Multivariable automatic arrays and transcendence

1. Problem Statement and Motivation

The paper addresses a fundamental question in transcendence theory regarding the arithmetic nature of real numbers defined by multidimensional automatic sequences.

• Context: The interaction between automata theory and Diophantine approximation has established that real numbers with "low combinatorial complexity" (specifically those generated by finite automata) often exhibit rigid arithmetic properties. A seminal result by Adamczewski and Bugeaud (2007) proved that for an integer base $b \ge 2$, if $(a_n)$ is a bounded automatic sequence of integers, the real number $\alpha = \sum_{n \ge 0} a_n b^{-n}$ is either rational or transcendental.
• The Gap: While the one-dimensional case is well-understood, the multidimensional analogue remained open. Specifically, given $r \ge 2$ multiplicatively independent integers $a_1, \dots, a_r$ and $r$ bounded automatic sequences $(p_n^{(i)})_{n \ge 0}$, what is the nature of the series:
  $$\alpha = \sum_{n_1, \dots, n_r \ge 0} \frac{f(p_{n_1}^{(1)}, \dots, p_{n_r}^{(r)})}{a_1^{n_1} \cdots a_r^{n_r}}$$
  where $f: \mathbb{Z}^r \to \mathbb{Z}$ is a function?
• Objective: The authors aim to extend the Adamczewski-Bugeaud result to this $r$-dimensional setting, proving that such numbers $\alpha$ are also either rational or transcendental.

2. Methodology

The proof synthesizes combinatorial properties of automatic sequences with deep tools from Diophantine approximation, specifically the Schmidt Subspace Theorem. The methodology proceeds in three main stages:

A. Combinatorial Properties: Stammering Sequences

The authors utilize the concept of stammering sequences (sequences containing arbitrarily long prefixes with large repeated blocks).

• Key Lemma: They establish that any $k$-automatic sequence is a stammering sequence (Proposition 2.9). This means for any automatic sequence, there exist sequences of words $(U_n)$ and $(V_n)$ such that $U_n V_n^w$ is a prefix of the sequence for some $w > 1$, with the ratio $|U_n|/|V_n|$ bounded and $|V_n|$ strictly increasing.
• Multidimensional Extension: For the $r$ sequences $(p_n^{(i)})$, they identify parameters $r_n^{(i)}$ (preperiod length) and $s_n^{(i)}$ (period length) derived from the stammering condition, ensuring that the sequences can be approximated by eventually periodic sequences with controlled error.

B. Construction of Auxiliary Rational Approximations

To apply Diophantine tools, the authors construct a sequence of rational numbers $\alpha_n$ that approximate $\alpha$.

• Approximation: They replace each automatic sequence $(p_k^{(i)})$ with an eventually periodic sequence $(p_k^{(i, n)})$ that matches the original sequence up to a certain point and then repeats.
• Rationality: Because the approximating sequences are eventually periodic, the resulting series $\alpha_n = \sum \frac{c_{n_1, \dots, n_r}^{(n)}}{a_1^{n_1} \cdots a_r^{n_r}}$ can be expressed explicitly as a rational number.
• Explicit Form: Proposition 4.1 shows that $\alpha_n$ takes the form:
  $$\alpha_n = \frac{P_n(a_1, \dots, a_r)}{\prod_{i=1}^r a_i^{r_n^{(i)}} (a_i^{s_n^{(i)}} - 1)}$$
  where $P_n$ is a polynomial with integer coefficients of bounded degree and bounded coefficients.

C. Diophantine Approximation: The Subspace Theorem

The core of the proof involves showing that if $\alpha$ were an irrational algebraic number, it would violate the Schmidt Subspace Theorem.

• Setup: They define a vector $x_n \in \mathbb{Q}^{2r+1}$ involving the denominators of the rational approximations and the polynomial $P_n$.
• Linear Forms: They construct linear forms $L_{v,i}$ over a number field $K$ (involving the primes dividing the bases $a_i$) such that the product of these forms evaluated at $x_n$ is extremely small.
• Estimation:
  • The "error" term $L_{\infty, 2r+1}(x_n)$ corresponds to $\alpha - \alpha_n$. Due to the stammering property (the exponent $w > 1$), this error decays faster than the height of the vector $x_n$ would allow for an algebraic irrational.
  • Specifically, they show $\prod |L_{v,i}(x_n)|_v \ll H(x_n)^{-(2r+1+\epsilon)}$.
• Application of Theorem: By the Schmidt Subspace Theorem, infinitely many $x_n$ must lie in a proper linear subspace. This implies a linear relation among the components of $x_n$ with rational coefficients.

3. Key Contributions and Results

Theorem A (Main Result):
Let $r \ge 2$, and let $a_1, \dots, a_r \ge 2$ be multiplicatively independent integers (i.e., $\log a_1, \dots, \log a_r$ are linearly independent over $\mathbb{Q}$). Let $(p_n^{(i)})$ be bounded $k_i$-automatic sequences and $f: \mathbb{Z}^r \to \mathbb{Z}$. Then the real number
$$\alpha = \sum_{n_1, \dots, n_r \ge 0} \frac{f(p_{n_1}^{(1)}, \dots, p_{n_r}^{(r)})}{a_1^{n_1} \cdots a_r^{n_r}}$$
is either rational or transcendental.

Technical Nuances:

• Rationality Condition: The proof demonstrates that if $\alpha$ is algebraic, the linear dependence derived from the Subspace Theorem forces $\alpha$ to be rational.
• Contradiction for Irrationality: If one assumes $\alpha$ is irrational algebraic, the linear dependence leads to a relation of the form $\sum c_I \prod a_i^{s_n^{(i)}} = 0$. Due to the multiplicative independence of the bases $a_i$ and the strictly increasing nature of the periods $s_n^{(i)}$, this leads to a contradiction (a non-trivial linear relation between logarithms of $a_i$ with integer coefficients, which is impossible).

4. Significance

1. Generalization of Adamczewski-Bugeaud: This work successfully lifts the landmark 2007 result from the 1-dimensional case to arbitrary dimensions $r \ge 2$. It confirms that the "rigidity" of automatic sequences (forcing numbers to be rational or transcendental) persists even in complex, multidimensional settings.
2. Interdisciplinary Synthesis: The paper provides a robust framework combining:
   • Automata Theory: Using the $k$-kernel and morphism properties to establish the "stammering" condition.
   • Combinatorics on Words: Analyzing the structure of prefixes and repetitions.
   • Diophantine Approximation: Utilizing the Schmidt Subspace Theorem and $S$-unit theorems to handle the arithmetic constraints.
3. Multidimensional Automatic Arrays: It formally treats arrays generated by multiple automatic sequences as a single object, defining them via a product automaton (Theorem 2.12), which is a necessary step for handling the multidimensional series.
4. Future Directions: The authors pose the question of whether this result holds for any bounded $[k_1, \dots, k_r]$-automatic array (not just those generated by independent sequences), suggesting a path for further research into the arithmetic nature of general multidimensional automatic series.

In summary, the paper resolves a significant open problem in transcendence theory by proving that the arithmetic nature of real numbers defined by multidimensional automatic data is binary: they cannot be irrational algebraic numbers.