An Alphabetical Approach to Nivat's Conjecture

Since techniques used to address the Nivat’s conjecture usually relies 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 $\eta \in A^{{\mathbb{Z}}^2}$ that contains all letters of a given finite alphabet $A$, if its complexity with respect to a quasi-regular set $\mathcal{S}\subset {\mathbb{Z}}^2$ (a finite set whose convex hull on ${\mathbb{R}}^2$ is described by pairs of edges with identical size) is bounded from above by $\frac{1}{2}\mathrm{\mid }\mathcal{S}\mathrm{\mid }+\mathrm{\mid }A\mathrm{\mid }-1$, then $\eta$ is periodic.