The structure of translational tilings in $\mathbb{Z}^d$

TL;DR

This paper investigates the structure of translational tilings in integer lattices, proving weak periodicity in certain cases, providing bounds on periodicity, and offering explicit formulas for universal periods, advancing understanding of tiling complexity.

Contribution

It establishes structural properties of translational tilings in $\

Findings

Level one tilings in $\

Counterexample of higher-level tilings not being weakly periodic

Polynomial bounds on periods and exponential bounds on tiling decision complexity

Abstract

We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of $\mathbb{Z}^2$ tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.