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Sharp Favard length of random Cantor sets

Alan Chang, Pablo Shmerkin, Ville Suomala

TL;DR

The paper resolves long-standing questions about the decay of Favard length for planar 1D random fractals by proving almost-sure convergence of Favard length for a broad class of grid-based models and identifying universal 1/log(1/r) decay in many cases. It develops a branching-process framework tied to Galton–Watson processes to connect expected decay with almost-sure behavior, and derives explicit limiting constants via a variance functional V(θ). The results distinguish non-degenerate and degenerate grid models, showing sharp 1/n decay in the former and log n/n in the latter, with some models exhibiting a slower log log/ log type decay, demonstrating non-universality. Overall, the work advances Besicovitch projection theory for random Cantor-like sets by providing precise asymptotics, almost-sure limits, and explicit constants that depend on directional variance structure.

Abstract

We show that for a large class of planar $1$-dimensional random fractals $S$, the Favard length $\operatorname{Fav}(S(r))$ of the neighborhood $S(r)$ is comparable to $\log^{-1}(1/r)$, matching a universal lower bound; up to now, this was only known in expectation for a few concrete models. In particular, we show that there exist $1$-Ahlfors regular sets with the fastest possible Favard length decay. For a wide class of planar one-dimensional "grid random fractals", including fractal percolation and its Ahlfors-regular variants, we further show that $\operatorname{Fav}(S(r))/\log(1/r)$ converges almost surely, and we identify the limit explicitly. Furthermore, we prove that for some $1$-dimensional Ahlfors-regular random fractals $S$, the Favard length of $S(r)$ decays instead like $\log\log(1/r)/\log(1/r)$, showing that the $1/\log(1/r)$ decay is not universal among random fractals, as might be expected from previous results.

Sharp Favard length of random Cantor sets

TL;DR

The paper resolves long-standing questions about the decay of Favard length for planar 1D random fractals by proving almost-sure convergence of Favard length for a broad class of grid-based models and identifying universal 1/log(1/r) decay in many cases. It develops a branching-process framework tied to Galton–Watson processes to connect expected decay with almost-sure behavior, and derives explicit limiting constants via a variance functional V(θ). The results distinguish non-degenerate and degenerate grid models, showing sharp 1/n decay in the former and log n/n in the latter, with some models exhibiting a slower log log/ log type decay, demonstrating non-universality. Overall, the work advances Besicovitch projection theory for random Cantor-like sets by providing precise asymptotics, almost-sure limits, and explicit constants that depend on directional variance structure.

Abstract

We show that for a large class of planar -dimensional random fractals , the Favard length of the neighborhood is comparable to , matching a universal lower bound; up to now, this was only known in expectation for a few concrete models. In particular, we show that there exist -Ahlfors regular sets with the fastest possible Favard length decay. For a wide class of planar one-dimensional "grid random fractals", including fractal percolation and its Ahlfors-regular variants, we further show that converges almost surely, and we identify the limit explicitly. Furthermore, we prove that for some -dimensional Ahlfors-regular random fractals , the Favard length of decays instead like , showing that the decay is not universal among random fractals, as might be expected from previous results.

Paper Structure

This paper contains 29 sections, 41 theorems, 199 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The minimal Favard length problem
  3. Main results
  4. Ideas of proof and connections to branching processes
  5. Structure of the paper and further results
  6. Preliminaries
  7. General notation
  8. Galton-Watson random fractals
  9. Grid models
  10. Examples and figures
  11. From expected to almost-sure Favard length
  12. The asymptotic limit
  13. Construction steps versus neighbourhoods of $S$
  14. Convexity of Favard length
  15. Proof of Theorem \ref{['thm:almost-sure-convergence']}
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\mathfrak X$ be an Ahlfors-regular uniform grid model. (The implied constants can depend on the model $\mathfrak X$.)

Figures (7)

  • Figure 1: $K_0, K_1, K_2$ for the four-corner Cantor set.
  • Figure 2: $S_0$ through $S_5$ for a grid model: $L=2$, Ahlfors-regular, $\mathfrak X$ is defined as a uniform choice among all pairs of disjoint $Q_1$ and $Q_2$.
  • Figure 3: $S_0$ through $S_5$ for a grid model (fractal percolation) with $L=2$. Each square is chosen independently with probability $1/2$. See \ref{['example:perco']}.
  • Figure 4: $S_0$ through $S_5$ for a grid model with $L=2$. $\mathfrak X$ selects one square uniformly and independently in each column, resulting in a vertically degenerate model. See \ref{['def:degenerate']}.
  • Figure 5: $S_0$ through $S_3$ for the model considered by Vardakis and Volberg, with $L=3$. See \ref{['example:VV-discs']}.
  • ...and 2 more figures

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3: Degeneracy
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Theorem 3.1: almost-sure convergence
  • ...and 81 more