Some variants of the periodic tiling conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913831525023744 |
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| author | Greenfeld, Rachel Tao, Terence |
| author_facet | Greenfeld, Rachel Tao, Terence |
| contents | The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$, that if there is a set $A$ that solves the tiling equation $\mathbb{1}_F * \mathbb{1}_A = 1$, there is also a periodic solution $\mathbb{1}_{A_{\mathrm{p}}}$. This conjecture is known to hold for some groups $G$ and fail for others. In this paper we establish three variants of the PTC. The first (due to Tim Austin) replaces the constant function $1$ on the right-hand side of the tiling equation by $0$, and the indicator functions $\mathbb{1}_F$ and $\mathbb{1}_A$ by bounded integer-valued functions. The second, which applies in $G=\mathbb{Z}^2$, replaces the right-hand side of the tiling equation by an integer-valued periodic function, and the functions $\mathbb{1}_F$ and $\mathbb{1}_A$ on the left-hand side by bounded integer-valued functions. The third (which is the most difficult to establish) is similar to the second, but retains the property of both $\mathbb{1}_A$ and $\mathbb{1}_{A_{\mathrm{p}}}$ being indicator functions; in particular, we establish the PTC for multi-tilings in $G=\mathbb{Z}^2$. As a result, we obtain the decidability of constant-level integer tilings in any finitely generated Abelian group $G$ and multi-tilings in $G=\mathbb{Z}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06757 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some variants of the periodic tiling conjecture Greenfeld, Rachel Tao, Terence Classical Analysis and ODEs Dynamical Systems Logic 52C23, 03B25, 37B52 The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$, that if there is a set $A$ that solves the tiling equation $\mathbb{1}_F * \mathbb{1}_A = 1$, there is also a periodic solution $\mathbb{1}_{A_{\mathrm{p}}}$. This conjecture is known to hold for some groups $G$ and fail for others. In this paper we establish three variants of the PTC. The first (due to Tim Austin) replaces the constant function $1$ on the right-hand side of the tiling equation by $0$, and the indicator functions $\mathbb{1}_F$ and $\mathbb{1}_A$ by bounded integer-valued functions. The second, which applies in $G=\mathbb{Z}^2$, replaces the right-hand side of the tiling equation by an integer-valued periodic function, and the functions $\mathbb{1}_F$ and $\mathbb{1}_A$ on the left-hand side by bounded integer-valued functions. The third (which is the most difficult to establish) is similar to the second, but retains the property of both $\mathbb{1}_A$ and $\mathbb{1}_{A_{\mathrm{p}}}$ being indicator functions; in particular, we establish the PTC for multi-tilings in $G=\mathbb{Z}^2$. As a result, we obtain the decidability of constant-level integer tilings in any finitely generated Abelian group $G$ and multi-tilings in $G=\mathbb{Z}^2$. |
| title | Some variants of the periodic tiling conjecture |
| topic | Classical Analysis and ODEs Dynamical Systems Logic 52C23, 03B25, 37B52 |
| url | https://arxiv.org/abs/2505.06757 |