三维空间中6个多立方体集合的平移平铺不可判定性
Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes
AI总结：
本文证明了用6个多立方体集合对三维整数格进行平移平铺的不可判定性，扩展了已知的二维结果，并推进了该领域的研究。
摘要：
本文关注用$k$个瓦片对$n$维空间$\mathbb{Z}^n$进行平移平铺的不可判定性。已知用8个瓦片的平移副本平铺$\mathbb{Z}^2$是不可判定的。Greenfeld和Tao在一系列工作中给出了强有力的证据，表明对于足够大的维度$n$，$\mathbb{Z}^n$的平移平铺问题可能仅用一个瓦片就不可判定。本文证明了用6个瓦片对$\mathbb{Z}^3$进行平移平铺的不可判定性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
