集和例外方向的投影的盒和包装尺寸的容量方法

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2019-01-30 DOI:10.4171/jfg/96

K. Falconer

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引用次数: 14

摘要

在[8,11]中引入了维度轮廓，给出了集$R^n$在几乎所有$m$维子空间上的正交投影的计数和填充维度的公式。然而，这些维度概要文件的定义是间接的，很难使用。在这里，我们首先给出了关于特定核的容量的维度轮廓的替代定义，这很容易导致投影的盒计数和包装维度，包括对投影维度小于典型值的子空间异常集的大小的估计。其次，我们认为，通过这种方法，不同类型维度的投影结果可以用统一的方式来考虑。第三，利用傅里叶变换方法进一步得到异常子空间大小的不等式。

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A capacity approach to box and packing dimensions of projections of sets and exceptional directions

Dimension profiles were introduced in [8,11] to give a formula for the box-counting and packing dimensions of the orthogonal projections of a set $R^n$ onto almost all $m$-dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of $E$ with respect to certain kernels, which lead to the box-counting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.