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arXiv · 2211.07140

A note on reduction of tiling problems

Tom Meyerovitch·Shrey Sanadhya·Yaar Solomon
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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 $d,N_1,\ldots,N_k \in \mathbb{N}$.

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Tom Meyerovitch, Shrey Sanadhya, Yaar Solomon. 2022-11-14. A note on reduction of tiling problems. https://arxiv.org/abs/2211.07140

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