This paper focuses on the undecidability of translational tiling of $n$-dimensional space $\mathbb{Z}^n$ with a set of $k$ tiles. It is known that tiling $\mathbb{Z}^2$ with translated copies with a set of $8$ tiles is undecidable. Greenfeld and Tao gave strong evidence in a series of works that for sufficiently large dimension $n$, the translational tiling problem for $\mathbb{Z}^n$ might be undecidable for just one tile. This paper shows the undecidability of translational tiling of $\mathbb{Z}^3$ with a set of $6$ tiles.

翻译：本文主要研究使用$k$个瓦片对$n$维空间$\mathbb{Z}^n$进行平移铺砌的不可判定性问题。已知使用$8$个瓦片通过平移副本铺砌$\mathbb{Z}^2$是不可判定的。Greenfeld和Tao在一系列工作中给出了有力证据，表明对于足够大的维度$n$，即使仅使用一个瓦片，$\mathbb{Z}^n$的平移铺砌问题也可能是不可判定的。本文证明了使用$6$个瓦片对$\mathbb{Z}^3$进行平移铺砌的不可判定性。