Undecidability of Translational Tiling with Three Tiles
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909428198932480 |
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| author | Yang, Chan Zhang, Zhujun |
| author_facet | Yang, Chan Zhang, Zhujun |
| contents | Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by showing that an aperiodic monotile exists in sufficiently high dimension $n$ [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, then 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 gives another supportive result for this conjecture by showing that translational tiling of the $4$-dimensional space with a set of three connected tiles is undecidable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10646 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Undecidability of Translational Tiling with Three Tiles Yang, Chan Zhang, Zhujun Combinatorics Computational Complexity Computational Geometry Metric Geometry Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by showing that an aperiodic monotile exists in sufficiently high dimension $n$ [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, then 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 gives another supportive result for this conjecture by showing that translational tiling of the $4$-dimensional space with a set of three connected tiles is undecidable. |
| title | Undecidability of Translational Tiling with Three Tiles |
| topic | Combinatorics Computational Complexity Computational Geometry Metric Geometry |
| url | https://arxiv.org/abs/2412.10646 |