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If a machine did it, it is probably transcendental (even $p$ -adically)

This paper establishes that $p$ -adic numbers with continued fractions generated by generalized automatic, periodic, or palindromic words are either algebraic of degree at most 2 or transcendental, thereby extending key results from the real setting to the $p$ -adic context.

Original authors: Laura Capuano, Sara Checcoli, Marzio Mula, Lea Terracini

Published 2026-07-14

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CC BY 4.0

Original authors: Laura Capuano, Sara Checcoli, Marzio Mula, Lea Terracini

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 magical machine that takes a number and breaks it down into a long, endless string of symbols, like a secret code. In the world of real numbers (the kind we use for measuring pizza slices), mathematicians have known for a long time that if this code follows a simple, repeating pattern, the number is usually "special" (like a square root of 2). But if the code is messy and chaotic, the number is likely "transcendental"—a fancy word for a number so wild and complex it can't be described by any simple algebraic equation.

Now, imagine a different kind of number system called p-adic numbers. Think of these as numbers that live in a strange, upside-down universe where the "size" of a number depends on how divisible it is by a specific prime number (like 3, 5, or 7) rather than how big it looks on a ruler. In this universe, the rules for breaking numbers into codes (called continued fractions) are much fuzzier. There isn't just one way to do it; there are infinitely many ways, and for a long time, nobody knew if the same "simple code = special number" rule applied here.

The Big Discovery
In this paper, Laura Capuano and her team built a bridge between the messy world of p-adic codes and the strict world of algebra. They asked a simple question: "If a machine generates a p-adic number using a code that has a specific, structured pattern (like repeating blocks or mirrored sections), is that number either a simple 'quadratic' number or a wild 'transcendental' one?"

Their answer is a resounding yes. They proved that if the code follows these specific patterns (which they call "property ♠" and "property ♣"), the resulting number cannot be a complex algebraic number of degree 3 or higher. It's an all-or-nothing situation: the number is either simple (degree 2 or less) or it's transcendental. There is no middle ground.

What They Ruled Out
The paper explicitly argues against the idea that you can find "complex" algebraic numbers (those that need complicated equations to solve) hiding inside these structured codes.

The "Unlikely Intersection" Argument: The authors explain that structured codes are rare, like finding a specific sentence in a library of all possible gibberish. Algebraic numbers of high degree are also rare. The paper suggests it is highly unlikely these two rare groups would ever meet unless the code was extremely simple (finite or perfectly repeating).
The "Machine" Limit: They show that even if you use a machine (a finite state machine) to generate the code, or if the code has "low complexity" (meaning it doesn't have too many different patterns), you still won't find those tricky, high-degree algebraic numbers. If the code is structured enough to be interesting, the number it produces is either too simple or too wild to be that specific "middle" type of algebraic number.

How Sure Are They?
The authors didn't just guess or run simulations; they proved it.

They used a powerful mathematical tool called the Subspace Theorem (a p-adic version of a famous theorem by Schmidt). Think of this theorem as a super-sensitive detector that can tell if a number is being approximated "too well" by simple fractions.
They showed that if the code has the right structure, the number is approximated so perfectly by a sequence of simpler numbers that it must be either quadratic or transcendental.
Their proof holds for any p-adic floor function (the rule the machine uses to pick the next symbol in the code), provided the symbols aren't too small in a specific p-adic sense. They even gave exact formulas for how big those symbols need to be (involving constants like $p^k$ ) to make the proof work.

The "Machine" Metaphor
Imagine the p-adic number as a song.

If the song is a simple, repeating loop, it's a "quadratic" number.
If the song is pure chaos, it's "transcendental."
The paper proves that you can't have a song that is a complex, non-repeating melody (like a jazz improvisation that follows a strict rule) and still be a "middle-ground" algebraic number. If the melody follows the specific "structured" rules the authors found, the song collapses into either a simple loop or total chaos.

Why This Matters
Before this, we only knew this rule for real numbers. In the p-adic world, things were messy because there are so many different ways to build the "floor function" (the machine's rulebook). This paper says, "It doesn't matter which rulebook you pick; as long as the code has these specific patterns, the result is the same." It extends a famous result by Bugeaud from the real world to the p-adic world, closing a gap in our understanding of how numbers and patterns interact in these strange mathematical universes.

In short: If a machine builds a p-adic number with a structured, repeating, or mirrored code, that number is either simple or transcendental. There is no "in-between" algebraic number hiding there.

Technical Summary: "If a machine did it, it is probably transcendental (even p-adically)"

Problem Statement
The paper investigates the algebraic nature of $p$ -adic numbers whose continued fraction expansions correspond to specific classes of words. While continued fractions in the real setting ( $\mathbb{R}$ ) provide a well-established bridge between the combinatorial structure of words and the algebraic properties of numbers (e.g., Lagrange's theorem linking periodic words to quadratic irrationals), the $p$ -adic setting ( $\mathbb{Q}_p$ ) presents significant structural challenges.

