Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes
<|reference_start|>Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes: 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.<|reference_end|>
@article{yang2024undecidability,
    title={Undecidability of Translational Tiling of the 3-dimensional Space with a
  Set of 6 Polycubes},
    author={Chao Yang, Zhujun Zhang},
    journal={arXiv preprint arXiv:2408.02196},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.02196},
    primaryClass={math.CO cs.CC math.MG}
}