论文标题
关于减少瓷砖问题的注释
A note on reduction of tiling problems
论文作者
论文摘要
我们表明，可以通过$ \ Mathbb {z}^d $中的翻译瓷砖问题有效地减少$ \ mathbb {z}^d $商的转化瓷砖问题。特别是，对于任何$ d \ in \ mathbb {n} $，$ k <d $和$ n_1，\ ldots，n_k \ in \ mathbb {n} $，存在于$ \ \ \ mthbb {z}^{z}^d-k}^{d-k}^{d-k} {d-k} {d \ timit（z \ z {z {z {z {z {z {z { \ times \ ldots \ times \ mathbb {z} / n_k \ mathbb {z}）$意味着在$ \ mathbb {z}^d $中存在上的aperiodic瓷砖。 Greenfeld和Tao最近以$ \ Mathbb {Z}^d $中的众所周知的周期性瓷砖猜想，用于足够大的$ d \ in \ Mathbb {n} $，通过在$ \ \ \ m m iathbb {n} $中构建aperiodic瓷砖，以$ \ mathbb {z}^{z} \ times \ ldots \ times \ times \ mathbb {z} / n_k \ mathbb {z}）$用于合适的$ d，n_1，\ ldots，n_k \ in \ mathbb {n} $。
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 $d,N_1,\ldots,N_k \in \mathbb{N}$.
