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Geometric Bounds for Favard Length

Rosemarie Bongers

Abstract

Given a set in the plane, the average length of its projections over all directions is called Favard length. This quantity measures the size of a set, and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. In this paper, we develop new geometric techniques for estimating Favard length. We will give a short geometrically motivated proof relating Hausdorff dimension to the decay rate of the Favard length of neighborhoods of a set. We will also show that the sequence of Favard lengths of the generations of a self-similar set is convex; this has direct applications to giving lower bounds on Favard length for various fractal sets.

Geometric Bounds for Favard Length

Abstract

Given a set in the plane, the average length of its projections over all directions is called Favard length. This quantity measures the size of a set, and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. In this paper, we develop new geometric techniques for estimating Favard length. We will give a short geometrically motivated proof relating Hausdorff dimension to the decay rate of the Favard length of neighborhoods of a set. We will also show that the sequence of Favard lengths of the generations of a self-similar set is convex; this has direct applications to giving lower bounds on Favard length for various fractal sets.

Paper Structure

This paper contains 3 sections, 6 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Dimension and Favard Length
  3. Self-similar Sets

Key Result

Theorem 1

Fix $s \in (0, 1)$ and suppose that $E \subseteq \mathbb{R}^2$ is measurable, and $A \subseteq S^1$ is measurable with positive (arc-length) measure. Suppose there exists a sequence of scales $r_n \to 0$ such that for some $C < \infty$. Then $\operatorname{dim}_{\mathcal{H}} E \le 1 - s$.

Theorems & Definitions (9)

  • Theorem
  • Theorem
  • Corollary
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof