Multivariable automatic arrays and transcendence

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.