Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes
<|reference_start|>Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes: Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [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, 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 shows that translational tiling of the $3$-dimensional space with a set of $5$ polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from $3$-dimensional space to $4$-dimensional space, we manage to show that translational tiling of the $4$-dimensional space with a set of $4$ tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the $n$-dimensional space with a monotile, for some fixed $n$.<|reference_end|>
@article{yang2024undecidability,
    title={Undecidability of Translational Tiling of the 4-dimensional Space with a
  Set of 4 Polyhypercubes},
    author={Chao Yang, Zhujun Zhang},
    journal={arXiv preprint arXiv:2409.00846},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.00846},
    primaryClass={math.CO cs.CC math.MG}
}