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Seventy Years of Fractal Projections

Kenneth J. Falconer

Abstract

Seventy years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For some time this paper attracted little attention, but over the past 40 years Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.

Seventy Years of Fractal Projections

Abstract

Seventy years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For some time this paper attracted little attention, but over the past 40 years Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.
Paper Structure (18 sections, 17 theorems, 84 equations, 5 figures)

This paper contains 18 sections, 17 theorems, 84 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General remarks
  3. Marstrand's projection theorem
  4. Exceptional sets of projections
  5. Other definitions of dimensions
  6. Box-counting dimensions
  7. Packing dimension
  8. Intermediate dimensions
  9. Assouad dimension
  10. Fourier dimension
  11. Projections in restricted directions
  12. Projections of IFS attractors: self-similar and self-affine sets
  13. Self-similar sets
  14. Self-affine sets
  15. Self-conformal sets
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $E \subset \mathbb{R}^2$ be a Borel or analytic set. Then (i) $\dim_{\rm {H}} \hbox{\rm proj}_\theta E \leq \min\{\dim_{\rm {H}} E, 1\}$ with equality for almost all $\theta \in [0,\pi)$, (ii) if $\dim_{\rm {H}} E >1$ then ${\mathcal{L}} (\hbox{\rm proj}_\theta E) > 0$ for almost all $\theta \in

Figures (5)

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Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 4.1
  • ...and 7 more