Improved bounds for the dimensions of planar distance sets

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2018-11-08 DOI:10.4171/jfg/97

Pablo Shmerkin

引用次数: 18

Abstract

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if $A$ has Hausdorff dimension $>1$, then the set of distances spanned by points of $A$ has Hausdorff dimension at least $40/57 > 0.7$ and there are many $y\in A$ such that the pinned distance set $\{ |x-y|:x\in A\}$ has Hausdorff dimension at least $29/42$ and lower box-counting dimension at least $40/57$. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input.

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平面距离集维数的改进边界

我们获得了距离集的Hausdorff维数和维度略大于$1$的平面Borel集的钉扎距离集的新下界，改进了Keleti和Shmerkin以及Liu在该机制中的最近估计。特别地，我们证明了如果$A$具有Hausdorff维数$>1$，那么$A$的点所跨越的距离集具有至少$40/57>0.7$的Hausdorf维数，并且A$中有许多$y\，使得A\｝$中的钉扎距离集$\｛|x-y|：x\具有至少$29/42$的Haussdorff维数和至少$40/57的下盒计数维数。我们使用了Keleti和Shmerkin早期工作的方法和许多结果，但将Guth、Iosevich、Ou和Wang最近工作的估计值作为额外的投入。

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.