论文标题
在$ \ mathbb {r}^d $中的一般二元格上
On the general dyadic grids in $\mathbb{R}^d$
论文作者
论文摘要
相邻的二元系统在分析和相关领域是通过二元组收集来研究连续对象的关键。在我们先前的工作（与江，奥尔森和魏的联合）中，我们描述了实际线上两个二元系统相邻的精确和充分条件。在这里，我们将这项工作扩展到所有维度，事实证明，由于$ d+1 $而不是$ 2^d $，网格是相邻的二元系统中$ \ mathbb {r}^d $中的最佳数字。作为副产品，我们表明，$ d+1 $二元系统的集合在$ \ mathbb {r}^d $中是相邻的，并且只有当其中任何两个坐标轴投射到任何坐标轴上都与$ \ mathbb {r} $相邻。在更高维度的概括中出现的基本几何结构本身就是有趣的对象，以使未来的研究成熟；这些使我们对我们的主要结果进行了紧凑的几何描述。我们描述了这些结构，以及相邻的二元组（和$ n $ adic，对于任何$ n $）的系统，从各种环境将它们与以前的工作联系起来，并说明了一个特定的示例。
Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson and Wei) we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that $d+1$, not $2^d$, grids is the optimal number in an adjacent dyadic system in $\mathbb{R}^d$. As a byproduct, we show that a collection of $d+1$ dyadic systems in $\mathbb{R}^d$ is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on $\mathbb{R}$. The underlying geometric structures that arise in this higher dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and $n$-adic, for any $n$) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific example.
