Multivariable automatic arrays and transcendence

Aadrita Paul; Anwesh Ray

arxiv: 2604.12468 · v1 · submitted 2026-04-14 · 🧮 math.NT · math.CO

Multivariable automatic arrays and transcendence

Aadrita Paul , Anwesh Ray This is my paper

Pith reviewed 2026-05-10 14:54 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords automatic sequencestranscendencemultidimensional arraysSchmidt subspace theoremDiophantine approximation

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The pith

Real numbers from multidimensional automatic arrays with independent bases are rational or transcendental.

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

The paper proves that real numbers formed as infinite sums over multidimensional arrays of bounded automatic sequences, weighted by powers of integer bases whose logarithms are linearly independent over the rationals, are either rational or transcendental. The construction uses a fixed integer-valued function of the sequence entries in each dimension. The authors combine the finite-value property of automatic sequences with Schmidt's Subspace Theorem to obtain the dichotomy. This extends the one-dimensional automatic sequence transcendence criterion to the multivariable setting.

Core claim

Let a1,...,ar >=2 be integers such that log a1,...,log ar are Q-linearly independent. Let (p_n(i)) be bounded automatic sequences for each i=1 to r and let f map Z^r to Z. Define alpha as the sum over n1,...,nr >=0 of f(p_n1(1),...,p_nr(r)) divided by a1^n1 ... ar^nr. Then alpha is rational or transcendental.

What carries the argument

The multidimensional weighted series alpha, whose transcendence or rationality follows from applying Schmidt's Subspace Theorem to the combinatorial constraints imposed by automatic sequences.

Load-bearing premise

The logarithms of the bases are linearly independent over the rationals and the sequences are bounded and automatic.

What would settle it

An explicit construction of bounded automatic sequences and bases with independent logs whose weighted sum alpha is an algebraic irrational number would disprove the claim.

read the original abstract

We study real numbers defined by multidimensional automatic arrays weighted by multiplicatively independent bases. Let $a_1, \dots, a_r\geq 2$ be integers such that $\log a_1, \dots, \log a_r$ are $\mathbb Q$-linearly independent. Given bounded automatic sequences $(p_n(i))_{n\geq 0}$ with $i=1, \dots , r$ and a function $f:\mathbb Z^r\rightarrow \mathbb Z$, we consider the associated series $\alpha = \sum_{n_1,\dots,n_r \geq 0} \frac{f(p_{n_1}(1),\dots,p_{n_r}(r))}{a_1^{n_1}\cdots a_r^{n_r}}$. Using combinatorial properties of automatic sequences and Schmidt's Subspace Theorem, we prove that $\alpha$ is either rational or transcendental. This extends a result of Adamczewski and Bugeaud to the multidimensional setting.

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.

Desk Editor's Note private letter to a colleague

This paper extends the Adamczewski-Bugeaud theorem to r-dimensional automatic arrays, proving the weighted sums are rational or transcendental under the stated conditions.

read the letter

The key takeaway is that the authors extend the Adamczewski-Bugeaud theorem from one dimension to multiple variables. They show that a real number alpha defined by a weighted sum over an r-dimensional automatic array, with multiplicatively independent bases, must be rational or transcendental. The new part is handling the multivariable case. In one dimension, automatic sequences give numbers whose base-a expansions have low complexity, leading to good rational approximations. Here, they generalize that to arrays where each dimension has its own automatic sequence, and f combines them. The proof adapts the combinatorial argument for the complexity of the digits and applies Schmidt's Subspace Theorem to the resulting approximations in the multi-base setting. The linear independence of the logs prevents the approximations from being dependent in a way that would allow algebraic numbers. This works well because the boundedness and automaticity are preserved in the product, and the weighting is integer-valued, keeping everything integral for the Diophantine part. The paper organizes a natural class of numbers that arise from higher-dimensional automata or tilings. The soft spots are not major. The result requires the specific assumptions on the bases and sequences, which are necessary but narrow the applicability. Without the full proof, one cannot check every step of how they construct the approximations from the partial sums, but the outline matches the 1D strategy without obvious gaps. No hidden parameters or circular reasoning appears. This paper is for specialists in transcendence theory and combinatorics on words. A reader who knows the original Adamczewski-Bugeaud paper will see the extension as a natural next step and get value from seeing how the tools carry over. It deserves a serious referee. The claim is precise, the method is standard but applied newly, and the area benefits from such generalizations. I recommend sending it for peer review.

