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On the periodic decompositions of multidimensional configurations

Pyry Herva, Jarkko Kari

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.

On the periodic decompositions of multidimensional configurations

Abstract

We consider -dimensional configurations, that is, colorings of the -dimensional integer grid with finitely many colors. Moreover, we interpret the colors as integers so that configurations are functions of finite range. We say that such function is -periodic if it is invariant under translations in 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 -periodicity of the functions in its periodic decomposition -- for any . The periodic decomposition theorem is then a special case of this result with . 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.
Paper Structure (7 sections, 14 theorems, 60 equations)

This paper contains 7 sections, 14 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Configurations and periodicity
  5. Algebraic concepts
  6. Periodic decomposition of configurations with specific annihilators
  7. Periodic decomposition of sparse configurations

Key Result

Theorem 2

Let $c\in \mathcal{A}^{\Z^d}$ be a configuration and assume that it has a non-trivial annihilator $f$. For all $\mathbf{u} \in \text{\rm supp}(f)$ there exist pairwise non-parallel vectors $\mathbf{v}_1,\ldots,\mathbf{v}_m \in \Z^d$ such that each $\mathbf{v}_i$ is parallel to $\mathbf{u}_i-\mathbf{

Theorems & Definitions (27)

  • Remark 1
  • Theorem 2: icalp, fullproofs
  • Theorem 3: Periodic decomposition theorem fullproofs
  • Remark 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7: DLT_invited_jarkko
  • Theorem 8
  • ...and 17 more