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Transversal families of nonlinear projections and generalizations of Favard length

Rosemarie Bongers, Krystal Taylor

Abstract

Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a \textit{transversality condition}. In this work, we establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four corner Cantor set, first established by Cladek, Davey, and Taylor.

Transversal families of nonlinear projections and generalizations of Favard length

Abstract

Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a \textit{transversality condition}. In this work, we establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four corner Cantor set, first established by Cladek, Davey, and Taylor.

Paper Structure

This paper contains 17 sections, 16 theorems, 133 equations, 4 figures.

Table of Contents

  1. Introduction and Main Results
  2. Nonlinear projections
  3. Visibility
  4. Favard curve length
  5. Favard surface length in $\mathbb{R}^d$
  6. Overview of transversality
  7. Main results
  8. Establishing Transversality
  9. Orthogonal projections
  10. Visibility
  11. Favard curve length
  12. Surface projections
  13. Energy techniques for pushforwards
  14. Applications and examples
  15. Proving Theorems \ref{['theorem:smooth_surface_visibility']} and \ref{['FavTheorem']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Fix $s \in (0, 1]$. If $F \subseteq \mathbb{R}^2$ is the support of a Borel probability measure with $\mu(B(x, r)) \le b r^s$ for all $x \in \mathbb{R}^2$ and $0 < r < \infty$, then if $s < 1$ and if $s = 1$.

Figures (4)

  • Figure 1: $\Phi_\lambda(\mathbf{a})$
  • Figure 2:
  • Figure 3: Case 1
  • Figure 4: Case 3

Theorems & Definitions (35)

  • Theorem 1.1: Favard lengths for neighborhoods; Mattila Mat90
  • Definition 1.2: Nonlinear projections
  • Definition 1.3: Transversality
  • Proposition 1.4: Nonlinear Marstrand theorem
  • Theorem 1.5: Average nonlinear projection length for neighborhoods
  • Theorem 1.6: Visibility for surfaces in $\mathbb{R}^n$
  • Theorem 1.7: Favard curve length of neighborhoods
  • Corollary 1.8: Visibility of $\mathop{\mathrm{\mathcal{K}}}\nolimits_n$
  • Corollary 1.9: Favard curve length of $\mathop{\mathrm{\mathcal{K}}}\nolimits_n$
  • Theorem 1.10: Favard surface length of neighborhoods
  • ...and 25 more