The central question addressed is whether families of "structured" words (generalizations of periodic, automatic, or palindromic words) correspond to algebraic numbers of degree greater than 2 in $\mathbb{Q}_p$ . In the real case, results by Bugeaud and Adamczewski-Bugeaud suggest that such structured words generally yield transcendental numbers, with the only algebraic exceptions being quadratic (periodic) or rational (finite). The authors aim to establish an analogous result for $p$ -adic numbers, overcoming the lack of a canonical floor function and the absence of a complete $p$ -adic analogue of Lagrange's theorem.

Methodology
The authors adapt the strategy used by Bugeaud in the real setting, utilizing the $p$ -adic Subspace Theorem (specifically the version by Schlickewei) as the primary tool. The methodology proceeds as follows:

Word Classes: The study focuses on two generalizations of periodic words defined by Bugeaud:
- $\spadesuit$ -words: Words containing prefixes of the form $W_n U_n V_n U_n$ (repeated blocks).
- $\clubsuit$ -words: Words containing prefixes of the form $W_n U_n V_n \tilde{U}_n$ (palindromic blocks, where $\tilde{U}_n$ is the reversal of $U_n$ ).
  These classes include automatic words and words of low complexity.
Approximation Strategy: The core idea is to leverage the repetitive or palindromic structure of the word to construct a sequence of approximating numbers ( $\alpha_n$ ) that are either quadratic or rational.
- For $\spadesuit$ -words, the authors construct $\alpha_n$ with periodic continued fractions.
- For $\clubsuit$ -words, they construct $\alpha_n$ with palindromic continued fractions.
  Because the original number $\alpha$ and the approximants $\alpha_n$ share a long common prefix in their continued fraction expansions, they are extremely close in the $p$ -adic metric.
Application of the Subspace Theorem: The authors define specific linear forms involving $\alpha$ and the continuants ( $A_n, B_n$ ) of the continued fraction. They demonstrate that if $\alpha$ were algebraic of degree $\ge 3$ , the quality of approximation provided by the structured words would violate the bounds imposed by the $p$ -adic Subspace Theorem. This leads to a contradiction, implying that $\alpha$ cannot be of degree 3 or higher.
Handling $p$ -adic Specifics: A significant portion of the technical work involves managing the non-canonical nature of $p$ -adic floor functions ( $s: \mathbb{Q}_p \to \mathbb{Z}[1/p]$ ). The authors impose mild conditions on the partial quotients (specifically bounds on their $p$ -adic and Euclidean norms) to ensure the convergence and applicability of the Subspace Theorem.

Key Contributions and Results
The paper's main contribution is Theorem 1.4 (and its explicit versions, Theorems 4.1 and 4.3), which states:

Let $p$ be an odd prime and $s$ be a $p$ -adic floor function. If $\alpha = [0, a_1, a_2, \dots]_s$ is a $p$ -adic number whose associated word satisfies condition $\spadesuit$ or $\clubsuit$ (with specific bounds on the growth of continuants and the size of partial quotients), then $\alpha$ is either rational, quadratic, or transcendental.

Specific findings include:

Generalization of Real Results: This provides a $p$ -adic analogue of Bugeaud's Theorem 1.3 for real numbers, extending the "unlikely intersections" philosophy to the non-archimedean setting.
Corollary 1.5: The result applies to classical $p$ -adic continued fraction algorithms by Ruban and Browkin, provided the partial quotients satisfy specific lower bounds on their $p$ -adic absolute values (e.g., $|a_n|_p \ge p^4$ for Ruban's function with repetitions).
Corollary 1.6: The result applies to automatic sequences on finite alphabets (specifically $\{n/p, m/p\}$ ), showing that for sufficiently large $p$ , the resulting $p$ -adic numbers are either rational, quadratic, or transcendental.
Refinement of Conditions: The authors provide explicit constants ( $C_\infty, C_p, k$ ) in Theorems 4.1 and 4.3 that define the precise thresholds for the partial quotients required to guarantee transcendence.

Significance and Claims
The authors claim that this work resolves Question 1.1 (whether structured words can correspond to algebraic numbers of degree $>2$ ) in the negative for a broad class of words in the $p$ -adic setting.

Overcoming Structural Difficulties: The paper highlights that the $p$ -adic case is more complex than the real case due to the infinite choices of floor functions and the lack of a universal Lagrange theorem. The authors' results are significant because they hold for any $p$ -adic floor function satisfying mild boundedness conditions, rather than being restricted to specific constructions.
Bridging Fields: The work reinforces the deep connection between word combinatorics and number theory, demonstrating that the "gap" between periodic words (algebraic degree 2) and general words (transcendental) persists in the $p$ -adic realm, even without the canonical properties of the real floor function.
Modesty: The authors note that while their results cover $\spadesuit$ and $\clubsuit$ words (which include automatic words), the question of whether all low-complexity words yield transcendental numbers remains open in general, as a direct analogue of Theorem 2.1 (linking low complexity to $\spadesuit$ ) for $\clubsuit$ words is not yet established.

In summary, the paper establishes that "machines" (finite state machines generating automatic words) producing structured $p$ -adic continued fractions generally produce transcendental numbers, with the only algebraic exceptions being those of degree at most 2.

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