A note on reduction of tiling problems

TL;DR

This paper demonstrates that tiling problems in quotient groups of integer lattices can be translated into tiling problems in standard integer lattices, linking the existence of aperiodic tiles across different group structures.

Contribution

It provides a method to reduce tiling problems in quotient groups to those in integer lattices, facilitating the transfer of results like the existence of aperiodic tiles.

Findings

Reduction of tiling problems from quotient groups to integer lattices

Implication of aperiodic tile existence in quotient groups for integer lattices

Connection to Greenfeld and Tao's disproval of the periodic tiling conjecture

Abstract

We show that translational tiling problems in a quotient of $\mathbb{Z}^d$ can be effectively reduced or ``simulated'' by translational tiling problems in $\mathbb{Z}^d$. In particular, for any $d \in \mathbb{N}$, $k < d$ and $N_1,\ldots,N_k \in \mathbb{N}$ the existence of an aperiodic tile in $\mathbb{Z}^{d-k} \times (\mathbb{Z} / N_1\mathbb{Z} \times \ldots \times \mathbb{Z} / N_k \mathbb{Z})$ implies the existence of an aperiodic tile in $\mathbb{Z}^d$. Greenfeld and Tao have recently disproved the well-known periodic tiling conjecture in $\mathbb{Z}^d$ for sufficiently large $d \in \mathbb{N}$ by constructing an aperiodic tile in $\mathbb{Z}^{d-k} \times (\mathbb{Z} / N_1\mathbb{Z} \times \ldots \times \mathbb{Z} / N_k \mathbb{Z})$ for suitable $d,N_1,\ldots,N_k \in \mathbb{N}$.