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PreprintCC BY 4.0arXiv

Visible parts and projections of the affine fractal percolation

  • Eeva Kuure
  • Markus Myllyoja
  • Aleksi Pyörälä
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Abstract

We study the visible parts and orthogonal projections of the affine fractal percolation. We show that for the affine fractal percolation E, conditioned on E ≠ ∅, almost surely the dimension of the visible part is equal to min{1, dimH E} from all directions except for the major contracting direction. The visible part from the major contracting direction is notably different from the other directions, and we provide a formula for its Hausdorff and box dimensions. In addition, we show that almost surely conditioned on E ≠ ∅, dimH π(E) =min{ 1, dimH E} for every orthogonal projection π excluding the one onto the major contracting direction, and given dimH E >1, these projections contain intervals.

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Cite

BibTeXRIS
BibTeX
@misc{kuure2026visible,
  title = {{Visible parts and projections of the affine fractal percolation}},
  author = {Eeva Kuure and Markus Myllyoja and Aleksi Pyörälä},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21768},
  eprint = {2610.21768},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21768},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Visible parts and projections of the affine fractal percolation
AU  - Eeva Kuure
AU  - Markus Myllyoja
AU  - Aleksi Pyörälä
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21768
UR  - https://arxiv.org/abs/2610.21768
AB  - We study the visible parts and orthogonal projections of the affine fractal percolation. We show that for the affine fractal percolation E, conditioned on E ≠ ∅, almost surely the dimension of the visible part is equal to min{1, dim_H E} from all directions except for the major contracting direction. The visible part from the major contracting direction is notably different from the other directions, and we provide a formula for its Hausdorff and box dimensions. In addition, we show that almost surely conditioned on E ≠ ∅, dim_H π(E) =min{ 1, dim_H E} for every orthogonal projection π excluding the one onto the major contracting direction, and given dim_H E >1, these projections contain intervals.
ER  -