On the periodic decompositions of multidimensional configurations

TL;DR

This paper advances the understanding of periodic decompositions of multidimensional integer grid configurations, providing new characterizations and extending results to sparse configurations with finitely many periodic components.

Contribution

It introduces two improvements to the periodic decomposition theorem, including a characterization of annihilators for guaranteed k-periodicity and a decomposition result for sparse configurations.

Findings

Characterization of annihilators guaranteeing k-periodicity

Decomposition of sparse configurations into periodic fibers

Extension of periodic decomposition to sparse 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. 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. 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 $k=1$. 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 non-zero values lie on a unique line.