Multivariable automatic arrays and transcendence

Abstract

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