Applications of dimension interpolation to orthogonal projections

TL;DR

This paper surveys the concept of dimension interpolation and explores its applications to the dimension theory of orthogonal projections, focusing on Fourier, intermediate, and Assouad spectra.

Contribution

It introduces the framework of dimension interpolation and discusses its novel applications to various spectral dimensions in the context of orthogonal projections.

Findings

Analysis of Fourier spectrum in projection dimensions

Applications of intermediate dimensions to projection problems

Insights into Assouad spectrum and projection behavior

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.