An Alphabetical Approach to Nivat´s Conjecture

Abstract
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, we consider an alphabetical version of the Morse-Hedlund Theorem. Following methods highlighted by Cyr and Kra [1], we show that, for a configuration  2 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 1 2jUj + jAj ? 1, then  is periodic.