Sharp Favard length of random Cantor sets

Abstract

We show that for a large class of planar $1$-dimensional random fractals $S$, the Favard length $\mathrm{Fav}(S(r))$ 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 $\mathrm{Fav}(S(r))/\log (1/r)$ 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.

v2: 43 pages, 8 figures, incorporated reviewer comments, to appear in GAFA