An alphabetical approach to Nivat’s conjecture

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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 that contains all letters of a given finite alphabet A, if its complexity with respect to a quasi-regular set (a finite set whose convex hull on is described by pairs of edges with identical size) is bounded from above by , then η is periodic.