论文标题
凸多角形锥的瓷砖和自植物瓷砖的拓扑特性
Tilings of convex polyhedral cones and topological properties of self-affine tiles
论文作者
论文摘要
令$ \ textbf {a} _1，\ dots，\ textbf {a} _r $是$ \ mathbb {r}^n $的半空间中的向量。我们称$$ C = \ textbf {a} _1 \ mathbb {r}^++ \ cdots+\ textbf {a} _r \ mathbb {r \ mathbb {r}^+$$ \ textbf {a} _r \} $ $ c $的生成器集。具有最小基数的发电机设置称为框架。我们研究了凸多面体锥的翻译瓷砖。 令$ t \ subset \ mathbb {r}^n $为一个紧凑的集合，使得$ t $是其内部的关闭，而$ \ mathcal {j} \ subset \ subset \ mathbb {r}^n $是一个离散的集合。我们说$（T，\ Mathcal {j}）$是$ c $的翻译瓷砖，如果$ t+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t $ t+t+\ t+\ m nathcal {J j} $的任何两个翻译是Lebesgue Mesuare in Lebesgue Mesuare n ebesgue Mathe n n ebesgue。 我们表明，如果$ c $的框架的基数大于$ \ dim c $，则$ c $的尺寸，则$ c $不承认任何翻译瓷砖；如果$ c $的框架的基数等于$ \ dim c $，则可以将$ c $的翻译瓷砖简化为$（\ mathbb {z}^+）^n $的翻译瓷砖。作为应用 多面角的角落，它概括了Odlyzko的结果[A。 M. Odlyzko，\ textit {位置号系统中的非阴性数字集}，proc。伦敦数学。 Soc。，\ textbf {37}（1978），213-229。]。
Let $\textbf{a}_1,\dots, \textbf{a}_r$ be vectors in a half-space of $\mathbb{R}^n$. We call $$C=\textbf{a}_1\mathbb{R}^++\cdots+\textbf{a}_r \mathbb{R}^+$$ a convex polyhedral cone, and call $\{\textbf{a}_1,\dots, \textbf{a}_r\}$ a generator set of $C$. A generator set with the minimal cardinality is called a frame. We investigate the translation tilings of convex polyhedral cones. Let $T\subset \mathbb{R}^n$ be a compact set such that $T$ is the closure of its interior, and $\mathcal{J}\subset \mathbb{R}^n$ be a discrete set. We say $(T,\mathcal{J})$ is a translation tiling of $C$ if $T+\mathcal{J}=C$ and any two translations of $T$ in $T+\mathcal{J}$ are disjoint in Lebesgue measure. We show that if the cardinality of a frame of $C$ is larger than $\dim C$, the dimension of $C$, then $C$ does not admit any translation tiling; if the cardinality of a frame of $C$ equals $\dim C$, then the translation tilings of $C$ can be reduced to the translation tilings of $(\mathbb{Z}^+)^n$. As an application, we characterize all the self-affine tiles possessing polyhedral corners, which generalizes a result of Odlyzko [A. M. Odlyzko, \textit{Non-negative digit sets in positional number systems}, Proc. London Math. Soc., \textbf{37}(1978), 213-229.].
