Sharp Favard length of random Cantor sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917162353950720 |
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| author | Chang, Alan Shmerkin, Pablo Suomala, Ville |
| author_facet | Chang, Alan Shmerkin, Pablo Suomala, Ville |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17753 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Favard length of random Cantor sets Chang, Alan Shmerkin, Pablo Suomala, Ville Classical Analysis and ODEs Metric Geometry Probability 28A75, 28A80, 60D05 (Primary) 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. |
| title | Sharp Favard length of random Cantor sets |
| topic | Classical Analysis and ODEs Metric Geometry Probability 28A75, 28A80, 60D05 (Primary) |
| url | https://arxiv.org/abs/2512.17753 |