Hausdorff dimension of intersections with planes and general sets

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2020-05-24 DOI:10.4171/jfg/110

P. Mattila

引用次数: 4

Abstract

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable sets $A\subset\Rn$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$, and with positive lower density. As an application we prove for measurable sets $A,B\subset\Rn$ with positive $s$- and $t$-dimensional measures, and with positive lower density that if $s + (n-1)t/n > n$, then $\dim A\cap (g(B)+z) = s+t - n$ for almost all rotations $g$ and for positively many $z\in\Rn$.

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平面与一般集相交的Hausdorff维数

我们给出了一个正交投影的一般族$P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$的条件，保证了Hausdorff维数公式$\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$对于具有正的和有限的$s$ -维Hausdorff测度，$s>m$和正的低密度的可测集$A\subset\Rn$是一般成立的。作为一个应用，我们证明了对于具有正的$s$和$t$维测度的可测集$A,B\subset\Rn$，并且具有正的低密度，如果$s + (n-1)t/n > n$，那么对于几乎所有的旋转$g$和对于正的许多$z\in\Rn$，如果$\dim A\cap (g(B)+z) = s+t - n$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.