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Exceptional projections of sets exhibiting almost dimension conservation

Ryan E. G. Bushling

TL;DR

The paper addresses how much fractal dimension can be lost when projecting sets in $\mathbb{R}^n$ onto $k$-planes, focusing on packing-dimension bounds for the exceptional set of projections under almost dimension conservation. It blends homogeneous and graph-directed fractal structures with Rams’ transversality framework and an energy-discretization approach on the Grassmannian to bound the size of the projection-exception set. The main contribution extends Orponen’s planar results to higher dimensions under structured hypotheses (homogeneous or graph-directed with finite transformation groups or dense orbits) and provides a quantitative mechanism (via two key lemmas and a discrete projection count) to relate fiber dimensions to the dimension of projected images. This advances the understanding of dimension drop under projections and offers tools for analyzing projection behavior in broader fractal geometries with potential applications in geometric measure theory and fractal geometry.

Abstract

We establish a packing dimension estimate on the exceptional sets of orthogonal projections of sets satisfying an almost dimension conservation law. In particular, the main result applies to homogeneous sets and to certain graph-directed sets. Connections are drawn to results of M. Rams and T. Orponen.

Exceptional projections of sets exhibiting almost dimension conservation

TL;DR

The paper addresses how much fractal dimension can be lost when projecting sets in onto -planes, focusing on packing-dimension bounds for the exceptional set of projections under almost dimension conservation. It blends homogeneous and graph-directed fractal structures with Rams’ transversality framework and an energy-discretization approach on the Grassmannian to bound the size of the projection-exception set. The main contribution extends Orponen’s planar results to higher dimensions under structured hypotheses (homogeneous or graph-directed with finite transformation groups or dense orbits) and provides a quantitative mechanism (via two key lemmas and a discrete projection count) to relate fiber dimensions to the dimension of projected images. This advances the understanding of dimension drop under projections and offers tools for analyzing projection behavior in broader fractal geometries with potential applications in geometric measure theory and fractal geometry.

Abstract

We establish a packing dimension estimate on the exceptional sets of orthogonal projections of sets satisfying an almost dimension conservation law. In particular, the main result applies to homogeneous sets and to certain graph-directed sets. Connections are drawn to results of M. Rams and T. Orponen.
Paper Structure (10 sections, 14 theorems, 75 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A theorem of Rams
  3. Orponen's packing dimension bound on exceptional sets
  4. Outline
  5. Definitions and notation
  6. Measures and dimensions
  7. Dimension conservation and homogeneous sets
  8. Almost dimension conservation and graph-directed sets
  9. Counting points on Gr(n,k)
  10. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $A \subseteq \mathbb{R}^n$ be analytic.

Figures (1)

  • Figure 1: A "checkerboard square" $R_\ell$ in the fat plane $T_i$ in the case $(n,k) = (3,1)$.

Theorems & Definitions (14)

  • Theorem 1.1: Marstrand's Projection Theorem
  • Theorem 1.2: Rams rams2002packing
  • Proposition 1.3
  • Proposition 1.4: Orponen orponen2015packing
  • Theorem 1.5
  • Corollary 1.6
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3: Furstenberg furstenberg2008ergodic
  • Proposition 2.4
  • ...and 4 more