Dimensions of popcorn-like pyramid sets

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2022-12-14 DOI:10.4171/jfg/135

Amlan Banaji, Haipeng Chen

引用次数: 1

Abstract

This article concerns the dimension theory of the graphs of a family of functions which include the well-known 'popcorn function' and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these graphs, as well as the intermediate dimensions, which are a family of dimensions interpolating between Hausdorff and box dimension. As tools in the proofs, we use the Chung$\unicode{x2013}$Erd\H{o}s inequality from probability theory, higher-dimensional Duffin$\unicode{x2013}$Schaeffer type estimates from Diophantine approximation, and a bound for Euler's totient function. As applications we obtain bounds on the box dimension of fractional Brownian images of the graphs, and on the H\"older distortion between different graphs.

查看原文本刊更多论文

爆米花状金字塔组的尺寸

本文讨论了一类函数图的维数理论，其中包括著名的“爆米花函数”及其金字塔状的高维类似物。我们计算了这些图的盒维和副维，以及中间维，它是在Hausdorff维和盒维之间插值的一组维。作为证明的工具，我们使用了概率论中的Chung$\unicode{x2013}$Erd\H{o}s不等式，Diophantine近似中的高维Duffin$\unicode{x2013}$Schaeffer型估计，以及欧拉的totient函数的界。作为应用，我们得到了图的分数布朗图像的盒维限，以及不同图之间的H\ \ old畸变。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

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.