盒维数剖面与中间维数

中文摘要 AI 辅助

对于非空有界集$E\subset\R^d$，其正交投影到几乎所有$m$维子空间的上、下盒维数为常数，这些几乎必然值分别称为$E$的上、下$m$维盒维数剖面。本文研究它们与中间维数的关系，中间维数通过限制覆盖集的相对大小在豪斯多夫维数与盒维数之间插值。我们建立了这两个量的若干等价刻画，得到用中间维数表示的盒维数剖面的上、下界，下界还涉及上Assouad谱；在拟Assouad维数的合适条件下，进一步推导出用盒维数和中间维数初始增长率表示的盒维数剖面显式公式。

英文摘要

For a non-empty bounded set $E\subset\R^d$, the upper and lower box dimensions of its orthogonal projections are constant onto almost all $m$-dimensional subspaces. These almost-sure values are called the upper and lower $m$-box dimension profiles of $E$, respectively. We study their relationship with intermediate dimensions, which interpolate between Hausdorff and box dimensions by restricting the relative sizes of covering sets. We establish several equivalent characterizations of both quantities and obtain upper and lower bounds for box dimension profiles in terms of intermediate dimensions, with the lower bounds also involving the upper Assouad spectrum. We further derive explicit formulas for the box dimension profiles in terms of the box dimension and the initial growth rate of the intermediate dimensions, under a suitable condition on the quasi-Assouad dimension.

发表机构

• School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）

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