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PreprintCC BY 4.0arXiv

Favard length of non-homogeneous random disc-like Cantor sets

  • Alan Chang
  • Damian Dąbrowski
  • Giacomo Del Nin
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Abstract

The Favard length of a planar set is the average length of its orthogonal projections. We provide sharp estimates for the Favard length decay of a random variant of the non-homogeneous four corner Cantor sets introduced by Garnett in the 1970s. As a corollary, we obtain examples which show that Mattila's classical lower bound on the Favard length in terms of the Riesz capacity is sharp for all possible decay rates. We also obtain a new family of examples of sets with positive analytic capacity and zero Favard length.

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Cite

BibTeXRIS
BibTeX
@misc{chang2026favard,
  title = {{Favard length of non-homogeneous random disc-like Cantor sets}},
  author = {Alan Chang and Damian Dąbrowski and Giacomo Del Nin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33557},
  eprint = {2609.33557},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33557},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Favard length of non-homogeneous random disc-like Cantor sets
AU  - Alan Chang
AU  - Damian Dąbrowski
AU  - Giacomo Del Nin
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33557
UR  - https://arxiv.org/abs/2609.33557
AB  - The Favard length of a planar set is the average length of its orthogonal projections. We provide sharp estimates for the Favard length decay of a random variant of the non-homogeneous four corner Cantor sets introduced by Garnett in the 1970s. As a corollary, we obtain examples which show that Mattila's classical lower bound on the Favard length in terms of the Riesz capacity is sharp for all possible decay rates. We also obtain a new family of examples of sets with positive analytic capacity and zero Favard length.
ER  -