An alphabetical approach to Nivat’s conjecture

Since techniques used to address the Nivat’s conjecture usually rely on Morse–Hedlund theorem, an improved version of this classical result may mean a new step towards a proof for the conjecture. In this paper, considering an alphabetical version of the Morse–Hedlund theorem, we show that, for a configuration η∈AZ2 that contains all letters of a given finite alphabet A, if its complexity with respect to a quasi-regular set U⊂Z2 (a finite set whose convex hull on R2 is described by pairs of edges with identical size) is bounded from above by 12|U|+|A|−1, then η is periodic.