论文标题
定期瓷砖猜想（公告）的反例
A counterexample to the periodic tiling conjecture (announcement)
论文作者
论文摘要
定期的瓷砖猜想断言，晶格$ \ mathbb {z^d} $的任何有限子集实际上是定期通过翻译的刻板。我们在这里宣布对足够大的$ d $的猜想有争议，这也意味着对欧几里得空间的相应猜想$ \ mathbb {r^d} $进行了证明。实际上，我们还获得了一组$ \ mathbb {z^2} \ times g_0 $的反例，用于某些有限的abelian $ g_0 $。我们的方法依赖于某些功能方程式编码特定类别的“ $ p $ - 构造函数”。
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. We announce here a disproof of 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 $G_0$. Our methods rely on encoding a certain class of "$p$-adically structured functions" in terms of certain functional equations.
