and Results

Definition of $\mathbb{Z}_U$ : The paper introduces $\mathbb{Z}_U$ as a topological group that generalizes the $p$ -adic integers to Pisot numerations. It is shown that elements of $\mathbb{Z}_U$ can be represented by infinite normalised sequences of digits.
Topological Isomorphism to a Torus:
- Theorem 1: If $U$ is a Pisot numeration associated with a Pisot number $\beta$ of degree $d$ and $U$ preserves zeros, there exists a continuous group epimorphism $\Phi: \mathbb{Z}_U \to \mathbb{T}^{d-1}$ .
- Unimodular Case: If $\beta$ is unimodular (the norm of $\beta$ is $\pm 1$ ), then $\Phi$ is a continuous group isomorphism. Thus, $\mathbb{Z}_U$ is topologically isomorphic to a $(d-1)$ -dimensional torus.
Characterization of Zero Preservation: The paper provides a decision procedure for when a numeration preserves zeros by linking it to Condition F. It confirms that standard $k$ -bonacci numerations (including Fibonacci) preserve zeros.
Geometric Realization: The group $\mathbb{Z}_U$ admits a geometric representation as a tiling of Euclidean space (or a fundamental domain of the torus). The map $\Phi$ sends the group elements to points on the torus, where the image of the normalised expansions forms a tiling (e.g., the Rauzy fractal for the Tribonacci case).

Significance and Claims
The authors position their work as filling a gap in the understanding of the algebraic structure of numeration systems. While previous work (e.g., Vershik, Rittaud, Vivier) had explored completions of specific numerations or connections to Wang tiles, this paper explicitly constructs the group structure for the general class of Pisot numerations satisfying zero preservation.

The paper claims that:

The construction of $\mathbb{Z}_U$ provides a rigorous algebraic framework for the "Fibonacci decomposition of naturals" and similar systems, moving beyond character theory to explicit group isomorphisms.
The condition of zero preservation is the precise analogue of Condition F in the context of constructing topological groups from numerations.
Unlike the $p$ -adic integers which are zero-dimensional, these groups are higher-dimensional manifolds (tori) when the underlying number is unimodular.
The results are specific to numerations and not merely substitutions, though the authors note that their results can be transferred to $\beta$ -substitutions to deduce that such shifts are almost automorphic (a result to be detailed in future work).

The paper explicitly notes limitations: the multiplicative structure of $\mathbb{Z}_p$ does not naturally extend to $\mathbb{Z}_U$ because standard integer multiplication is not compatible with the topology (it does not preserve zeros), and alternative multiplications (like Knuth's) lack distributivity. Thus, the focus remains strictly on the additive topological group structure.