Applications of dimension interpolation to orthogonal projections.

IF 1.2 3区数学 Q1 MATHEMATICS

Research in the Mathematical Sciences Pub Date : 2025-01-01 Epub Date: 2025-01-25 DOI:10.1007/s40687-025-00496-9

Jonathan M Fraser

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引用次数: 0

Abstract

Dimension interpolation is a novel programme of research which attempts to unify the study of fractal dimension by considering various spectra which live in between well-studied notions of dimension such as Hausdorff, box, Assouad and Fourier dimension. These spectra often reveal novel features not witnessed by the individual notions and this information has applications in many directions. In this survey article, we discuss dimension interpolation broadly and then focus on applications to the dimension theory of orthogonal projections. We focus on three distinct applications coming from three different dimension spectra, namely, the Fourier spectrum, the intermediate dimensions, and the Assouad spectrum. The celebrated Marstrand-Mattila projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. This result has inspired much further research on the dimension theory of projections including the consideration of dimensions other than the Hausdorff dimension, and the study of the exceptional set in the Marstrand-Mattila theorem.

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维度插值是一项新颖的研究计划，它试图通过考虑介于豪斯多夫维度、盒维度、阿苏阿德维度和傅里叶维度等已被充分研究的维度概念之间的各种光谱，来统一对分形维度的研究。这些频谱往往揭示了单个维度概念所不具备的新特征，这些信息在很多方面都有应用价值。在这篇文章中，我们将广泛讨论维度插值，然后重点讨论正交投影维度理论的应用。我们将重点讨论来自三种不同维度谱的三种不同应用，即傅立叶谱、中间维度和阿苏阿德谱。著名的马斯特兰-马蒂拉（Marstrand-Mattila）投影定理给出了几乎所有正交投影在欧几里得空间中的博尔集合正交投影的豪斯多夫维度。这一结果激发了人们对投影维度理论的进一步研究，包括对豪斯多夫维度以外的维度的考虑，以及对马斯特兰-马蒂拉定理中例外集的研究。

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来源期刊

Research in the Mathematical Sciences Mathematics-Computational Mathematics

CiteScore

2.00

自引率

8.30%

发文量

期刊介绍： Research in the Mathematical Sciences is an international, peer-reviewed hybrid journal covering the full scope of Theoretical Mathematics, Applied Mathematics, and Theoretical Computer Science. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to the research areas of both theoretical and applied mathematics and theoretical computer science. This journal is an efficient enterprise where the editors play a central role in soliciting the best research papers, and where editorial decisions are reached in a timely fashion. Research in the Mathematical Sciences does not have a length restriction and encourages the submission of longer articles in which more complex and detailed analysis and proofing of theorems is required. It also publishes shorter research communications (Letters) covering nascent research in some of the hottest areas of mathematical research. This journal will publish the highest quality papers in all of the traditional areas of applied and theoretical areas of mathematics and computer science, and it will actively seek to publish seminal papers in the most emerging and interdisciplinary areas in all of the mathematical sciences. Research in the Mathematical Sciences wishes to lead the way by promoting the highest quality research of this type.