Abstract We show that for a large class of planar 1-dimensional random fractals S S, the Favard length Fav (S (r) ) Fav (S (r) ) of the neighborhood S (r) S (r) is comparable to ^-1 (1/r) 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 Fav (S (r) ) / (1/r) 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 S, the Favard length of S (r) S (r) decays instead like (1/r) / (1/r) log log (1 / r) / log (1 / r), showing that the 1/ (1/r) 1 / log (1 / r) decay is not universal among random fractals, as might be expected from previous results.
Chang et al. (Thu,) studied this question.
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