Nivat's conjecture holds for sums of two periodic configurations

TL;DR

This paper proves Nivat's conjecture for configurations that are sums of two periodic configurations, confirming that low complexity configurations are necessarily periodic in this case.

Contribution

It demonstrates that Nivat's conjecture holds for sums of two periodic configurations, providing an alternative proof for the case of low complexity configurations.

Findings

Nivat's conjecture holds for sums of two periodic configurations.

Configurations with low complexity are necessarily periodic.

Provides an alternative proof for a known result on periodicity.

Abstract

Nivat's conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps $\mathbb Z^2 \rightarrow \mathcal A$ where $\mathcal A$ is a finite set of symbols. Such configurations are often understood as colorings of a two-dimensional square grid. Let $P_c(m,n)$ denote the number of distinct $m \times n$ block patterns occurring in a configuration $c$. Configurations satisfying $P_c(m,n) \leq mn$ for some $m,n \in \mathbb N$ are said to have low rectangular complexity. Nivat conjectured that such configurations are necessarily periodic. Recently, Kari and the author showed that low complexity configurations can be decomposed into a sum of periodic configurations. In this paper we show that if there are at most two components, Nivat's conjecture holds. As a corollary we obtain an alternative proof of a result of Cyr and Kra: If there exist $m,n \in \mathbb N$ such that $P_c(m,n) \leq mn/2$, then $c$ is periodic. The technique used in this paper combines the algebraic approach of Kari and the author with balanced sets of Cyr and Kra.