Referee Report

0 major / 3 minor

Summary. The paper considers real numbers α given by the multidimensional series ∑ f(p_{n1}(1),…,p_{nr}(r)) / (a1^{n1}⋯ar^{nr}), where the bases a_i ≥ 2 have Q-linearly independent logarithms, the sequences (p_n(i)) are bounded and automatic, and f maps to integers. It proves that any such α is either rational or transcendental by combining the low-complexity combinatorial structure of automatic sequences (to produce strong Diophantine approximations) with an application of Schmidt’s Subspace Theorem in the multi-base setting. This extends the one-dimensional Adamczewski–Bugeaud theorem.

Significance. If the proof is correct, the result supplies a natural multidimensional generalization of a central theorem on the transcendence of automatic series. It shows that the Diophantine-approximation method based on automatic-sequence complexity continues to work when several independent bases are present, provided the logarithmic independence hypothesis prevents collapse of the approximations. This strengthens the link between automata theory and Diophantine approximation and may serve as a template for further extensions to other classes of low-complexity multidimensional sequences.

minor comments (3)

The abstract and introduction should explicitly recall the definition of a k-automatic sequence (or at least cite a standard reference such as Allouche–Shallit) so that readers outside the subfield can follow the combinatorial argument without external lookup.
The precise statement of Schmidt’s Subspace Theorem invoked in the proof (including the precise height and degree bounds used) should be stated as a numbered lemma or proposition in the preliminaries, rather than left implicit.
Notation: the indexing of the sequences as (p_n(i))_{n≥0} with i=1,…,r is slightly ambiguous; a clearer notation such as (p^{(i)}_n)_{n≥0} would avoid any confusion between the multi-index n and the base index i.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the main result and its relation to the one-dimensional Adamczewski–Bugeaud theorem. As no specific major comments appear in the report, we have no points requiring detailed rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity; proof relies on external theorems

full rationale

The derivation applies combinatorial properties of automatic sequences (standard in the literature) together with Schmidt's Subspace Theorem—an external Diophantine result—to conclude that the constructed α is rational or transcendental. The Q-linear independence of the log a_i, boundedness of the sequences, and integrality of f are independent assumptions that do not reduce the target statement to a fitted parameter or self-definition. The argument extends the one-dimensional Adamczewski–Bugeaud result without invoking any load-bearing self-citation or renaming of known patterns as new derivations. No equation or step equates the conclusion to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on the standard Schmidt Subspace Theorem and established combinatorial facts about automatic sequences; no new free parameters or invented entities are introduced.

axioms (1)

standard math Schmidt's Subspace Theorem
Invoked to control Diophantine approximations and conclude transcendence or rationality.

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discussion (0)

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Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

On the complexity of algebraic numbers

[AB07] Boris Adamczewski and Yann Bugeaud. On the complexity of algebraic numbers. I. Expansions in integer bases. Ann. of Math. (2), 165(2):547–565, 2007. [AC03] Boris Adamczewski and Julien Cassaigne. On the transcendence of real numbers with a regular expansion.J. Number Theory, 103(1):27–37, 2003. [All00] Jean-Paul Allouche. Nouveaux résultats de tran...

work page 2007

[1] [1]

On the complexity of algebraic numbers

[AB07] Boris Adamczewski and Yann Bugeaud. On the complexity of algebraic numbers. I. Expansions in integer bases. Ann. of Math. (2), 165(2):547–565, 2007. [AC03] Boris Adamczewski and Julien Cassaigne. On the transcendence of real numbers with a regular expansion.J. Number Theory, 103(1):27–37, 2003. [All00] Jean-Paul Allouche. Nouveaux résultats de tran...

work page 2007