Hölder coverings of sets of small dimension

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2017-02-03 DOI:10.4171/JFG/78

Eino Rossi, Pablo Shmerkin

引用次数: 4

Abstract

We show that a set of small box counting dimension can be covered by a H\"older graph from all but a small set of directions, and give sharp bounds for the dimension of the exceptional set, improving a result of B. Hunt and V. Kaloshin. We observe that, as a consequence, H\"older graphs can have positive doubling measure, answering a question of T. Ojala and T. Rajala. We also give remarks on H\"older coverings in polar coordinates and, on the other hand, prove that a Homogenous set of small box counting dimension can be covered by a Lipschitz graph from all but a small set of directions.

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小尺寸集合的Hölder覆盖

我们证明了一组小盒计数维数可以被一个H\“旧图从除一小组方向之外的所有方向覆盖，并给出了例外集维数的锐界，改进了B.Hunt和V.Kaloshin的结果。因此，我们观察到，H\”旧图可以具有正加倍测度，回答了T.Ojala和T.Rajala的问题。我们还对极坐标系中的H\“older覆盖给出了注记，另一方面，我们证明了小盒计数维数的齐次集可以被Lipschitz图从除一小组方向之外的所有方向覆盖。

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.