Nivat's Conjecture Holds for Sums of Two Periodic Configurations

Nivat’s conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps Z → A where A is a finite set of symbols. Such configurations are often understood as colorings of a two-dimensional square grid. Let Pc(m,n) denote the number of distinct m × n block patterns occurring in a configuration c. Configurations satisfying Pc(m,n) ≤ mn for some m,n ∈ 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 ∈ N such that Pc(m,n) ≤ 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.