On the periodic decompositions of multidimensional configurations

Abstract

We consider $d$-dimensional configurations, that is, colorings of the $d$-dimensional integer grid $\mathbb{Z}^d$ with finitely many colors. Moreover, we interpret the colors as integers so that configurations are functions $\mathbb{Z}^d \to \mathbb{Z}$ of finite range. We say that such function is $k$-periodic if it is invariant under translations in $k$ linearly independent directions. 1-periodic functions are called periodic. It is known that if a configuration has a non-trivial annihilator, that is, if some non-trivial linear combination of its translations is the zero function, then it is a sum of finitely many periodic functions. This result is known as the periodic decomposition theorem. We prove two different improvements of it and discuss some applications of these improvements. The first improvement gives a characterization on annihilators of a configuration to guarantee the $k$-periodicity of the functions in its periodic decomposition -- for any $k$. The periodic decomposition theorem is then a special case of this result with $k=1$. We discuss an application of this result concerning translational tilings. The second improvement concerns so called sparse configurations for which the number of non-zero values in patterns grows at most linearly with respect to the diameter of the pattern. We prove that a sparse configuration with a non-trivial annihilator is a sum of finitely many periodic fibers where a fiber means a function whose support (that is, the set of points where the function gets non-zero values) is contained in a unique line. As an application of this result, we show that $\mathbb{R}$-configurations with uniformly discrete supports that have non-trivial annihilators are necessarily periodic.