# An Alphabetical Approach to Nivat's Conjecture

## TL;DR

This paper introduces an alphabetical version of Morse-Hedlund Theorem and demonstrates that configurations with bounded complexity over a quasi-regular set are necessarily periodic, advancing the understanding of Nivat's conjecture.

## Contribution

It proposes an improved, alphabetically adapted Morse-Hedlund Theorem and links bounded complexity to periodicity in configurations, offering a new approach towards Nivat's conjecture.

## Key findings

- Configurations containing all alphabet letters with bounded complexity are periodic.
- An improved version of Morse-Hedlund Theorem is established for an alphabetic setting.
- Bounded complexity condition implies periodicity for configurations over quasi-regular sets.

## 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.