论文标题
$ \ mathbb {z}^d $中翻译瓷砖的结构
The structure of translational tilings in $\mathbb{Z}^d$
论文作者
论文摘要
我们通过有限瓷砖在$ \ mathbb {z}^d $中的周期性功能的翻译砖中获得结构性结果。特别是，我们表明，$ \ mathbb {z}^2 $中周期性设置的任何一个层次都必须是弱周期性的（单独定期在一个方向上的集合的连接结合），但呈现一个$ \ mathbb {z}^2 $的较高级别平铺的反例，但未能实现周期性的周期性。我们还建立了二维周期性瓷砖猜想的定量版本，该版本断言，$ \ mathbb {z}^2 $中的任何有限瓷砖都必须通过在此期间提供多项式绑定来承认定期瓷砖；这也给出了指数类型的限制，即确定$ \ mathbb {z}^2 $ tiles的给定有限子集的计算复杂性。作为我们结构理论的副产品，我们还为一维图块的所有瓷砖都获得了普遍时期的明确公式。
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of $\mathbb{Z}^2$ tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.
