标题： 三个瓷砖的平移铺砌的不可判定性 显示英文标题

标题： Undecidability of Translational Tiling with Three Tiles

摘要： 是否存在一个固定维度$n$，使得在$\mathbb{Z}^n$上使用单块进行平移铺砌是不可判定的？ 最近的几个结果支持对此问题的肯定回答。 Greenfeld 和 Tao 通过证明在足够高的维度$n$中存在非周期性单块，从而否定了周期性铺砌猜想 [Ann. Math. 200 (2024), 301-363]。 在另一篇论文 [即将发表于 J. Eur. Math. Soc.] 中，他们还表明，如果维度$n$是输入的一部分，那么在$\mathbb{Z}^n$的子集上使用一块进行平移铺砌是不可判定的。 这两个结果为以下猜想提供了强有力的证据：对于某个固定的$n$，$\mathbb{Z}^n$上使用单块进行平移铺砌是不可判定的。 本文通过证明使用三个连通瓷砖在$4$维空间中的平移铺砌是不可判定的，为该猜想提供了另一个支持性结果。 显示英文摘要

摘要： Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by showing that an aperiodic monotile exists in sufficiently high dimension $n$ [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, then the translational tiling for subsets of $\mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper gives another supportive result for this conjecture by showing that translational tiling of the $4$-dimensional space with a set of three connected tiles is undecidable.