标题： 关于用一组$3$个多面体在$3$维空间中铺砌的不可判定性 显示英文标题

标题： On the Undecidability of Tiling the $3$-dimensional Space with a Set of $3$ Polycubes

摘要： 翻译结果： 可翻译的平移铺砌问题是数学各个领域中最基本和最具代表性的不可判定问题之一。 Greenfeld和Tao近年来在平移铺砌的不可判定性方面取得了两个显著的结果。 一个是存在一个足够高维空间中的非周期单体。 另一个是，当空间的维度作为输入的一部分时，用单个瓷砖对空间的周期子集进行平移铺砌是不可判定的。 这两个结果支持以下猜想：存在一个固定的维度$n$，使得用单个瓷砖进行平移铺砌是不可判定的。 解决这个猜想的一种策略是证明在固定维度空间中使用一组$k$个瓷砖进行平移铺砌的不可判定性，其中正整数$k$尽可能小。 在本文中，展示了使用一组$3$个多立方体在$3$维空间中进行平移铺砌是不可判定的。 显示英文摘要

摘要： Translational tiling problems are among the most fundamental and representative undecidable problems in all fields of mathematics. Greenfeld and Tao obtained two remarkable results on the undecidability of translational tiling in recent years. One is the existence of an aperiodic monotile in a space of sufficiently large dimension. The other is the undecidability of translational tiling of periodic subsets of space with a single tile, provided that the dimension of the space is part of the input. These two results support the following conjecture: there is a fixed dimension $n$ such that translational tiling with a single tile is undecidable. One strategy towards solving this conjecture is to prove the undecidability of translational tiling of a fixed dimension space with a set of $k$ tiles, for a positive integer $k$ as small as possible. In this paper, it is shown that translational tiling the $3$-dimensional space with a set of $3$ polycubes is undecidable.