2025/12/19 by Alan Chang, Pablo Shmerkin, Chang, Alan +3 · 1 citation
Mathematics · #28A75 #28A80 #60D05 (Primary) #Advanced Combinatorial Mathematics #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.2512.17753
openalex publication_date 2025/12/19 · openalex created_date 2025/12/23 · openalex updated_date 2026/07/28
We show that for a large class of planar 1-dimensional random fractals S, the Favard length 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 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 loglog(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.