论文标题
定期瓷砖猜想的反例
A counterexample to the periodic tiling conjecture
论文作者
论文摘要
定期的瓷砖猜想断言，晶格$ \ mathbb {z}^d $的任何有限子集实际上是定期通过翻译的刻板。在这项工作中，我们将这一猜想反驳了足够大的$ d $，这也意味着对欧几里得空间的相应猜想$ \ mathbb {r}^d $的相应猜想。实际上，我们还获得了一组$ \ mathbb {z}^2 \ times g_0 $的反例，用于某些有限的abelian $ 2 $ -group $ g_0 $。我们的方法依靠编码一个“ sudoku拼图”，其行和其他非冬季线被约束在某些类别的“ 2 $ aped Iped Iparthenteral个函数”中，就某些功能方程式而言，这些方程又可以用作单个平铺方程，然后证明对这种sudoku puzzu puzzue puzzle puzzle puzzuzz puzzuze puzzuze upeal butynonnon-periodic cy nor-periodic concodect。
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.
