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.

翻译：是否存在一个固定的维度$n$，使得$\mathbb{Z}^n$的单砖平移铺砌问题是不可判定的？若干近期研究结果支持对该问题的肯定回答。Greenfeld与Tao通过证明在足够高的维度$n$中存在非周期单砖，从而否定了周期铺砌猜想[Ann. Math. 200(2024), 301-363]。在另一篇论文[即将发表于J. Eur. Math. Soc.]中，他们还证明当维度$n$作为输入的一部分时，$\mathbb{Z}^n$子集的单砖平移铺砌问题是不可判定的。这两项结果为“存在某个固定$n$使得$\mathbb{Z}^n$的单砖平移铺砌不可判定”的猜想提供了强有力的证据。本文通过证明四维空间中三个连通砖块的平移铺砌问题具有不可判定性，为该猜想提供了另一项支持性结果。