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(2) -> A where 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 mxn block patterns occurring in a configuration c. Configurations satisfying P-c(m, n) <= mn for some m, n is an element of 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 is an element of 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.