On Configurations of Order 2

TL;DR

This paper investigates configurations on the 2D integer lattice with non-trivial annihilators, revealing how their periodic structures can be detected and decomposed, with implications for tilings and their order bounds.

Contribution

It establishes a method to detect periodic directions from annihilators, relates the order of weakly periodic configurations to their decomposition, and provides bounds on tiling orders based on geometric properties.

Findings

Order of a weakly periodic configuration equals the number of components in its minimal decomposition.

Configurations of order 2 can be expressed as sums of two 1-periodic configurations.

Tilings with prime square tiles have points of order at most 2 in their orbit closure.

Abstract

Let $c:\mathbb Z^2\to \{0, 1\}$ be a configuration with a non-trivial annihilator. We show that if $c$ is weakly periodic then the directions of periodicity in a minimal weakly periodic decomposition of $c$ can be detected from the annihilator ideal associated to $c$. We show that the order of a weakly periodic configuration is same as the number of components in any minimal decomposition into $1$-periodic elements. We then give an upper bound on the order in terms of the support of any of its annihilators. In the special case of tilings this gives an upper bound on the order of any tiling in terms of a geometric quantity associated to the tile. We prove that if $c:\mathbb Z^2\to \{0, 1\}$ is a configuration having a non-trivial annihilator and has order $2$ then it can be written as a sum of two $1$-periodic configurations valued in $\{0, 1\}$. Lastly we show that any tiling of $\mathbb Z^2$ by a tile of cardinality the square of a prime has a point of order at most $2$ in its orbit closure.