A counterexample to the periodic tiling conjecture
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909308105523200 |
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| author | Greenfeld, Rachel Tao, Terence |
| author_facet | Greenfeld, Rachel Tao, Terence |
| contents | The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_15847 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A counterexample to the periodic tiling conjecture Greenfeld, Rachel Tao, Terence Combinatorics Dynamical Systems 05B45, 52C22, 52C23 The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic. |
| title | A counterexample to the periodic tiling conjecture |
| topic | Combinatorics Dynamical Systems 05B45, 52C22, 52C23 |
| url | https://arxiv.org/abs/2211.15847 |