Nivat’s Conjecture Holds for Sums of Two Periodic Configurations

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 $P_c(m,n) \le mn$ 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 $P_c(m,n) \le mn/2$, 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.

M. Szabados—Research supported by the Academy of Finland Grant 296018.