Sharp Favard length of random Cantor sets

TL;DR

This paper investigates the decay rate of Favard length for various random planar fractals, establishing universal bounds, convergence properties, and examples with different decay behaviors.

Contribution

It proves that Favard length decay of certain random fractals is comparable to 1/log(1/r), and identifies conditions for different decay rates, including almost sure convergence and explicit limits.

Findings

Favard length decays like 1/log(1/r) for many random fractals

Favard length divided by log(1/r) converges almost surely for grid fractals

Some fractals exhibit decay like log log(1/r)/log(1/r), showing non-universality

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.