Indistinguishable asymptotic pairs and multidimensional Sturmian configurations

Two asymptotic configurations on a full ${\mathbb{Z}}^d$-shift are indistinguishable if for every finite pattern the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of ${\mathbb{Z}}^d$. We prove that indistinguishable asymptotic pairs satisfying a “flip condition” are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together the two results provide a generalization to ${\mathbb{Z}}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction.