集与例外方向投影的盒维数与填充维数的容量方法
A capacity approach to box and packing dimensions of projections of sets and exceptional directions
摘要：
在文献[8，11]中引入了维数剖面，给出了一个关于集合$R^n$在几乎所有$m$维子空间上的正交投影的计盒维数和填充维数的公式。然而，这些尺寸轮廓的定义是间接的，并且很难使用。在这里，我们首先根据$E$相对于某些核的容量给出维度轮廓的替代定义，这可以相当容易地得出投影的盒计数和填充维度，包括对异常子空间集的大小的估计，其中投影的维度小于典型值。其次，我们认为，这种方法的投影结果为不同类型的尺寸可以被认为是在一个统一的方式。第三，我们利用傅里叶变换方法得到了关于例外子空间大小的进一步不等式。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
