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arXiv · 1904.04897

An Alphabetical Approach to Nivat's Conjecture

Cleber F. Colle·Eduardo Garibaldi
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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, 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 $\frac{1}{2}|\mathcal{S}|+|A|-1$, then $\eta$ is periodic.

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Cleber F. Colle, Eduardo Garibaldi. 2019-04-09. An Alphabetical Approach to Nivat's Conjecture. https://doi.org/10.1088/1361-6544/ab7638

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