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Applications of dimension interpolation to orthogonal projections

Jonathan M. Fraser

TL;DR

The paper surveys dimension interpolation as a unifying framework for fractal dimensions and develops its impact on the dimension theory of orthogonal projections. It introduces three spectra—Assouad spectrum, intermediate dimensions, and Fourier spectrum—alongside their elementary properties, end-point behavior, and sample computations, illustrating how interpolation yields finer geometric information than endpoint dimensions. It then shows how these spectra influence projection theorems: Marstrand–Mattila results for Hausdorff dimension, dimension profiles for box-type dimensions, and novel Marstrand-type results and lower bounds for projections derived from Assouad and intermediate spectra, with further use of the Fourier spectrum to bound exceptional projection sets. The work also develops capacity-based and Fourier-analytic tools to connect projections with projection-exception phenomena, including continuity results for the dimension of exceptional sets and sharp bounds in terms of spectral quantities. Overall, it highlights both settled results and open problems, notably in extending Marstrand-type phenomena to the Fourier and Assouad spectra and in exploiting interpolation to sharpen projection estimates.

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.

Applications of dimension interpolation to orthogonal projections

TL;DR

The paper surveys dimension interpolation as a unifying framework for fractal dimensions and develops its impact on the dimension theory of orthogonal projections. It introduces three spectra—Assouad spectrum, intermediate dimensions, and Fourier spectrum—alongside their elementary properties, end-point behavior, and sample computations, illustrating how interpolation yields finer geometric information than endpoint dimensions. It then shows how these spectra influence projection theorems: Marstrand–Mattila results for Hausdorff dimension, dimension profiles for box-type dimensions, and novel Marstrand-type results and lower bounds for projections derived from Assouad and intermediate spectra, with further use of the Fourier spectrum to bound exceptional projection sets. The work also develops capacity-based and Fourier-analytic tools to connect projections with projection-exception phenomena, including continuity results for the dimension of exceptional sets and sharp bounds in terms of spectral quantities. Overall, it highlights both settled results and open problems, notably in extending Marstrand-type phenomena to the Fourier and Assouad spectra and in exploiting interpolation to sharpen projection estimates.

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.

Paper Structure

This paper contains 14 sections, 14 theorems, 94 equations, 1 figure.

Table of Contents

  1. Dimension theory
  2. Dimension interpolation
  3. The Assouad spectrum
  4. The intermediate dimensions
  5. The Fourier spectrum
  6. Simple example: revisited
  7. Dimension theory of orthogonal projections
  8. Hausdorff dimension and the Marstrand--Mattila theorem
  9. Dimension profiles and box dimension
  10. Further Marstrand--type theorems?
  11. Using the Assouad spectrum to study box dimension profiles
  12. Using the intermediate dimensions to study box dimension profiles
  13. Using the Fourier spectrum to study the exceptional set for Hausdorff dimension
  14. Continuity of the dimension of the exceptional set

Key Result

Theorem 3.1

falconerprofile$(i)$ Let $1\leqslant k<d$ be an integer. For almost all $V \in G(d,k)$, if $X\subseteq \mathbb{R}^d$ is bounded $(ii)$ For $0<s<k$, if $X\subseteq \mathbb{R}^d$ is bounded

Figures (1)

  • Figure 1: Complete interpolation: plots of the Fourier spectrum (left), the intermediate dimensions (centre), and the Assouad spectrum (right) as functions of $\theta \in (0,1)$ for the simple example $X =\{1/n\}_{n \in \mathbb{N}} \times [0,1]$.

Theorems & Definitions (22)

  • Theorem 3.1
  • Theorem 4.1
  • Corollary 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 5.1
  • Lemma 5.2
  • proof
  • Theorem 5.3
  • ...and 12 more